AEDC-TR-7S-127 c':.-~\,4- ANALYTICAL MODEL OF SUPERSONIC, TURBULENT, N EAR-WAKE FLOWS. [97t)

Transcription

1 AEDC-TR-7S-127 c':.-\,4- Ill (i 1 1,U II " [97t) ANALYTICAL MODEL OF SUPERSONIC, TURBULENT, N EAR-WAKE FLOWS ENGINE TEST FACILITY ARNOLD ENGINEERING DEVELOPMENT CENTER AIR FORCE SYSTEMS COMMAND ARNOLD AIR FORCE STATION, TENNESSEE September 1976 Final Reprt fr Perid July December 1975 Apprved fr public release; distributin unlimited, t'j F'uu:",\' Prepared fr DIRECTORATE OF TECHNOLOGY (DYR) ARNOLD ENGINEERING DEVELOPMENT CENTER ARNOLD AIR FORCE STATION, TENNESSEE 37389

2 NOTICES 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. 1 J 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., This reprt has been reviewed by the Infrmatin Office (01) and is releasable t the Natinal Technical Infrmatin Service (NTIS). At NTIS, it will be available t the general public, including freign natins. APPROVAL STATEMENT This technical reprt has been reviewed and is apprved fr publicatin. FOR THE COMMANDER //JL. t::: l/?r ELTON R. THOMPSON Research & Develpment Divisin Directrate f Technlgy...t'7 _ /2 -..# '1!!<J,? ROBERT O. DIETZ Directr f Technlgy

3 I. REPORT NUMBER AEDC-TR-76-l27 REPORT DOCUMENTATION PAGE UNCLASSIFIED \2. READ INSTRUCTIONS BEFORE COMPLETING FORM GOVT ACCESSION NO. 3. RECIIEN'''S CATALOG NUMBER 4. TITLE (and Subtitle) 5. TyPE OF REPORT 1\ PERIOD COVERED ANALYTICAL MODEL OF SUPERSONIC, TURBULENT, NEAR-WAKE FLOWS Final Reprt - July December PERFORMING ORG. REPORT NUMBER 7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s) C. E. Peters and W. J. Phares, ARO, Inc. 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK AREA a WORK UNIT NUMBERS Arnld Engineering Develpment Center (DY) Air Frce Systems Cmmand Arnld Air Frce Statin, Tennessee II. CONTROLLING OFFICE NAME AND ADDRESS Prgram Element 65807F 12. REPORT DATE Arnld Engineering Develpment September 1976 Center (DYFS), Air Frce Systems Cmmand 13. NUMBER OF PAGES Arnld Air Frce Statin, Tennessee : MONITORING AGENCY NAME /I ADDRESS(II dillerent Irm Cntrlllns Office) 15. SECURITY CL ASS. (f this reprt) UNCLASSIFIED 16. DISTRIBUTION STATEMENT (01 thl. Reprt) ISa. DECL ASSI FICATION/ DOWNGRADING SCHEDULE / N A Apprved fr public release; distributin unlimited. 17. DISTRIBUTION STATEMENT (f the abstrsct entered In Blck 20, II dillereni Irm Reprt) 18. SUPPLEMENTARY NOTES Available in DDC 19. KEY WORDS (Cntinue n reverse side Jf necessary and Identify by blck number) ;ma thema ticai mdel bundary layer transprt prperties -- supersnic free stream turbulence turbulent flw Mach number interactins flws (near-wake) base pressure viscus flw 20. ABSTRACT (Cntinue n reverse side II necessary and Identify by blck number) An analytical mdel fr planar and axisymmetric, supersnic, turbulent, near-wake flws is presented. The viscus regin behind the blunt base is described by the integral frm f the bundary-layer equatins, and the inviscid uter-flw regin, including the remnant f the initial turbulent bundary layer, is cmputed with the rtatinal methd f characteristics. The slutin f the tw regins is fully cupled. A saddle-pint DO FORM I JAN EDITION OF 1 NOV 65 IS OBSOLETE UNCLASSIFIED

4 UNCLASSIFIED 20. ABSTRACT (Cntinued) singularity, similar t the Crcc-Lees critical pint, ccurs dwnstream f the rear stagnatin pint. The base pressure is btained by iteratin f the initial cnditins until the flwfield slutin will pass thrugh the singularity. Base bleed f a gas different frm the uter-stream gas is included in the frmulatin, and prvisin is made t treat equilibrium chemical reactins in the viscus wake. Hwever, an unreslved prblem has been encuntered in the slutin f the species cnservatin equatins. Therefre, results fr nly single gas flws are presented. The analytical mdel is shwn t adequately predict the effect f free-stream Mach number and initial bundary layer n the planar base pressure. In additin, the planar flw-field structure is well predicted. Axisymmetric' base pressure and flw-field structure are reasnably well predicted fr freestream Mach numbers greater than 2.0, but the turbulent transprt mdel used yields nly fair results fr Mach numbers less than 1.7. The effect f base bleed n the axisymmetric base pressure is well predicted. AFSC Arnld AFS Til"" UNCLASSIFIED

5 PREFACE The wrk reprted herein was cnducted by the Arnld Engineering Develpment Center (AEDC), Air Frce Systems Cmmand (AFSC). The results were. btained by ARO, Inc. (a subsidiary f Sverdrup & Parcel and Assciates, Inc.), cntract peratr f AEDC, AFSC, Arnld Air Frce Statin, Tennessee. The Prgram Element number was 65807F. The wrk was dne under ARO Prject Ns. RF423, R33P-60A, and R33A-02A. The authrs f this reprt were C. E. Peters and W. J. Phares, ARO, Inc. The manuscript (ARO Cntrl N. ARO-ETF-TR-76-69) was submitted fr publicatin n July I,

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7 AE DC-TR CONTENTS 1.0 INTRODUCTION 1.1 Prblem Definitin 1.2 Review f Earlier Analytical Wrk Imprtant Features f a Realistic Analytical Mdel 2.0 DEVELOPMENT OF ANALYTICAL MODELS 2.1 Basic Assumptins 2.2 Basic Equatins Shear Layer Prfiles Cmputatin f Density in the Viscus Flw 2.5 Turbulent Transprt Terms 2.6 Transfrmatin f the Integral Equatins 2.7 Slutin f the Outer Flw and Cupling with the Viscus Layer 2.8 Expansin f the Initial Bundary Layer 2.9 Characterizatin f the Initial and Bundary Cnditins 2.10 Tw-Layer Treatment f the Outer Flw 2.11 Methd f Slutin Mdel fr the Turbulent Reynlds Number 3.0 DISCUSSION OF RESULTS 3.1 Cncentratin Field Slutin Prblems 3.2 Planar Base Pressure 3.3 Planar Flw-Field Structure 3.4 Axisymmetric Base Pressure 3.5 Effect f Base Bleed n Axisymmetric Base Pressure 3.6 Axisymmetric Flw-Field Structure 3.7 Dividing Streamline Prperties 4.0 CONCLUDING REMARKS REFERENCES ILLUSTRATIONS Figure 1. Features f the Near-Wake Flw Field 2. Examples f Supersnic Near-Wake Flws 3. Nmenclature fr Near-Wake Analysis 4. Cupling f the Shear Layer and Outer-Flw Slutins

