BOUNDARY LAYERS with FLOW REVERSAL John F. Nush

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 characteristics 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 reproduced 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, U t /c = 2.0,Uot/c = 4.0 . . 71 OP 27. WallShear Stress Distributions for Various Ampl itudes;Oscillatory Flow, U t /c = 2.0,Uot/c = 5.0 . . 72 OP 28. DisplacementThickness Distributions for Various Amplitudes;Oscillatory Flow, U t /c= 2.0,Uot/c = 5.0 . . 73 OP 29. Variation of WallShear Stress with Time; OscillatoryFlow, U t /c = 2.0, x = 0.6% ...... 74 OP 30. Variation of DisplacementThickness with Time; OscillatoryFlow, U t /c = 2.0, x = 0.6% ...... 75 OP 31. Movement ofthe Reversal Point with Time; OscillatoryFlow, U t /c = 2.0 ...... 76 OP 32. The Functionf(t) for Oscillatory Flows of Various Ampl i tudes; U t /c = 4.0 ...... 77 OP 33. Variation of PressureGradient at x = x with Time; 0 OscillatoryFlow, U t /c = 4.0 ...... 78 OP 34. WallShear Stress Distributions for Various Amplitudes;Oscillatory Flow, U t /c = 4.0, Uot/c = 4.0 . . 79 OP

viii Figure Page

35. DisplacementTh cknessDistributions for Various Amplitudes; Osc 1 latory Flow, U t /c = 4.0, U t/c= 4.0 . . 80 OP 0 36. Movement of the ReversalPoint with Time: Oscillatory Flow, U t /c = 4.0 ...... 81 OP 37. VelocityProfiles at U t/c = 1.0; OscillatoryFlow, 0 U t /c = 2.0, A = 0.5 ...... 82 OP 38. VelocityProfiles at U t/c = 1.5; OscillatoryFlow, 0 U t /c = 2.0, A = 0.5 ...... 83 OP 39 Velocity Profi les at U t/c = 2.0; OscillatoryFlow, - 0 U t /c = 4.0, A = 0.5 ...... 84 OP 40. VelocityProfiles at U t/c = 3.0; OscillatoryFlow, 0 U t /c = 4.0, A = 0.5 ...... 85 OP 41. Velocitiesat y = 0.16;Oscillatory Flow, A = 0.5 .... 86

42. Computed VelocityProfil s Over a FlatPlate; LaminarFlow, U c/v = 10E ...... 87 0 43. Vellcity Profiles at U t/c = 2.0; LaminarFrozen Flow, U t /c = 2.0, ffo= 0.5 88 of ...... 44, VelocityProfiles at U t/c = 4.0;Laminar Frozen Flow, U t /C 2.0, f O= 0.5 89 of - f ...... 45. VelocityProfiles at Uot/c = 8.0; LaminarFrozen Flow, U t /c = 2.0, ff = 0.5 90 of ...... 46. Wall Shear StressDistributions for Increasing Time;Laminar Frozen Flow, U /c = 2.0, ff = 0.5 91 oft .... 47. DisplacementThickness Distributions for Increasing Time;Laminar Frozen Flow, U t /c = 2.0, ff = 0.5 92 of .... 48. Variation of WallShear Stress with Time; Frozen Flow; U t /c = 2.0, ff = 0.5, x = 0.65~ 93 of ...... 49. Variation of DisplacementThickness with Time; FrozenFlow, U t /c = 2.0, ff = 0.5, x - 94 of 0.65~...... 50. VorticityContours at U t/c = 2.0; FrozenLaminar Flow,Uotf/c = 2.0 ...... 0 95

51. VorticityContours at Uot/c = 4.0;Frozen Laminar Flow, U t /c = 2.0 96 of ...... 52. VorticityContours at U t/c = 8.0;Frozen Laminar 0 Flow, U t /c = 2.0 97 of ......

~ INTRODUCTION

Extensivecalculations have beenperformed, over the last several years, to'study the characteristics of time-dependent turbulent boundary layers.

These calculations havetaken the form of numerical experiments, yielding datawhich -- althoughinferior in obvious respects to reliable wind-tunnel results -- havenevertheless served to fill thevoid which still exists because ofthe scarcity of comprehensivemeasurements. indeed, much of what is known atthis time about unsteady boundary layers has been derived fromtheory rather than from measurement,and, whilethe need for good wind-tunnelexperiments to confirm the calculated results (or at least to validatethe theoretical models) remains as acuteas ever it was, the informationcontent of calculations seems likelyto "stay ahead" ofthat of measurements for some timeto come.

