NASA CONTRACTOR REPORT FURTHER STUDIES OF UNSTEADY BOUNDARY LAYERS WITH FLOW REVERSAL John F. Nush Prepared by SYBUCON, INC. Atlanta, Ga. 303 39 for Ames Research Center TECH LIBRARY KAFB, NU NASA CR-2767 I I 4. Title nd Subtitle I 5. Report Date "Further Studies of Unsteady Boundary Layers with . December 1976 Flow Reversalll Organization 6. Performing Code 7. Author($) 8. Performing Orgnization Report No. John F. Nash 10. Work Unit No. 9. Performing Orpmization Nama and Address Sybucon, Inc. 11.Contract or GrantNo. 9 960) Perimeter Place,N.W. (Suite 2-8771 Atlanta, Georgia 30339 - NAS 13.Type of Report andPeriod Covered 12. Sponsoring myName md Address 6 Contractor Report- National Aeronautics Space Administration 14.Sponsorirg Aqmcy Code Washington, D. C. 20546 I 15.Supplementary Notas 16. Abstract Further computational experiments have been conducted to study the charac- teristics of flow reversal and separation in unsteady boundary layers. One set of calculations was performed using thefirst-order, time-dependent turbulent boundary-layer equations, and extended earlier work by Nash and Pate1 to a wider rangeof flows. Another set of calculations was performed for laminar flow using the time-dependent Navier-Stokesequati.ons. 1 The results of the calculations confirm previous conclusions concerning the existence of a regime of unseparated flow, containing an embedded regionof reversal, which is accessible to first-order boundary-layer theory.However certain doubts are caston the precise natureof the events which accompany the eventual breakdownof the theory due to singularity onset.The earlier view that the singularity appearsas the final event in a sequence involving rapid thickeningof the boundary layer and the formationof a localized region of steep gradients, is called into questionby the present results. It appears, first, that singularity onsetis not necessarily preceded by rapid boundary-layer thickening, or even necessarily produces immediate thickening. Furthermore, the formation of a region of steep gradients could not be reproduced in the solutionsof the Navier-Stokes equations, and may, itself, prove to bea feature of first-order boundary-layer theory.and not part of a more complete descriptionof the flow. 17. Key Words 1Supgsct.d by Auh(s)) 18. Dimibution Statement Turbulent Flow Calculation Navier-StokesEquationsUNCLASSIFIED - UNLIMITED Unsteady Boundary Layers STAR Category 02 18. Scurity Oamif. (of this report) 20. Security Mf.lot this m) 21. No. of Pages 22. Rice' UNCLASSIFIED UNCLASSIFIED $4.00 97 *For sale by the National Technical Information Slnica, Sprirqfimld, Virginia 22161 CONTENTS .Page ... FOREWORD ............................. i i I SUMMARY ............................. iv SYMBOLS ............................. V LISTOFFIGURES ......................... vii INTRODUCTION ........................... 1 PART I . TURBULENTFLOW CALCULATIONS ............... 4 1.1NATURE OF THE FLOWSCONSIDERED ............... 4 1.2 COMPUTATIONALEXPERIMENTS .................. 7 Frozen Flows ........................ 7 OscillatoryFlows ..................... 12 1.3 THE NATURE OF UNSTEADY SEPARATION .............. 19 PART I I . LAM I NAR FLOW CALCULATIONS ............... 24 11.1 NATUREOFTHE FLOWS CONSIDERED ............... 24 I 1.2 COMPUTATIONAL EXPERIMENTS ................. 29 CONCLUDINGREMARKS ........................ 32 APPENDIXA:CALCULATION METHODFOR TIME-DEPENDENT TURBULENTBOUNDARY LAYERS .............. 35 APPENDIXB:SOLUTION METHODFOR TIME-DEPENDENT NAVIER-STOKESEQUATIONS ............... 40 REFERENCES ............................ 44 FIGURES ............................. 46 ii FOREWORD Thisreport was preparedunder Contract NAS2-8771 by Sybucon, I nc. Scientific andBusiness Consultants 9 PerimeterPlace, N.W. (Suite 960) Atlanta,Georgia 30339 forthe UnitedStates Army Air Mobility Researchand Development Laboratory Ames Directorate MoffettField, California 94035. The contract was administered by theNational Aeronautics and Space Administration, Ames ResearchCenter. The Technical Monitor was Dr. Lawrence W. Carr. SUMMARY Furthercomputational experiments have been conducted to studythe char- acteristicsof flow reversal andseparation in unsteady boundary layers. One set of calculations was performedusing the first-order, time- dependentturbulent boundary-layer equations, and extended earlier work byNash and Pate1 to a widerrange of flows.Another set of calcula- tions was performed for laminar flow usingthe time-dependent Navier- Stokesequations. The resultsof the calculations confirm previous conclusions concerning theexistence of a regime ofunseparated flow, containing an embedded region of reversal,which is accessible to first-order boundary-layer theory. However, certaindoubts are cast on theprecise nature of. the