8 Figure 5. Crner Expansin f the Initial Bundary Layer Bundary-Layer Prfile Expnent fr Adiabatic Flw alng a Flat Plate Bundary-Layer Mmentum Thickness Rati fr Adiabatic Flw alng a Flat Plate Tw-Layer Treatment f the Supersnic Flw Field 9. Sub critical and Supercritical Slutins Turbulent Reynlds Number fr Planar Shear Layers with N Secndary Flw 11. Cnfiguratins fr Lw-Speed Separating and Reattaching Flw Experiments Turbulent Reynlds Number Distributin Upstream f the RSP in the Reattachment Regime Turbulent Reynlds Number Dwnstream f the RSP Influence f the Turbulent Reynlds Number at the RSP n the Predicted Base Pressure Turbulent Reynlds Number at the RSP as a Functin f the Pressure Gradient Parameter Effect f the Initial Bundary Layer n the Base Pressure in Planar Flw (Mal = 1.5) Effect f the Initial Bundary Layer n the Base Pressure in Planar Flw (Mal = 2.0) Effect f the Initial Bundary Layer n the Base Pressure in Planar Flw (Mal = 3.0) Mach Number Effect n the Base Pressure fr Planar Flw Predicted Base Pressure fr Planar Flw Ttal Pressure at the Edge f the Shear Layer fr Planar Flw 22. Centerline Pressure Distributin fr Planar Wake Flw (Mal = 2.0) Centerline Pressure Distributin fr Planar Wake Flw (Mal = 3.05) Centerline Mach Number Distributin fr Planar Wake Flw (Mal = 3.05) Pitt Pressure Prfiles fr Planar Wake Flw (Mal = 3.05) Effect f the Initial Bundary Layer n the Base Pressure in Axisymmetric Flw (Mal = 2.0)

9 Figure 27. Effect f the Initial Bundary Layer n the Base Pressure in Axisymmettic Flw (Mal = 3.9) Mach Number Effect n the Base Pressure fr Axisymmetric Flw 29. Effect f Base Bleed n Axisymmetric Base Pressure 30. Ttal Pressure at the Edge f the Shear Layer fr Axisymmetric Flw Centerline Pressure Distributin fr Axisymmetric Wake Flw (Mal = 3.91) Pitt Pressure Prfiles fr Axisymmetric Wake Flw (Mal = 3.91) Centerline Pressure Distributin fr Axisymmetric Wake Flw (Mal = 1.92) Centerline Mach Number Distributin fr Axisymmetric Wake Flw (Mal = 1.92) Centerline Pressure Distributin fr Axisymmetric Wake Flw (Mal = 2.03) Ttal Pressure alng the Dividing Streamline fr Planar Flws Ttal Pressure alng the Dividing Streamline fr Axisymmetric Flws APPENDIX A. COEFFICIENTS OF THE INTEGRAL EQUATIONS NOMENCLATURE

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11 AEDC TR INTRODUCTION 1.1 PROBLEM DEFINITION Turbulent near wakes behind blunt bases in supersnic external flw ccur in a variety f flws f practical interest. The near-wake flw prcess (Fig. I) is characterized by a strng interactin between the inviscid external flw and the turbulent prtin f the flw; therefre, cmputatin f the entire flw field requires a cupled slutin fr the viscus and inviscid regins. Several features f the near-wake flw field are shwn in Fig. 1. First, a recirculatin regin exists just dwnstream f the base. With n bleed flw frm the base, the velcity n the centerline is zer at the base plane. When flw is bled int the base, the nset f recirculatin is delayed until the bleed flw has been entrained int the turbulent shear layer. If the bleed flw is sufficiently large, the recirculatin regin can be eliminated. As shwn in Fig. 1, the pressure just dwnstream f the base is cnsiderably lwer than the pressure n the bdy surface upstream f the base. The centerline static pressure rises slightly with distance until the bleed flw is cmpletely entrained, then tends t decrease slightly with distance as the recirculatin velcities becme larger. At sme distance dwnstream f the base, the shear layer merges with the axis and the flw begins t recmpress. The recmpressin regin is characterized by large pressure gradients and by acceleratin f the flw alng the centerline. Dwnstream f the rear stagnatin pint (RSP) the velcities thrughut the viscus layer are psitive. If the flw is planar, and if the external flw upstream f the base is effectively infinite and parallel t the centerline, then the centerline static pressure in the recmpressin regin mntnically rises t the free-stream static pressure. Hwever, in an axisymmetric flw, the pressure in the recmpressin regin "vershts" the free-stream pressure (Fig. 1). Dwnstream f the recmpressin regin the flw relaxes tward a self preserving turbulent far-wake structure, in which static pressure gradients are negligible. Experience has shwn that the base pressure and the near-wake flw structure are dependent upn (1) the thickness and shape f the bundary-layer velcity prfile just upstream f the base, (2) the inviscid external flw prperties (Mach number, flw angle, etc.) at the base plane, and (3) the gas prperties and flw rate f the base bleed. The near-wake flw structure als depends n the lateral extent f the supersnic external flw. If the external flw is bunded by a slid wall r by a free bundary, then waves which are generated by the expansin prcess at the base plane will be reflected frm the bundary. These reflected waves can influence the near-wake structure if they intersect the viscus wake upstream f the pint where the flw n the axis has reaccelerated t supersnic speed. 7

12 Recmpressin Shck Wave /' /' RSpJ Recirculating Flw /,,// I " / I?------?> C 7" t<"'> 0") 4 u_ v t Turbulent Shear Layer Static Pressure n Bdy Ahead f Base ---III..A Q) ::l V'l V'l Q) 0- W +-' r +-' (/'). 0 Q) +-' C Q) U Base Pressure Base Bleed Velcity AXiSymmetric /- L Planar Q) > Q) c Q) +-' C Q) u RSP Axial Dista nce frm Base Figure 1. Features f the near-wake flw field. 8