Themost recent work on unsteady turbulent boundary layers, done by the presentauthor and hisco-workers, has beenconcerned withthe effects of time-dependenceon reversal and separation onset. It was shown in

References [1,2] thattime-dependence results in a. del ay ofreversal onset, and thatthis delay could not be relatedin anysimp1 emanner to an alleviationof the pressure gradients; indeed, some measu re of delay was observedeven when thepressure gradients were augmented by theeffects ofthe unsteadiness. it was shown further,that the reversal point in an unsteadyturbulent boundary layer is not a singularpoint -- anobserva- tionwhich confirmed earlier statements which had been made concerning laminar flow [3,4,5,6] -- and thatthe boundary layer remains thin even thoughreversal had takenplace [l]. Subsequentwork was directed to

studying a class of unsteadyturbulent boundary layers in which a region

of embedded reversal was allowedto develop but in which, under certain

conditions, a singularityoccurred sometime later [7,8]. Thissingu-

larityis related to thefinal separation of theboundary layer, in the

sense of detachment of theouter flow from the bodysurface, and repre-

sentsthe limit ofvalidity of first-orderboundary-layer theory. However,

theregime between reversal and separation appears to beaccessible to

first-ordertheory [i'], and thecalculations which were performed in this

regimeare believed to be meaningful.

Thepresent work, which forms a sequel tothat of Reference [71, hastwo mainobjectives:

1. toextend the calculations of unsteady turbulent boundary layers

to a widerrange of flow configurations, and

2. to tryto elucidate the mechanism of separationonset by perform-

inghigher-order calculations which would not break down asthe

resu 1 t of thedevelopment of a singularity.

In pursui t of thefirst objective, calculations have beenperformed for two add i t iona typesof flow: flows in which an unsteadyretardation is first imposed butwhere the external velocity distribution is subse- quentlyfrozen allowing a relaxationtowards steady-state conditions, and oscillatoryflows in which reversal occurs during part of the cycle. The results of thesecalculations are presented in Part I ofthis report.

Thesecond objective was pursuedby programming a simplemethod for solv-

ingthe unsteady Navier-Stokes equations, and performing some calculations

2 forlaminar flow. It was recognizedthat the characteristics of turbu-

lentflows would not be represented appropriately by this means, but it was hoped thatthe mechanism ofseparation would have sufficient generality

to make theselaminar calculations useful. The resultsof this study are

presentedin Part It ofthe report.

3 PART I.- TURBULENTFLOW CALCULATIONS

1.1 NATURE OF THE FLOWS CONSIDERED

The turbulentboundary-layer studies reported here were carried out for incompressible,time-dependent flow over a two-dimensionalsurface of largeor infinite radius. As inthe earlier work [1,7], orthogonal coordinatesare erected on the surface, with y measurednormal to it and x measuredalong the surface from some origin where theboundary layer is alreadyturbulent and of known properties:corresponding to steady, constant-pressureconditions. Themain features of the calculation method arereviewed in Appendix A. The calculationsrelate to the boundary layer developingover a "plate" of chord c whichextends downstream from the origin.

The externalvelocity: Ue, overthe plate, is assumed tovary in a prescribed manner with x andtime, t. Specifically, it is assumed that

ue - uo, for t -< 0 and all x (1)

'e X (1 f(t)l, for t -> 0 and 0 "< x < x -= - 0 uQ -x0

x -x 'e 1 "- 1" (1 - f(t)>,for t - 0 and x < x < c (3) X 0 - "0 0 where is some referencevelocity, and x and x areprescribed values uo 0 1 of x: x = 5c/7 ( = 0.714c), x1 = 2x . InEquations (2,3), f(t), which 0 0 isthe value of U /U at x = x is chosen to be a functionof time, with eo 0'

4 therestriction that f = 1 at t = 0 inorder to satisfy Equation (1).

The flowsdiscussed in Reference [71 corresponded to

f = 1 - wxot/c, (4) where w = constant,which imposed a distortionof the external flow which was monotonic in time.

Here,the emphasis is onpatterns of distortion of theexternal velocity field which are of two othertypes:

(1) patternsin which an initial distortion, extending over a finite

interval (0 "< t < tf), is fol lowed by freezing of the external

velocityfield,

(2) osci 1 latoryexternal velocity fields.

The "frozen"flows were generated by taking

f = 1 - (1 - ff)t/tf, for 0 -< t 5 tf (5)

f = ff, for t tf (6)

Values of ff = 0 and 0.5 wereused, together with values of t f correspondingto t /c = 0.2, 1.0, and 2.0. U of

The oscillatory flows were of a triangular waveform,and were generated by taking

t - to f = 0.5 - A 4{ - 1) P over the first half of the cycle: to -< t -< t + t /2, and OP t - to f = 0.5 + AC4 - 33 P overthe second half of the cycle:. t + t /2 < t to. Here, t isthe 0 p" P periodof the oscillation, and t refersto the beginning of the cycle. 0 It will benoted that the mean valueof f is 0.5, except"for the first half-cycle of the motion (0 -< t < t /2) wherean initial transient sat- -P isfiesthe requirement: f = 1 for t = 0; thistransient is defined by

t f=l - (1+2A)t. (9) P

The periodof the motion, which in dimensionless terms is U t /c, was OP varied between 2.0 and 8.0 inthe calculations. The constant A, in Equa- tions (7,8,9), controlsthe amplitude of the oscillatory external velocity distribution;values between 0 and 0.5 werechosen forthe calculations.