eventswhich accompany theeventual breakdown ofthe theory due to singularityonset. The earlierview that the singularity appears as the finalevent in a sequenceinvolving rapid thickening of the boundary layer and theformation of a localizedregion of steepgradi ents,is calledinto question by thepresent results. It appears, fi rst,that singularityonset is not necessarily preceded by rapid boundary-layer thickening, or even necessarilyproduces immediate thickening. Further- more, theformation of a regionof steep gradients could not be repro- duced inthe solutions of the Navier-Stokes equations, and may, itself, proveto be a featureof first-order boundary-layer theory and notpart of a morecomplete description of the flow. iv Ll ST OF SYMBOLS A, AmA, Imbalancebetween production and dissipation of turbulent kineticenergy (see Equation A8) A,B,C,D,E Coefficientsappearing in Equation (814) Empiricalfunctions in the shear-stress model (Equation A3) alya2 C Chordlength of theplate, and length of theintegration doma i n F functionGeneral f Functiondefined by Equation (3) j Integers,n appearing in Equat ion(AI I) R integerTime-level L Dissipationlength m, n Node-pointindexing integers P pressureStatic 2222 Resultantfluctuating velocity(q =u +v +w ) R,R, 'R2 Functionsappearing inEquations (B14,Bl5,Bl6) S integrationtheHeight of domain t Ti me U,V,W Ensembleaverage velocity components inthe x-, y-, z-d i rections , respectively UYVYW Fluctuatingvelocity componentscorresponding to U,V,W, respectively X,Y Y Cartesiancoordinates fixed inthe plate: x measuredalong theplate, y measurednormal to it, and z measured laterally. Parametersappearing inEquations (2, 3) xo~xl Q Shear-stressgradient Equation(see A10) r, @ Parametersgppearing Equationsin (A3,A5) Steplength in the numerical method Boundary-layerthickness 01 U Displacementthickness (6” = I (1 - -) dy) 0 ‘e Vorticity component inthe z-direction Constant inthe Law of theWall Kinematicviscosity Dens i ty Shearstress Subscripts C Convect ion values e Value atthe edge of theboundary layer m Value atthe matching station 0 Values at x = 0 (except xo, whichsee) P Penetrationvalue S Stagnationvalue W Value at the wall vi LIST OF FIGURES Figure Page. 1. Definitionof the External Velocity Distributions ..... 46 2. WallShear Stress Distributions for Increasing Time; FrozenFlow, U t /c = 0.2, ff = 47 of 0 ............. 3. DisplacementThickness Distributions for Increasing Time; FrozenFlow, U t /c = 0.2, ff = 0 ...........48 of 4. WallShear Stress Distributions for Increasing Time;Frozen Flow, U t /c = 2.0, ff = 0 49 of .......... 5. DisplacementThickness Distributions for Increasing Time;Frozen Flow, U t /c = 2.0, f = 0 .......... 50 of f 6. WallShear Stress Distributions for Increasing Time; Frozen F low, tf = 2.0, f = 0.5 51 f ........... ... 7. -.Displacement Thickness Distributions for Increasing I Ime; FrozenFlow, tf = 2.0, f = 0.5 52 f ........ ... 8. Variation of WallShear Stress with Time; Frozen Flow ... 53 9. Variation of Displacement Thickness with Time; Frozen F 1 ow ..................... ... 54 10. Movement ofthe Reversal Point with Time;Frozen Flow ..- 55 11. Predictionof Singularity Onset, Using the Criterion ofReference [7]; FrozenFlow ............... 56 12. The Function f (t) for Oscillatory Flows of VariousPeriods; A = 0.5 .............. ... 57 13. Variationof Pressure Gradient at x = x with Time; 0 OscillatoryFlow, A = 0.5 .............. ... 58 14. WallShear Stress Distributions for Increasing Time; U t /c = 2.0, A = 0.5.Oscillatory Flow ....... 59 OP 15. DisplacementThickness Distributions for Increasing Time; U t /c = 2.0, A = 0.5. Osci 1 latoryFlow .... ... 60 OP 16. WallShear Stress Distributions for Increasing Time; OscillatoryFlow, U t /c = 4.0, A = 0.5 .......... 61 OP 17. DisplacementThickness Distributions for Increasing Time; OscillatoryFlow, U t /C = 4.0, A = 0.5 .... ... 62 OP vi i Figure Page 18. WallShear Stress Distributions for Increasing Time; OscillatoryFlow, U t /c = 8.0, A = 0.5 ....... 63 OP 19. DisplacementThickness Distributions for Increasing Time; OscillatoryFlow, U t /c = 8.0, A = 0.5 ....... 64 OP 20.Movement ofthe Reversal Point with Time; OscillatoryFlow, U t /c = 2.8 .............. 65 OP 21. Variation of WallShear Stress with Time ......... 66 22. Variationof Displacement Thickness with Time; OscillatoryFlow, U t /c = 2.8 .............. 67 OP 23. The Function f (t) for Osci llatory Flows of Various Ampl itudes; U t /c = 2.0 ................. 68 OP 24. Variation of Pressure Grad ientat x = x with Time; Oscillatory Flow, U t /c = 2.0 ..............0 69 OP 25. Wall Shear StressDistributions for Various Amplitudes;Oscillatory Flow, U t /c = 2.0, Uot/c = 4.0 . 70 OP 26. DisplacementThickness Distributions for Various Amplitudes;Osci 1 latoryFlow,