13 AEDC TR Several examples f supersnic near-wake flws which are f interest at the AEDC are shwn in Fig. 2. The classic turbulent near-wake flw is that behind bdies in supersnic flight (Fig. 2a). Truncated plug exhaust nzzles (Fig. 2b) have been used fr bth turbine engines and rckets. An accurate predictin f the plug base pressure is required fr the analytical assessment f the thrust perfrmance f such nzzles. The supersnic external flw is limited in lateral extent, and wave reflectins frm the free bundary must be taken int accunt in the predictin f the near-wake structure. Truncated plug nzzles are ften perated with base bleed. The annular ejectr (Fig. 2c) is ften used in place f the traditinal ejectr, which has a centrally lcated supersnic driving nzzle. The supersnic external flw in the annular ejectr is bunded by a slid wall, and wave reflectins frm the bundary strngly interact with the near-wake structure. A highly cmplex near-wake flw structure is invlved with the external burning prpulsin cncept (Fig. 2d) which was first prpsed by Strahle (Ref. I). Accrding t Strahle, waves are generated by the mixing and cmbustin prcess; these waves interact with the near wake and cause the base pressure t be much higher than that with n fuel injectin. Indeed, Strahle suggests that the base pressure can be increased t a level greater than free-stream static pressure, perhaps t a level where net thrust can be btained with a slender frebdy. This external burning prpulsin cncept has been the subject f several experimental investigatins (e.g., Ref. 2). It is clear that an adequate analysis f this cmplex flw must be based n a physically pe,rceptive analysis f the simpler flw shwn in Fig. 2a. In view f the numerus practical flws in which supersnic near-wake mechanisms ccur, it was decided that a generalized analysis fr such flws was needed at the AEDC. The generalized analysis shuld include the fllwing factrs: (1) arbitrary turbulent initial bundary layers; (2) arbitrary inviscid external flw prfiles at the base plane; (3) external flws which are bunded by a slid wall r by a cnstant pressure bundary; (4) arbitrary base bleed, including gases which can react chemically with the external flw; and (5) planar r axisymmetric gemetry. The analysis shuld prvide reasnably accurate predictins nt nly f the grss features f the flw, such as the base pressure, but als f the detailed flw structure in the viscus wake and in the adjacent inviscid flw field. 9

14 l // / / E-""-...'\ -----e. "-\ '" '" ---'" \ '- '\ """- "' a. Bdies in supersnic flight COWI b. TnJlncated plug nzzles Free Bundary» m () :0 C!l I\J --.J,... //////// (/ '701 / / / / /... Pumped Flw ////// / -. I njectin Shck Wave M> L/-- / ;. "''-::,::*«;s:.;;s::::../?:ju;{;i L Fuel Generatr Mixi ng and Cmbustin Zne c. Annular ejectrs d. External burning prpulsin Figure 2. Examples f supersnic near-wake flws.

15 1.2 REVIEW OF EARLIER ANALYTICAL WORK The analysis f turbulent near-wake flws has received much attentin duling the past twenty-five years. Mst f the wrk has been based either n the viscus interactin mdel prpsed by Crcc and Lees (Ref. 3) r n the cmpnent mdel first develped by Krst fr turbulent flws (Refs. 4 and 5). Mst investigatrs have cncentrated n planar flws, which are inherently simpler than axisymmetric flws. Crcc-Lees Mdel - This integral methd is based n slving the integral frm f the bundary-layer equatins fr the viscus prtin f the flw, including the reversed flw regin. The planar external flw field is cupled t the viscus regin by the use f the Prandtl-Meyer equatins, which relate the lcal static pressure t the flw angle at the edge f the viscus layer. The viscus layer grwth rate is cmputed with an empirical entrainment cefficient. A majr prblem in the riginal Crcc-Lees analysis is establishing the relatinship f the integral parameters f the viscus flw prfiles t the ther variables f the prblem. A majr feature f the Crcc-Lees slutin is the appearance f a saddle-pint singularity (the critical pint) in the recmpressin regin. This singularity is interpreted as being analgus t chking in the thrat f a nzzle, i.e., the wake flw upstream f. the singularity is subcritical and can transmit disturbances in the upstream directin. The wake flw dwnstream f the singularity is supercritical, i.e., supersnic n the average. The crrect near-wake slutin is taken as that which will pass thrugh the singularity. Althugh applicable t a brad class f strng viscus interactin prblems, the riginal Crcc-Lees analysis yields rather pr results fr planar turbulent near wakes. The first successful extensin f the Crcc-Lees framewrk t the planar turbulent near-wake prblem was made by Alber (Refs. 6 and 7). Alber btained the integral parameters f the viscus flw prfiles by using, in a transfrmed incmpressible flw, the Stewartsn slutins f the Falkner-Skan equatins. The empirical mixing cefficient f the riginal Crcc-Lees analysis was replaced by the slutin f the first mment f mmentum equatin (the mechanical energy equatin); the dissipatin integral in this equatin was evaluated by the use f an empirical eddy viscsity mdel. Prandtl-Meyer cupling with the external flw was retained. It is nly recently that an attempt has been made t extend the Crcc-Lees framewrk t axisymmetric turbulent wake flws. Strahle and his cwrkers (Refs. 8 and 9) have been develping an axisymmetric versin f Alber's analysis, with the bjective f extending. it t the external burning p pulsin prblem. The external flw is assumed t be isentrpic and is treated with an apprximate methd f characteristics technique. (Prandtl-Meyer cupling is nt applicable t axisymmetric flw). Only flws withut base 11