When A = 0, theflow degenerates from an oscillatory to a "frozen"flow

(as defined above) with ff = 0.5 and tf = tp/2.

When A = 0.5, inthe oscillatory calculations, the external flow stagnates momentarily(i.e. U = 0), atthe point on the surface: x = x once e 0' duringevery cycle: namely, at t = t + t /2. With ff = 0, inthe OP "frozenflows," stagnation of the outer flow occurs at x = x for a1 1 0 timessubsequent to t = t Stagnation does notoccur at all if A < 0.5, f' inthe oscillatory flows, and if ff > 0 inthe frozen f lows.

Figure 1 illustratesthe features of the two typesof f low,and provides a graphicaldefinition of the parameters f tf, t and A. f' P

". "If A = 0.5 the mean valueof f is 0.5 evenduring the first half-cycle.

6 1.2 COMPUTATIONALEXPERIMENTS

Frozen F 1 ows

A number ofcalculations were done forflows in which the external velocitydistribution was frozen,at t = tf, following an initial dis-

tortion starting at t = 0.

Figures 2 through 7 show thespacia 1 distributionsof wall shear stress anddisplacement thickness, respect ively,at various time levels, for

thefollowing cases:

U0tf/C ff

0.2 0 2.0 0 2.0 0.5.

In the first two cases, a regionof reversedflow, of substantial extent, has a1 ready formedon the plate by the time (t = t ) theexternal flow f isfrozen. Inthe third, reversal has not yettaken place at the time of f reez i ng butoccurs later. With ff = 0 , stagnation of theouter flow occursat x = x and t > tf, and thedisp lacementthickness takes on a 0' - locallyinf initevalue. However, thisinfinity reflects the division by zeroin the definitionof displacement thickness anddoes notnecessarily

indicatethe presence of a singularity in the solution of theboundary-

layerequations.

Variouscriteria were described in Reference [7] foridentifying the onset of a singularity:steepening of the gradients of wall shear stress and displacementthickness, versus x, the development of locally high displacement-thickness maxima,and breakdown of theboundary-layer approxima- tioned. A surrogatecriterion was alsosuggested: 6'' = O.lc,for flows of thepresent type and at these Reynolds numbers, whichappeared to

correlate with the other criteria, andwhich helped to pinpoint the loca-

tion of thelocally high gradients betweennode pointsin the calcula-

tion.This surrogate criterion will beused here also, for the moment,

although.itsusefulness is called into question later in the discussion.

In anycase, it is necessary torepeat the cautionary statements made in

Reference [71 tothe effect that the real measure ofsingularity onset is

steepeningof the velocity gradients rather than the attainment of some arbitraryvalue of 6", and thatthe surrogate only has validityinsofar as it helpsto locate the formation of the steep gradients.

Figures 2 through 5, whichrelate to the twocases where ff = 0, indicate

thatincipient singular conditions form at a timecorresponding to Uot/c= 2.0 for U t /c = 0.2, and at a timecorresponding to U t/c = 3.0 for U t /c = 2.0. of 0 of Thus thelower initial rate of distortion of the outer flow, in the latter case, resultsin a delayin the final onset of thesingularity. Figures 8,

9, inwhich the wall shear stress and the displacement thickness are plotted versustime, emphasize this delay. The smalldip in one ofthe curves of

6': versustime (Figure 9) resul ts from a forward movement of the 6': maximum onthe plate. It isinteresting to note that, for these two cases with f =0, f thereis a measure of similarity in the development of each solutiontowards

.L singularconditions: the two sets of results(for T or 6") wouldroughly W coincide if theywere plotted against t - where t refersto the onset of tS , S thesingularity. This similarity does notextend to the case where f,=O.5

(Figures 6,7), wherethe approach tothe singularity is somewhat slower

(Figures8,g). The slowerapproach to singular conditions presumably has to do withthe smaller gradient of Ue withrespect to x, when ff=0.5, but direct comparison of theresults for this case with those for ff=O is difficult

8 because thesingularity forms furtheraft when ff = 0.5: at around

x = 0.6c, ratherthan x = 0.25~ with f = 0. f

In Reference [71 a semi-empiricalmodel was proposedfor predicting the

onset of thesingularity. The procedure was to compare thevelocity of

penet rat ion of the reversal point, upstream into the oncoming boundary

layer, w ith anaverage "convection" velocity in the reversed-flow region.