16 bleed are cnsidered. The slutin fr nly ne axisymmetric flw field has been presented (Ref. 9). Krst mdel - The Krst cmpnent mdel (Ref. 5) is much simpler than the Crcc-Lees mdel and can be cnsidered a first-rder slutin. The cmpnent prcesses invlved in the near-wake flw field (inviscid external flw, turbulent mixing, recmpressin) are cnsidered separately. The results frm the cmpnent analyses are then cmbined t yield a slutin fr the base pressure and fr the ther glbal prperties f the recirculating flw regin. Three majr assumptins are invlved in the basic Krst analysis: (1) the recirculatin regin is quasi-stagnant, (2) the flw alng the "discriminating" streamline (that which separates the flw which passes thrugh recmpressin frm that which is recirculated upstream) isentrpically recmpresses, and (3) the flw alng the discriminating streamline stagnates at the peak f the recmpressin pressure distributin. Althugh experimental evidence is available t shw that the first and third f these assumptins are quite incrrect, the basic Krst analysis (especially the simplest versin in which initial bundary-layer effects are neglected) yields surprisingly gd predictins f the base pressure fr planar and quasi-planar flws. The analysis has been extended t include base bleed, asymmetrical tw-stream flws, nnadiabatic flws, and flws with chemical reactins in the near wake (Ref. 10). Numerus attempts have been made t imprve the cmpnent analyses in the Krst mdel. Nash (Ref. 11) intrduced an empirical factr t accunt fr the fact that the discriminating streamline des nt stagnate at the peak f the pressure distributin. Other investigatrs (e.g., Lamb and Hd, Ref. 12) have develped cntrl vlume analyses fr the recmpressin regin. The basic assumptins in the basic Krst analysis are nt very realistic. Because f cmpensating effects, the simplest Krst analysis fr negligible initial. bundary layers yields fairly satisfactry base pressure predictins fr experimental flws with initial bundary layers f mderate thickness (0 less than the base height). Attempts t prperly include the influence f the initial bundary layers in the Krst analysis, using techniques such as the apparent mixing layer rigin shift cncept f Hill and Page (Ref. 13), have nt been very successful. Hwever, these methds wrk reasnably well when applied t a mixing prcess which des nt invlve a large expansin at the base plane. The discrepancy fr near-wake flws is related t the fact that the rigin shift cncept includes the change in the bundary-layer mean flw prfile when it passes thrugh the' crner expansin prcess, but des nt take int cnsideratin that the bundary-layer turbulence decays rapidly when it passes thrugh the expansin. This rapid decay f the turbulent fluctuatins in the "remnant" f the initial bundary layer was bserved experimentally by Lewis and Behrens (Ref. 14) and Page and Sernas (Ref. 15). These 12

17 bservatins lead t the cnclusin that the bundary-layer flw after the crner expansin shuld nt be taken as the initial cnditin fr the subsequent develpment f the free turbulent mixing layer. Instead, the remnant f the bundary layer behaves as a rtatinal but inviscid external flw, and a new turbulent layer begins t develp at the edge f this rtatinal flw_ Several attempts have been made t apply the Krst methd t the axisymmetric base pressure prblem. If the methd f characteristics is used t cmpute the inviscid flw adjacent t the cnstant pressure prtin f the wake flw, then the slutin cannt be extended t the axis. Therefre, several investigatrs (e.g., Ref. 16) have used the assumptin that the flw recmpresses, in a lcally planar prcess, n an imaginary sting which extends frm the base. Mueller (Ref. 17) assumed that recmpressin ccurs either n a sting with a radius f ne-half f the base radius, r n a sting equivalent in radius t the experimental radius f the wake after recmpressin; the latter radius was determined as an empirical functin f the free-stream Mach number. The simple recmpressin cncepts f the riginal Krst thery are prly suited fr the cmputatin f axisymmetric near-wake flws. Hwever, there are enugh free parameters in the axisymmetric extensins f the Krst mdel s that reasnably adequate values fr the base pressure can be btained. But these mdels really are used t crrelate experimental base pressure measurements rather than t predict them. Mrever, the Krst-type mdels d nt prvide any infrmatin abut the detailed flw structure in the recmpressin regin. Other methds - McDnald (Ref. 18) tk a unique apprach t the predictin f the planar base pressure fr flws which reattach t a slid wall. (The centerline f Fig. I is replaced by a wall). McDnald als applied his analysis t axisymmetric flws in which a relatively large sting extends dwnstream frm the base (Ref. 19). The base pressure is uniquely determined as that which causes the reattached flw dwnstream f recmpressin t becme a fully develped flat plate bundary layer. Althugh McDnald prduced reasnable predictins f the base pressure, his mdel can be criticized n physical grunds. Numerus experiments have shwn that the planar near-wake structure fr reattachment t a plane f symmetry is nt significantly different frm that fr reattachment t a wall. Therefre, wall viscus effects cannt be a majr cntrlling factr in the near-wake structure. Chw (Ref. 20) has develped a tw-layer integral treatment f recmpressin in planar flw; in many aspects, this physically realistic mdel cmbines features f bth the Crcc-Lees and Krst mdels, Chw fund it necessary t include the lateral pressure gradients in the recmpressing viscus layer; hwever, this need t include lateral pressure gradients is prbably related t the assumed gemetry f the dividing streamline. 13

18 This recmpressin analysis is applicable nly t the regin upstream f the RSP (Fig. I). The predictins f the Chw mdel are discussed in Sectin 3.0. Recently, Chw and Spring (Ref. 21) develped a methd fr predicting the wake develpment dwnstream f the RSP. In additin, they imprved the cmputatin f the turbulent mixing upstream f the nset f recmpressin, and quite gd predictins f planar base pressures were btained. Weng (Ref. 22) has extended the Chw recmpressin mdel t axisymmetric wake flws in which a sting extends frm the base; it seems that this mdel shuld, in plinciple, als apply t flws withut a sting. In cmmn with the mdels f Chw and f Chw and Spring, the finite recirculatin velcities are neglected in the flw upstream f the recmpressin regin. The external flw is cmputed by the methd f charactelistics dwnstream t the nset f recmpressin; Prandtl-Meyer cupling is used in the recmpressin regin. The initial bundary-layer prfile after the base expansin prcess is taken as the initial cnditin fr the develpment f the free shear layer. Weng used Prandtl mixing length thery t cmpute the turbulent shear stresses in the flw upstream f the nset f recmpressin; in the recmpressin regin, he used an arbitrary spatial variatin f the eddy viscsity. Weng btained reasnably gd predictins f the effect f sting-base radius rati n the base pressure fr free-stream Mach numbers frm 1.5 t 2.5. Summary f Earlier Analytical Wrk - Of the available analytical mdels fr planar wake flws, thse f Alber (Refs. 6 and 7) and Chw and Spring (Ref. 21) are the mst perceptive. Bth mdels prvide a predictin r the flw-field structure as well as f the base pressure. The Chw and Spring mdel prvides the best predictin f the effect f initial bundary layer n the base pressure ver the Mach number range investigated, but Alber's mdel is attractive because finite reverse flw velcities are included in the regin upstream f the nset f recmpressin. Neither f these mdels has been applied t flws with base bleed; indeed, nly the Krst-type mdels have been used t predict the effects f base bleed. Of the available mdels fr axisymmetric wake flws, nly thse f Strahle, et al. (Refs. 8 and 9) and Weng (Ref. 22) are based n a reasnably adequate descriptin f the detailed flw prcesses. Strahle, et ai., have nt yet included initial bundary-layer effects in their mdel. As discussed in Sectin 3.0, Weng's methd f cmputing the initial bundary-layer effect will prbably be inadequate fr free-stream Mach numbers f three r grea ter. 14

19 It is clear that even the recently develped mdels fr axisymmetric flw d nt satisfy the critella established in Sectin 1_1 fr a generalized near-wake analysis. The vid between the requirements and the capabilities f available mdels was even mre prnunced at the time that the wrk described in this reprt was initiated. Therefre, it was decided t undertake the develpment f an analytical mdel which wuld be applicable t a brad range f supersnic near-wake flws (Sectin 1.1). 1.3 IMPORTANT FEATURES OF A REALISTIC ANALYTICAL MODEL When ne cnsiders the earlier analytical wrk, as well as the available experimental infrmatin,' it becmes clear that a perceptive mdel f the near-wake structure shuld include several features. First, adequate treatment f the external flw in the axisymmetric case requires the use f the methd f characteristics. Secnd, the evidence n quenching f the turbulence in the expanding bundary layer suggests that the remnant f the initial bundary layer can be cmputed with the rtatinal methd f characteristics. Third, as shwn by Weinbaum (Ref. 23), the viscus frces are negligible cmpared t inertial frces during the turning f the initial bundary layer, and the expansin f the supersnic prtin f the initial bundary layer can be cmputed with the rtatinal methd f characteristics. (A streamtube methd can be used t cmpute the expansin f the subsnic prtin f the initial bundary layer). Furth, a new turbulent shear layer shuld be cnsidered t begin after the crner expansin prcess. It shuld be nted that the cncept f a new shear layer which develps dwnstream f the crner expansin has been suggested by Krst (Ref. 24) and has been used in the planar analysis f Chw and Spring (Ref. 21). The available experimental evidence suggests that the near-wake regin between the nset f recirculatin and the beginning f recmpressin can be cnsidered t be a cnstant pressure regin, as has been assumed by mst investigatrs. Hwever, the experimental evidence des nt suggest that the reverse flw velcities can be neglected, as has been dne in the Krst-type mdels; the finite recirculatin velcities must be included. The analytical mdel which is described in Sectin 2.0 incrprates the features which have been discussed. The framewrk f the analysis is cnceptually similar t the Crcc-Lees apprach. Hwever, the detailed treatment f the flw prcesses is clsely related t the methd used by the authrs fr dueted tw-stream mixing flws (Refs. 25 and 26). 15

20 2.0 DEVELOPMENT OF ANALYTICAL MODEL The gemetry and nmenclature fr the near-wake analysis is illustrated in Fig. 3. The flw dwnstream f the base is divided int three distinct regimes: (1) the "jet regime" which extends frm the base plane t the statin where recirculatin begins, (2) the "wake regime II which extends frm the nset f recirculatin dwnstream t the nset f recmpressin, and (3) the "fully develped" regime which begins at the nset f recmpressin and extends dwnstream int the fully develped far-wake regin. / </ / Z 7 Upper Bundary fexternal / / / /' /' Flw (Prescribed Pressure r / 6] PI / </./ Prescribed Gemetry) /: /' Prfile f I us/ I Axial Velcity /' /// Pb - - _ Us /' -;:/ / / / _ /'/'//1 uil - b - _ /' /' /' / / / f--rs--- /' / / /,u s _ r r ---- U c I rm I ' ll - - /- Beginning f Recirculatin--'-/ L RSP ime Wake Regime Beginning f Press ure Rise Fully Develped ----t+-- Regime Figure 3. Nmenclature fr near-wake analysis. 2.1 BASIC ASSUMPTIONS The majr assumptins used in the analysis are: 1. The flw is steady and is either planar r axisymmetric. 2. All gases bey the perfect gas law. 3. The integral frm f the bundary-layer equatins describes the flw between the axis and the uter edge f the turbulent mixing layer, rs. 4. The turbulent Prandtl and Schmidt numbers are unity. 16

21 5. The free shear layer which frms dwnstream f the base expansin prcess is fully turbulent (negligible laminar transprt) and has a negligible initial thickness_ 6. The flw utside f the free turbulent shear layer (r > rs) is supersnic, adiabatic, and inviscid. 7. The upper bundary f the supersnic external flw is inviscid and is either a prescribed gemetry r has a prescribed cnstant pressure. 8. The nrmalized prfiles f the axial cmpnent f velcity in the turbulent shear layer (between ri and rs) are similar and are represented by a csine functin. 9. The element species and axial cmpnent f velcity between the axis and ri are ne-dimensinal. 10. The centerline static pressure in the jet regime is an isentrpic functin f u e. 11. The centerline static pressure is cnstant in the wake regime. 12. The initial bundary layer is turbulent and is characterized by its thickness and a pwer law velcity prfile. 13. Pressure feedback effects in the bundary layer upstream f the base are negligible. 14. The expansin f the initial bundary layer at the base plane is inviscid. 15. The base bleed flw is unifrm and subsnic at the base plane. 16. Chemical reactins in the viscus wake flw, if they ccur, are assumed t be in equilibrium. 17. If the planar flw reattaches t a wall rather than t a plane f symmetry, then viscus effects alng the wall are negligible. The use f the bundary-layer apprximatins (assumptin 3) is cnsistent with Crcc-Lees and implies that (1) there are negligible lateral pressure gradients in the viscus wake flw, and (2) streamwise turbulent transprt mechanisms are negligible. The assumptin f negligible lateral pressure gradients is reasnably valid in the jet and wake regimes because the curvature f the streamlines is small in the high-speed prtin f the 17

22 viscus layer; this assumptin is less valid in the recmpressin regin f the fully develped regime. The use f a csine prfile in the shear layer, alng with a unifrm reverse flw prfile, was suggested by Green (Ref. 27) fr incmpressible wake flws. It shuld be nted that, even thugh the cmmn term "base bleed" is used in this reprt, the mmentum f the stream issuing frm the base is nt neglected; therefre, the. analysis is applicable t "bleed" velcities in the high subsnic range. 2.2 BASIC EQUATIONS The basic bundary-layer equatins are: Cntinuity (1) where Axial Mmentum a a 0 fr planar flw and 1 fr axisymmetric flw a au a au a ( a) pur ax + pvr a-; = ar rr (2) where Element Species Cnservatin T is the turbulent shear stress a ac a ac a ( a) pur - + pvr _ = - qr ax ar ar (3) where q is a turbulent species flux. Equatins (1) thrugh (3) are integrated (Ref. 25) t btain five integral equatins: (1) a cntinuity equatin fr the flw between the centerline and r s, (2) a mmentum equatin fr the flw between the centerline and r s, (3) a mmentum equatin fr the flw between the centerline and rm (where fr == fi + b/2), (4) a species cnservatin equatin fr the flw between the centerline and r s,.and (5) a species cnservatin equatin fr the flw between the centerline and rm. The integral equatins are: 18