' Forwant of a betterdefinition, this convection velocity was takento

beone-half of the maximum negativevelocity in the region at the particu-

lartime level. The predictioncriterion proved useful in correlating

the resul ts of Reference [71, and it was the reforeof interest to see

whether it wou Id be validhere. To this end , estimateswere made ofthe

penetrationve locity,for the three frozen f lowsconsidered here, by plot-

tingthe movement ofthe point of flowreversal (F igure lo), and differ-

entiatingthe curves graphically. The valuesobta inedin this manner

are shown inFigure 11, andcompared withdata for theconvection velocity,

as definedabove.

The pointsof intersection of the curves corresponding to the pairs of

velocitiesyields the predicted times at which the singularity is sup-

posed to form.These predictions are compared withthe approximate

observedvalues (indicated by 6'' = 0.1~)in the following table:

Case: PredictedObservedonsetonset

U0tf/C ff Uot/c U0t/C

0.2 2.0 0 1.2

2.0 0 2.2 3.0

2.0 0.5 3.9 4.8.

9 It will beseen thatthere is a constantdiscrepancy in Uot/c, of about

0.8, betweenthe predictions and theobservations, and the criterion has

no more than qua 1 i tat ive Val idi ty for these resul ts. The reason why it

does notapply here is not clear, although some suggestionscan be made.

It mightbe suggested that the assumed ratioof convection velocity to maximum negativevelocity needs to bereduced. However thediscrepancies

aretoo large to be resolved in such a manner: theratio would have to be

reduced virtually to zero to make thepredictions andobservations agree.

Indeed, forthe present frozen flows, the singularity does not seem to

formuntil the forward movement ofthe reversal point has ceased: i .e. untilthe penetration velocity has fallento zero. In the flows discussed

in Reference [71 , the singular i ty was observed to develop whi le the

reversalpoint was still movingforward. Another suggestion, antici- pating some ofthe later discussion in this report, is that the real singularity,in the sense ofsteepening of the gradients, doesform at thepredicted times, and thatthe discrepancies simply measure the failure of the surrogate criter ionto pinpoint itsformation. Since the surrogate criterion was used inthe derivat ionof the heuristic model, in

Reference [7], the imp1 ication mustbe thatthe rates of increase of 6"' withtime were so much greaterthan they are here that any differences

intiming between truesingularity onset and thecondition: 6" = O.lc,

-1. weremasked. It istrue that the rates of increase of 6" weregenerally higher,in Reference 171, thanthey are in the present flows, and so the suggestion has some merit. On theother hand thediscrepancies, between thepredicted andthe observed singularity onset are rather larger than can be explainedentirely in this way,and forthe moment theconclusion

10 must remainthat the heuristic model, proposed in Reference [71, is not sufficientlygeneral to make quantitativepredictions outside the class of flowsfor which it was originallydeveloped.

11 Osci 1 latory Flows

(a) Two setsof calculations were performed for oscillatory flows; the first set was concernedwith the effects of variation of frequency, and includedthe following cases:

u t /c A OP - 2.0 0.5

4.0 0.5 4.0

8.0 0.5.

The functionf(t), corresponding to each of these cases, is plotted in

Figure 12.The pressuregrad ientover the plate: ap/ax, isgiven by

.=-" at

and varieswith both x and t. The pressuregradient has a discontinuity - at x = x * however,the value at x = x is representativeof conditions 0' 0 over the forward portion of the plate: 0 -< x 5 xo, and canbe expressed as a functionof f and itsderivative f' (= df/dt)by:

c ap c C "=- f(1 - f) - f' . 2 ax 2 x0 0 "0

Now, sincefrom Equations (7.8), f' = -2/tover the first half of the P' - cycle, and - 2/tp,over the second half, ap/ax at x = x canbe determined 0 asa functionof time, and is shown inFigure 13 forthe three cases def i ned above.

12 The distributionsof wall shearstress and displacement thickness, for

thethree cases, are presented in Figures I4 through 19. Theemphasis,

here, is ondemonstrating the failure of the boundary layer to reach a

stableoscillatory condition. By,contrast, when reversal does notoccur

duringpart of the cycle, stable conditions are reached after a fewcycles

(seeReference [g]). For U t /c = 2.0 (Figures l4,15), thedistributions OP atstages through the first cycle are compared withthose at correspond-

ingstages during the second cycle. It will beseen thatthe wall shear

stressis typically lower, andthe displacement thickness typically

higher,during the second cyclethan the first. The differencesare most

conspicuousover the range of x correspondingto the reversed-flow region.