23 Overall Cntinuity _ p v r a s s s (4) Overall Mmentum -p v u r a - s s s s (a + I) (5) Half-Radius Mmentum dx (a + I) (6) Overall Species (7) Half-Radius Species (8) Because the flw is ne-dimensinal fr 0 < r < ri, the integral equatins can be rewritten as fllws: Overall Cntinuity Overall Mmentum r s a! ax (pu)ra dr (9) 1 a+1 a+ 1 d Pc rs r i d 2 - P vu r a (a+ 1 ) dx(pcuc) s s s s dx (a + 1) (10) Half-Radius Mmentum dp r a + 1 T ra _ c m m m d;- (a + 1) a+ 1 a+1 rid _r_i_ ( u 2 ) (p u ) (a + 1) dx Pc c + urn (a + 1) c c (11) 19

24 AE DC-TR Overall Species Half-Radius Species a+1 f i d -- - (p U C ) (a + 1) dx e e e (12) a+1 fi d a+1 fi d qmr - (a + 1) dx (PeueCJ + Cm (a + 1) dx (Peu e) (13) Equatins (9) thrugh (13) apply t all three regimes f Fig. 3. Of curse, ri ges t zer in the fully develped regime, the pressure gradient terms are zer in the wake regime, and C e = Cj = I in the jet regime. 2.3 SHEAR LAVER PROFILES The free shear layer velcity prfile is given by u - u e 1 1 (. f-f i) = cs 1Tu - u 2 2 b s e (14) The velcity at the half-radius surface, rr (= ri + b/2), is given by 1 +-(u -u) 2 s e (15) If the shear layer were at cnstant pressure, with cnstant values f Us and u e, then the element species cncentratin prfile wuld be given by r c - C e c - c s e u-u e u - u s e (16) which is the well knwn Crcc integral slutin btained frm Eqs. (2) and (3). In the near-wake prblem, hwever, the pressure gradient terms are negligible nly in the wake regime. In additin, C e, u e, and Us vary with x, even in a regin f cnstant pressure. Therefre, Eq. (16) cannt be used. The apprach which has been taken in this study is t assume that the deviatins frm Eq. (16) can be expressed as c - C e (17) where K can vary with x. It shuld be nted that a pwer law relatin similar t Eq. (17) has been used t cmpensate fr the deviatins frm the Crcc relatin which are 20

25 caused by nn unity transprt cefficient ratis in free turbulent flws (Ref. 28). Since C s is defined t be zer, Eq. (17) becmes C=c -c, {U_U)K c c u _ u c s c 2.4 COMPUTATION OF DENSITY IN THE VISCOUS FLOW Fr unity turbulent Prandtl and Schmidt numbers, the stagnatin enthalpy and the gas prperties are uniquely related t C, and the density field is calculated with the techniques described in Ref. 25. The stagnatin temperature, specific heat, and gas cnstant are specified as a functin f C, which varies frm zer (pure uter-stream gas) t ne (pure base bleed gas). Fr chemically frzen flw, nly the T, c p and R values at C = 0 and at C = 1 need be specified. Fr flw with equilibrium chemical reactins, T, c p and R are specified as functins f C. At any pint in the shear layer, u is given by Eq. (14), C is given by Eq. (18), and T, c p, and R are then determined. The density can be btained frm the energy equatin and the perfect gas law. Therefre, the density can be expressed functinally as 2.5 TURB,ULENT TRANSPORT TERMS (18) P = P (u, c, p) (19) The turbulent shear stress, T m, which appears in Eq. (11) is given by r t- au I m - Pm ar m (20) where is the turbulent eddy viscsity. The eddy viscsity is given by t = _1_ b lu - ucl RT S (21) where RT is a turbulent Reynlds number based n the eddy viscsity, the shear layer width and the velcity difference acrss the layer; the mdel used t btain RT is described in Sectin The equatin fr T m becmes, fr the csine velcity prfile, r = _TT_ P lu - u I (u - u ) m 2RT m s c S c (22) 21

26 The turbulent species flux, qm, is given by r (23) 2.6 TRANSFORMATION OF THE INTEGRAL EQUATIONS Jet regime - Assuming that the flw cnditins alng rs can be specified, sufficient infrmatin is available t relate the terms in Eqs. (9) thrugh (13) t fur flw-field variables. Since there are nly fur unknwns in the jet regime, Eq. (13) is nt used. The fur dependent variables were selected t be Pc, ri, band K. Equatins (9) thrugh (12) can be transfrmed int the fllwing system (Refs. 25 and 29) (24) where N == F fr the cntinuity equatin, N == G fr the verall mmentum equatin, N == H fr the half-radius mmentum equatin, and N == I fr the verall species cnservatin equatin. The equatins fr the cefficients are presented in Appendix A. The numerus auxiliary equatins necessary fr cmputatin f the cefficients are develped in the same manner as described in Ref. 25. Equatins (24) are slved fr the derivatives (dpc/dx, drddx, db/dx, dk/dx) by use f a matrix factrizatin technique. The resulting equatins fr the derivatives are numerically integrated with a mdified Euler technique (variable step size). Wake regime - The five dependent variables selected in the wake regime are uc, ri, b, K and C e. Equatins (9) thrugh (13) are transfrmed int the fllwing system: duc dri db dk de c N 1 dx + N 2 dx + N 3 dx + N 4"";h + N 5 dx = N6 (25) where N == J fr the half-radius species equatin. The equatins fr the cefficients are presented in Appendix A. 22