Thistrend continues through the third cycle, and isevidently associa-

tedwith the gradual approach towards eventual separation and theforma-

tionof a singularity.NevertheTess, during the latter part of the first

cycle andthe early part of the second,the reversed-flow region shrinks

andvanishes, and forward flow is temporarily reestablished over the

whole ofthe plate. This intermittent behavior which will bediscussed

more fullylater, is repeated for atleast twomore cycles

Forthe intermediate frequency: U t /C = 4.0 (Figures 16, 17), reversed OP flowpersists throughout the first cycle and a singularity formstowards

the end ofthe cycle. It will be recalledfrom Figure 13 that ap/ax < 0

duringthe second half of the cycle, and so we havethe interesting

situationhere in which reversed flow exists on the plate in spite of a

favorablepressure gradient. With U t /c = 8.0, a singularity forms OP duringthe first half-cycle andforward flow is never recovered once

reversalis established.

L The movement ofthe reversal point, with time, is plotted in Figure 20,

forthe three cases. In all three cases, reversal first occurs near

x = x and then moves rapidlyupstream with increasing time. At the 0 highestfrequency considered (U t /c = 2.0), thereversal point reaches OP a furthestupstream position at the mid point of the cycle and then moves

aftagain, with reversed-flowvanishing at a pointon the plate somewhat upstreamof x . At thetwo lower frequencies (U t /c = 4.0 and 8.0),the 0 OP reversalpoint never moves appreciablyaft from its furthest-upstream position. The approximatetimes at which 6'' first reachesthe value 0.lc

(otherthan when U + are shown onthe curves in Figure 20. e 0)

Inorder to observe an intermediate behavior, between that for U t /c=2.0 OP inwhich reversed flow temporarily vanishes during part of thecycle, and thatfor U t /c = 4.0 when it persistsindefinitely, anadditional calcu- OP lation wasmade for U t /c = 2.8and theresults are shown bythe dashed OP curvein Figure 20. Here, forward flow is reestablished for a short time,at the end of thefirst cycle, but reversed flow persists during the second cyclewith 6" reachingthe value 0.lc about half-way through the second cycle.

Figures 21,22 show thevariation, with time, of thewall shear stress and displacementthickness at two neighboring positions on the plate, forthis last case. The displacementthickness reaches 0.lc at about

U t/c = 4.0, althoughsteep gradients of wall shear stress occur some- 0 what earlier.It is ofinterest to notethe pronounced phase shift betweenthe displacement thickness and the external velocity gradients

(whichare symmetrical about the mid point of the cycle) even during

14 the first cycle when theflow is still far from developing a singularity.

The magnitude of this phaselag isconsiderably greater than the lags found inReference [9] forflows which did not suffer reversai during part of the cycle.

The 1 imited data presented here suggest that there is some connection between thepersistence of reversal during the cycle, even when the pressuregradient becomes negative, and theappearance of the singularity.This may simply mean that a singularity can onlydevelop when reversedflow is present; but, if so theseresults are at variance with suggestionswhich have been made that a singularity canoccur upstream ofreversal. Alternatively, the incipient formation of a singularity maysomehow inhibitthe shrinkage and disappearanceof the reversed-flow regionwhich would otherwise take place.

(b) The secondset ofcalculations for oscillatory flow was concerned

followingcases:

u t /c A op -

2.0 0 2.0 0.1 2.0 0.3 2.0 0.5

4.0 0 4.0 0.1 4.0 0.3 4.0 0.5.

The twocases with A = 0.5 arethe same astwo of the oscillatory flows mentionedabove, and the case: A = 0, U t /c = 4.0 is the same as the OP frozenflow, where U t /c = 2.0 and f = 0.5, which was discussed of f

15 earlier. The functionf(t) is plotted in Figure 23, for thecases where

U t /c = 2.0,and inFigure 32 for-the cases where U t /c = 4.0. The OP OP- correspondingplots of thepressure gradient, at x = X are shown in 0' Figures 24 and 33, respectively.

The distributions of wallshear stress and displacementthickness, over

theplate, for thecases where U t /c = 2.0,are shown inFigures 25, OP 26 for a timecorresponding to U t/c = 4.0: i.e.,at the end of two com- 0 pletecycles, and in Figures 27,28 for U t/c = 5.0, whichcorresponds 0 tothe mid point of the third cycle. In Figures 29, 30 thewal 1 shear

stress and displacementthickness at one point on the plate, are plotted

versustime, and Figure 31 shows the movement ofthe reversal point with

time.