27 AEDC TR Fully develped regime - The five dependent variables selected in the fully develped regime are Pc, Uc, b, K and Cc. Equatins (9) thrugh (13) are transfrmed int the fllwing system: (26) The equatins fr the cefficients are presented in Appendix A. 2.7 SOLUTION OF THE OUTER FLOW AND COUPLING WITH THE VISCOUS LAVER The supersnic flw in the regin r > rs is cmputed with the rtatinal methd f characteristics. The "reference plane" technique, in which the supersnic flw parameters are established alng planes nrmal t the axis, is used. The axial lcatin f the reference planes cincides with the axial statins used in the numerical integratin f the viscus layer equatins. Many terms which appear in the cefficients,n, f the viscus layer equatins depend n the flw cnditins (and their derivatives with respect t x) alng rs, the interface between the viscus regin and the uter supersnic flw. Therefre, the slutin f the tw regins must be cupled. The technique used fr cupling the tw regins is illustrated in Fig. 4. At sme statin, x, the flw in bth regins is fully specified. In the integratin technique fr the viscus layer equatins, the step size, &, is given, as are the tentative values f p and rs at (x + L'lx). A left running characteristic, alng which the flw is fully defined, is cnstructed frm the the last bundary pint (Fig. 4). The right running characteristic and the streamline, each f which passes thrugh the new bundary pint and intersects the left running characteristic, are cnstructed; the equatins fr the streamline and the right running characteristic are slved in the usual iterative manner t define the entrpy and flw angle at the new bundary pint. Then the density and velcity at the new bundary pint are cmputed, and the derivatives with respect t x f these flw prperties alng rs are cmputed with an upstream differencing technique. (These derivatives are assumed t be zer at x = 0). Therefre, sufficient infrmatin is available abut the flw prperties alng rs s that the cefficients f the viscus layer equatins at (x + L'lx) can be evaluated. Within the framewrk f the Euler integratin technique, this bundary pint prcedure is repeated, with different values f Pc and rs at (x + L'lx); until the slutin f the viscus equatins cnverges. The slutin then prceeds dwnstream t a new axial statin. In this dwnstream marching prcedure, develpment f the slutins fr the tw flw regins is fully cupled. 23

28 Streamline Which Passes thrugh New Bundary Pint,/ '",/ /' /'./ /' /,,/"'''-- Left Running Characteristic /' /' alng Which Flw Is Fully Defined "': / Right Running Characteristic '" "- / "- "- " Upper Edge f Turbulent Shear Last Bundary Pint Layer, rs(x) f--fj.x X New Bundary Pint Figure 4. Cupling f the shear layer and uter-flw slutins. Even thugh the uter flw is treated with the rtatinal methd f characteristics, n shck waves are allwed t develp. Hwever, tw shck waves exist even in simple near-wake flw fields, the lip shck and the recmpressin shck. The lip shck, which has been experimentally investigated by Hama (Ref. 30), results frm the verturning f the lw Mach number prtin f the initial bundary layer in the base expansin prcess (Ref. 23). In neglecting the lip shck, we have treated it as an' isentrpic cmpressin which is justified since experiments have shwn that the lip shck is rather weak, at least fr free-stream Mach numbers up t 2.0 r 3.0. The series f cmpressin waves which result frm the recmpressin prcess calesce t frm the recmpressin shck wave (Fig. I). Hwever, the shck is frmed t far dwnstream t affect the near-wake clsure mechanism. Therefre, neglecting the recmpressin shck wave influences nly the dwnstream prtin f the recmpressin prcess. Inclusin f the lip and recmpressin shck waves in the present analysis is quite feasible, but wuld require a significant increase in the cmputatin time. 2.8 EXPANSION OF THE INITIAL BOUNDARY LAYER The expansin f the supersnic prtin f the initial bundary layer (Fig. 5) is cmputed with the rtatinal methd f characteristics. Since pressure fccdback effects are neglected, the supersnic expansin prcess is initiated at x = 0 with a Prandtl-Meyer 24

29 AE DC-TR turn (frm PI t Pb) at the snic pint in the bundary layer. Expansin f the subsnic prtin f the initial bundary layer is cmputed with an isentrpic stream tube mdel. The subsnic layer is divided int ten streamtubes, each f which expands isentrpically frm PI t Pb. The flw directin f the stream tubes after expansin is taken t be the same as that f the initially snic streamline after it passes thrugh the Prandtl-Meyer turn. The flw in the wall streamtube, after expansin, is the initial edge cnditin (at x = 0) fr the new turbulent shear layer which develps dwnstream f the base. T [) ual t r--- Velcity Prfile in Initial Su ndary Layer -./ -,/..--./..--- /././ Supersnic 1:;// Prandtl-Meyer Expansin.... at x :: 0 (frm PI t Pb) Sublayer Velcity Prfile after Turn I nitial Val ue f Velcity at Edge f Shear Layer Figure 5. Crner expansin f the initial bundary layer. Fr values f Pb /PI greater than the snic static-ttal-pressure rati, ne r mre f the streamtubes will be subsnic after the expansin prcess. In such cases, the Mach number f the subsnic streamtubes is arbitrarily adjusted t be slightly greater than unity. The effect f this adjustment prcess is very small fr typical initial bundary-layer thicknesses because the mass flw in the adjusted streamtubes is small and is quickly entrained by the turbulent shear layer. 25

30 AE DC-TR CHARACTERIZATION OF THE INITIAL AND BOUNDARY CONDITIONS The turbulent near-wake flw field is fully determined by the fllwing cnditins, which must be specified: 1. The upper bundary f the supersnic external flw, 2. The base bleed flw rate, 3. The ttal temperatures and gas prperties f the external flw and the bleed flw, 4. Mixture gas prperties and ttal temperature as a functin f C, if chemical reactins ccur (Sectin 2.4), 5. Thickness, 8, and velcity prfile shape f the adiabatic initial bundary layer, and 6. The prfiles at the base plane f the static pressure, ttal pressure and flw angle, between the edge f the bundary layer and the upper bundary. Fr cmputatin f near-wake flws embedded in external flws f large lateral extent (effectively infinite), the upper bundary is specified as a wall parallel t the centerline and lcated far enugh frm the near wake s that reflected waves frm the crner expansin prcess d nt interact with the recmpressin regin. The initial bundary-layer velcity prfile is assumed t be given by a pwer law where Ual is the velcity at the edge f the bundary layer, 8 is the thickness f the layer, y is the distance frm the wall (Fig. 3), and n is specified. The initial bundary-layer characteristics fr near-wake expeiiments are usually specified in terms f the mmentum thickness and the mmentum thickness Reynlds number, Re. Therefre, a methd was develped t estimate n as a functin f Mal and Re. Maise and McDnald (Ref. 31) shwed that supersnic bundary-layer prfiles n an adiabatic flat plate can be well represented by the Cles law-f-the-wall/law-f-the-wake prfiles, alng with the Van DIiest transfrmed velcities and the Spalding-Chi skin fiictin values. The 8/8 values shwn by Maise and McDnald as a functin f Ma 1 and Re, were matched with 8/8 values fr pwer law prfiles (Ref. 32) t determine n as a functin f Mal and Re (Fig. 6). The relatin f n, Mal, and 8/8 is shwn in Fig