It wi 11 benoted from Figure 29 that,for A = 0 and0.1, reversal does

notoccur until some timeduring the second cycle. However,once it

does takeplace, it persistsuntil a singularitydevelops in the third

cycle(Figure 28). In contrast, for A = 0.3and 0.5, reversal appears

duringthe first cyclebut subsequently vanishes again, later in the

cycle,leaving forward flow reestablished over the whole plate. This

intermittentformation andsubsequent disappearance of reversed flow is

repeatedfor at least twomore cycles(Figure 29). The displacement

thicknessreaches a valueof 0.lc at around the mid point of thethird

cycle, when A = 0.3,but there is no evidenceof singular behavior any- where inthe first three cycles when A = 0.5 (thecalculation was not

continuedbeyond Uot/c = 6: the end ofthe third cycle); the large values

of p at U t/c = 5 (Figure28) result from the stagnation of the outer 0

16 flow anddo notindicate the onset of a singularityin the solution. The

differencein behavior between the two1,ow-amplitude cases, inwhich

reversalpersists throughout the second and subsequentcycles, and the

twohigh-amp1 i tude cases, where reversa1 is intermittent, correlates

withthe sign of ap/ax duringthe second halfof the cycle (Figure 24):

positive for A = 0, 0.1and negative for A = 0.3, 0.5. On theother hand

it has already beenobserved that reversalcan persist even when thepres-

suregradient is favorable, and, clearly,the mechanism whichcontrols the

appearanceand disappearance ofthe reversed-flow region is not simply a

function of thepressure gradient. Indeed, as alreadymentioned, it was

shown inReference [l] thattime-dependence caused a delayin the first

appearance of reversal even when aU/at made apos it ive contribution to

the pressure gradient.

Figures 34 through 36 presentdata for the lower frequency calculations

(U t /c = 4.0).Here, it will beseen thatforward flow is notrecovered OP for any ofthe amplitudes considered; once formed, reversal persists

until 6" reaches 0.lcnear the end of the first cycle.Specifically,

singularityonset (according to this criterion) occurs earliest for the

high-amplitudecase: at U t/c = 3.7, and latestfor A = 0: atabout 0 Uot/c = 4.7. Thisorder of onset, in terms of the values of A, is oppo-

siteto the order observed at the higher frequency. The differenceis

probablyassociated with the failure of the reversal point to move aft,

duringthe second half of the cycle, when U t /c = 4.0 (Figure 36). At OP thelower frequency, the reversal point moves forward,during the first

half-cycle, and subsequentlyremains near itsfurthest-forward position,

leaving an extensiveregion of reversedflow, for the duration of the

motion. It isinteresting to note the wide variation in the terminal

positionsof the reversal point, over the range of amplitudes considered:

L fromabout x = 0.6c, when A = 0, to about 0.34~ when A = 0.5 (Figure

36).

It isclear that, at this relatively lowfrequency, the behavior of the boundarylayer is strongly influenced by theadverse pressure gradients presentduring the first half cycle. The alleviationof these gradients, duringthe second halfcycle, hasless effect than it does athigher frequencies,although the effects are still noticeable. This canbe seen by comparingthe movement ofthe reversal point for the case:

U t /c = 4.0, A = 0.5, inFigure 36, withthe dashedcurve which corre- OP sponds tothe frozen flow: Uotf/c = 2.0, ff = 0; the twoflows suffer the same retardation during the interval 0 -< U t/c < 2.0, but different 0 - retardationsthereafter. 1.3 THE NATURE OF UNSTEADY SEPARATION

Some importantconclusions concerning the nature of unsteadyseparation

of theturbulent boundary layer are suggested by the results of the compu-

tat ional experiments presented above.

First:the relaxation of a turbulentboundary layer, which hasbeen

subjectedto an initial retardation strong enough to causereversal,

towards a steadystate separation, is achieved in a finitetime. It was

suggested, in Reference [l],that the relaxation time is of the same order

asthe time taken for the fluid to beconvected offthe part of the sur-

face ahead of theultimate point of separation. The presentresults

supportthis hypothesis: the calculations for the frozen flow inwhich

U t /c = 0.2, ff = 0, indicateincipient separation at x = 0.27~ in a of time U t/c = 2.0, implyingthat all of the fluid with anaverage velocity 0 exceed i ng 0.1 5U0 had been replaced by new f 1 u i d entering the boundary

layerafter time tf. Inthe case, just mentioned, the initial retarda-

tion was abrupt;in cases where thereta.idation is more gradualthe time

necessary toreach incipient separation is longer. However, thefinal

divergencetowards singular conditions, as separation is approaches, is

rapid,with the displacement thickness increasing roughly exponentially

withtime.

Second: theapproach to the separation singularity, in oscillatory flows,

appears to be a functionof both the long-time-average, and theinstan-

taneous,pressure gradients. At lowfrequencies, the effects of the

adverseinstantaneous pressure gradients seem to dominate,and separation

L canoccur during the first half-cycle if thegradients are sufficiently

severe. At higherfrequencies, the approach to separationappears to

dependmore onthe time-average pressure gradients, while the nature of

theinstantaneous gradients may besuch as to have a delayingeffect on

separationonset. In one of the cases examined (U t /c = 2.0, A = 0.5), OP threecomplete cycles were completed without evidence of imminent separa-

tion eventhough the average gradients were large enough to provokesepa-

rationafter two cycles(as can be seen from the results for U t /c=2.0, OP A = 0).

Thedelay of separation onset, observed in the oscillatory cases, where

theamplitude in large, is evidently associated with the alleviation of

thepressure gradients during the parts of the cycle when theexternal velocitiesare increasing with time: instantaneously favorable gradients can exist,at sufficiently high amplitudes andfrequencies, which offset thetime-average adverse gradients. On theother hand, theexistence of favorableinstantaneous pressure gradients is not enough toguarantee the avoidanceof separation -- or even toguarantee the reestablishment of forwardflow in place of a reversed-flowregion.

In some casesthe reversed-flow region did vanish during the phases of themotion when thepressure gradients were favorable. In others, not onlydid the reversed-flow region persist during such phases, but the extent of theregion actually increased. This type of behavior seemed tooccur when theflow in the boundary layer was alreadyretarded to a degreewhere separation could be expected to occurshortly: the change to favorablepressure gradients was then"too late" to save the boundary

20 layerfrom separating. Figures 37 through 40 contrast,in more detail,

thecases where recovery was achieved,with those where it was not achieved.Figures 37,38 show a typicalrecovery from reversed flow to

forwardflow, as thepressure gradient changes signfrom positive to

negative.These results, which are for the oscillatory flow: U t /c = OP 2.0, A = 0.5, show thevelocity profiles before and afterthe recovery.

The boundary-layerthickness increases as a resultof the recovery, althoughthe displacement thickness decreases (Figure 15). Figures 39,

40 show a casewhich fails to recover (U t /c = 4.0, A = 0.5) despite a OP similar change inthe pressure gradient; instead, the reversal point moves furtherupstream, accompanied byan abruptincrease in the boundary-

layerthickness over the reversed-flow region. This latter effect is

reminiscentof the flow patterns, reported in Reference [71, depicting

theapproach to singular conditions, although the onset of a singularity

(according to the surrogate criterion: 6" - 0.1c) has not yet been

reached inthe solution plotted in Figure 40. A more detailed examina-

tionof the results indicates, however, thatthe solution in Figure 40

alreadycontains a singularity.Figure 41 shows a plotof the velocity at y = 0.1,versus x, for the two sets of results, before and afterthe

change inpressure gradient. In the case where recovery is achieved,the

progressionfrom reversed to forward flow is accompanied bya smooth,

continuousvariation of velocity with x. In contrast, in the case where

recoverydid not take place, a discontinuityin velocity occurs, and the

partof the solution where U > 0 losescontact with the part where U < 0.

The largevalues of alJ/ax, acrossthe discontinuity, give rise to large

valuesof aV/ay,and correspondingly large values of V, whichare

21 reflectedin the growth of the boundary-layer thickness seen in Figure

40, butthey have not yet been translatedinto values of 6" exceeding

0.1~. Nordoes thesolution imply an excessively large normal pressure

gradient,at this stage. Two conclusionstherefore seem to emerge:

(1)that the failure of a boundarylayer to recover forward flow upon a

change inthe pressure gradient is indicative of an incipientsingularity,

and (2) thatthe onset of a singularity may notalways result in an immediate

grossthickening of the boundary layer. This second conclusion needs to

be examinedmore carefully in future workbecause singularityonset has

normally been identifiedwith imminentseparation. The presentcomputa-

tional frameworkwould not be particularlyappropriate for such an examina-

tion because ofthe coarseness of the mesh in the x- y plane, andbecause

there may be some suppressionof large temporal gradients (even though

shorttime steps were used). Some analyticalstudies would appear to offer moreprospect of success.

Meanwhile,the possibi 1 ity exists that a sequence ofevents takes place,

priorto the final separation of the flow, in which the extent of flow

reversalincreases, and thickening of the boundary layer occurs, regard-

less of any alleviationof the separation-provoking pressure gradients.

In none ofthe casesdiscussed here was a retreatfrom imminent separation

observed,although it may bedemonstrated by further work that the imposi-

tionof sufficiently strong favorable gradients would bring this about.

It isinteresting to draw a parallel,here, with the hysteresis effect observedin dynamic stall: the recovery of the flow on the airfoil is

notachieved until the incidence is reduced substantially below the level

reachedbefore stall occurred [lo]. The flowaround a pitchingairfoil

22 isclearly morecomplicated than the flows discussed in this report; in particular,the vortex shedfrom the leading edge of the airfoil plays an importantrole in the hysteresis process. Nevertheless, a generalrule may beemerging that, once the separation mechanism isinitiated, it is necessaryto take drastic steps, in terms of achange ofexternal-velocity conditions, to restorethe boundary layer to an attachedcondition. Some furthercalculations, on the present lines could usefully be done topur- sue this particular objective. PART II. LAMINAR FLOW CALCULATIONS