31 AEDC TR The pwer law prfile, as well as the prfile used by Maiseand McDnald, des nt smthly jin the external stream velcity prfile. The resulting II crner ll n the ttal-pressure prfile was fund t cause cmputatinal prblems in the viscus layer slutin when the crner is entrained int the shear layer. T alleviate this prblem, the uter edge f the initial bundary-layer velcity prfile is smthed with a cubic functin which extends frm (rb ) t (rb ). This smthing has a negligible effect n the mmentum thickness. 0.20F c c 0.16 Cl> c 0 c. >< UJ... Cl> 0.14 >- -' '" I >- '- '0 '" 0.12 c ::l 0 c:> 0.10 U/U a l = (y/)n )'a = Free-stream Mach Number, Mal 4 5 Figure 6. Bundary-layer prfile expnent fr adiabatic flw alng a flat plate c 'E Cl> c c. x UJ 0.14 >- :3 & -E O. 12 ::l CCl 0.10 U/U a l = (y/)n )'a = 1.4 Mmentum Thickness Rati, 8/0 Figure 7. Bundary-layer mmentum thickness rati fr adiabatic flw alng a flat plate. 27

32 2.10 TWO-LAYER TREATMENT OF THE OUTER FLOW It was discvered early in this investigatin that the entrpy prfile in the initial bundary layer tends t spread int the uter flw, which is usually irrtatinal. Of curse, the diffusin f the entrpy field acrss streamlines is incrrect since the flw is assumed t be inviscid. Apparently, this false diffusin in the methd f characteristics flw field is related t the rapid expansin f the highly rtatinal bundary-layer flw. The false diffusin effect is nt unique t the present methd f characteristics frmulatin; ther widely used methd f characteristics cmputer prgrams were fund t yield similar results_ T eliminate the spurius entrpy diffusin, a tw-layer technique is used (Fig. 8). The streamline which riginates at the uter edge f the inital bundary layer is established, and the ttal pressure n this streamline is specified t be P a' Therefre, the entrpy field in the supersnic flw is divided int tw layers, and the entrpy gradients frm the initial bundary layer are cnfined t the lwer layel Left Running Characteristics Which Originate at Prandtl-Meyer Expansin / /,/ / Velcity Prfile in / // // // Initial Bundary Laye7 ;' //,// // Subsnic Layer 7 / / / // ---'------:/-"---,t._ " / / /:;/ 1/// /1// Streamline Which Originates at Outer Edge f Initial Bu ndary Layer, Figure 8. Tw-layer treatment f the supersnic flw field METHOD OF SOLUTION As in the Crcc-Lees analysis, a saddle-pint singularity (the critical pint), which is intrinsic t the slutin, ccurs in the reattachment regin. The behavir f the slutin is illustrated by the axial distributin f centeriine velcity (Fig. 9). The crrect slutin is that which passes smthly thrugh the singularity. If the assumed base 28

33 pressure is t lw the slutin will be supercritical and "blw up", i.e., in terms f Cramer's rule, the determinant f the cefficients fr Eqs. (26) will g t zer befre the numeratr determinants d. If the assumed base pressure is t high, the slutin will be sub critical and is characterized by the centerline velcity reaching a maximum, then decreasing with distance (Fig. 9). By iterating n the initial cnditins (base pressure), ne can cnverge n the set f initial cnditins which will cause the flw-field slutin t pass smthly thrugh the singularity. (+) Critical Pi nt U :::J... 'g Q,) > Q,) c:... Q,)... c: Q,) u (-) 0 / S upercritical Slutins (Base Pressure T Lw) / Axial Distance frm Base, x S ubcritical Sl utins (Base Pressure T High) Figure 9. Subcritical and supercritical slutins. The sub critical slutins are identified by due /dx ging t zer. The supercritical slutins are identified by an inflectin in the centerline velcity distributin, i.e., by d 2 U c /dx 2 ging t zer, which implies that the determinant is appraching zer. A halving technique, with the bunds established by the last subcritical and the last supercritical values f Pb, is used t cnverge t the crrect value f Pb. With a reasnable first guess n the bunds fr Pb, say within 20 percent f the crrect value, the cnverged value f Pb can be identified t within 0.5 percent with apprximately ten iteratins f the flw-field slutin. Althugh it is nt difficult t define the initial cnditins with practical engineering accuracy, establishing the slutin thrugh and beynd the critical regin is very difficult. The nature f a saddle-pint singularity is such that very small changes in the initial cnditins result in large changes in the slutin as the singularity is apprached. Tw 29

34 AE DC-TR techniques have been used t btain slutins which pass thrugh the critical regin. The first is a "brute frce II apprach; the crrect initial cnditin is established with a very small tlerance, typically t within ne part in At a statin just upstream f the critical regin, the centerline velcity is perturbed upward by a small amunt (typically ne percent) and the slutin will usually prceed smthly thrugh the critical regin. This prcedure has been fund t wrk fairly well fr planar flws, and fr axisymmetric flws with high external stream Mach numbers. Hwever, fr axisymmetric flws with mderate external stream Mach numbers (2.0 r 3.0), this prcedure has nt been successful, even when the initial cnditins are cnverged t within ne part in An alternate apprach fr develping the slutin thrugh the critical regin is t cnverge the initial cnditins fairly tightly, perhaps t within ne part in 10 3 r 10 4, then t extraplate the centerline velcity distributin. Results at tw axial statins near the RSP, which is well upstream f the critical pint, are used t establish an expnential extrplatin equatin fr uc(x). One equatin, the cntinuity equatin, is remved frm Eqs. (26), and the singular behavir f the slutin is remved. Experience has shwn that the expnential extrplatin can be extended dwnstream, with gd results, t the peak f the centerline pressure distributin in axisymmettic flw. Results btained with bth f the techniques fr develping the slutin thrugh the critical regin are presented in Sectin 3.0. Frtunately, the engineer is usually interested nly in a predictin f the initial cnditins (the base pressure). The initial cnditins, and the flw field dwnstream t the vicinity f the RSP, can be determined with reasnable accuracy withut develping the slutin thrugh the critical regin. The ccurrence f the saddle-pint singularity in the slutin is a result f the elliptic nature f the near-wake flw field. That is, an essentially elliptic prblem is treated as an iterative initial value prblem, and the critical pint is the mechanism whereby the dwnstream wake clsure cnditins are manifested. The near-wake analysis has been prgrammed fr numetical slutin with an IBM cmputer. A typical cmputatin, fr ne assumed set f initial cnditins, which prceeds frm the base plane t dwnstream f the RSP requires a cmputatin time f frm 30 t 60 secnds. The determinatin f the base pressure t practical engineering accuracy (less than ne percent uncertainty) invlves a cmputatin time f five t ten. minutes. Fr near-wake flws in which there is n base bleed, r in which the base bleed gas has the same ttal temperature and gas prperties as the uter flw, a simplified versin 30