11.1 NATUREOFTHE FLOWS CONSIDERED

The laminar-flowstudies were carried out for incompressible, two-dimensional,time-dependent flow over a flatsurface. The viscousportion of

theflow was takento be boundary-layer-like,insofar as the longitudinal extentof the flow was assumed to belarge compared withits transverse extent, and the potential-flow velocity was prescribedalong the upper edge ofthe integration domain,which ran parallel to the wall. On theother hand, noassumption wasmade aboutthe smallness of second derivatives in thelongitudinal direction (except at the downstreamend ofthe domain), and no restriction was placedupon the variation of staticpressure normal to thewall: the solution in the interior of the domaincorresponds tothe full unsteadyNavier-Stokes equations. Detai 1s ofthe equations, and the numerical scheme used tosolve them, aregiven in Appendix B.

It was assumed thatthe upstream edge ofthe plate, uponwhich the flow was developing, was a sharpleading edge. At the downstream end ofthe plate, it was assumed that(regardless of its earlier condition, further upstream)the flow had recoveredto a formwhich was consistentwith the first-orderboundary-layer approximations; this assumption wasmade inorder toavoid the need forspecifying the downstream velocity and vorticity profiles. The regionof interest was theportion of the flow over the central partof the plate, where reversal was inducedby suitable choice of the externalvelocity distributions. These distributionswere similar to those def i ned by Equations(1,2,3) , above,and the same Values of x /c and x kc 0 1

24 4 wereused. The Reynolds number: U c/v, was takento be IO . 0

Resultsare presented here for a singlecase, viz. a "frozenflow," of thetype considered in Part I, above with U t /c = 2.0, and f = 0.5. of f These resultsare of interestboth in themselves and insofaras they may becompared.and contrasted with the results of thecorresponding turbulent calculation which hasalready been described.

The initialconditions, at t = 0, correspond to steady,constant-pressure flow, andwere generated by means of a time-relaxationcalculation; an approximatesolution was assumed, andthen the calculation was advanced in timeuntil the steady-state solution was obtained; At thehigh Reynolds number, forwhich the calculation was performed, it would be expectedthat theexact sdlution of theNavier-Stokes equations would be closeto the

Blasiussolution of thelaminar boundary-layer equations, the only signifi- cantdifference in boundary conditions lying in the confined domain in whichthe present flow was assumed to be developing;the height of the domain was takento be 0.lc at t = 0. The velocityprofiles producedby thetime relaxationprocess are compared withthe Blasius profiles in Figure 42.

Calculationswere done for three different mesh densities:

(a) Ax = O.lc, Ay = 0.01~ (11 x 11 node points)

(b) Ax = 0.05~~Ay = 0.005~ (21 21x node points)

It will beseen from Figure 42 thatthe agreement with the Blasius solution is good forthe finest mesh, tolerable for themedium-density mesh, andpoor forthe coarse mesh, particularlyover the upstream half of the plate.There are two main reasons for the poor agreement, in this last

25 case: first,the number of node pointsthrough the boundary layer is toosmall to give proper resolution of the higher derivatives in the y-direction, andsecond, thewide mesh spacingin the x-direction leads toinadequate resolution of the longitudinal derivatives associated with theflow near the leading-edge.

Some comments need to be made hereconcerning the artificial viscosity, introduced by the finite mesh size,and its effect on the quality of the solution. The appropriate measuresof artificialviscosity are the two cell Reynoldsnumbers: UAx/v andVAy/v, which multiply secondderiva- tivesin the x- and y-directions,respectively. If thesecell Reynolds numbers areeverywhere small compared to 2,the solution can beregarded as valid;otherwise the effective Reynolds number ofthe solution is reduced, atleast in the regions of the flow wherethose conditions are notsatisfied. Now, inthe present flow, at t = 0, thetransverse cell

Reynoldsnumber: VAy/v isat mostabout unity, even for the coarse mesh, and so i tseffects on the derivatives: a2/ay2, cannot beconsidered serious. The longitudinalcell Reynolds number:UAx/v, is much higher than 2, even forthe finest mesh,and canscarcely bereduced to 2 without placing an enormousnumber of node pointsin the x-direction. However, it affectsonly the second derivativesin x which,fortuitously are small forthe constant-pressure flow; these derivatives are neglected altogether inthe Bl.asius solution. Consequently, the solution for t = 0 appears to be relativelyfree of the effects of artificial-viscosity. The same cannot be saidof the subsequent solutions for t > 0, however.There,

VAy/v wouldbegin to exceed 2, even for the finest mesh, and, also, second derivativesin x startto become significant. Thus artificial- viscosity becomes a more seriousproblem as timeincreases, and thisfact needs to beborne in mindduring the discussion of theresults throughout thissection.

On thebasis of the comparisons shown inFigure 42,and for reasons of computational economy, it was decidedto use the medium-density mesh for thetime-dependent calculations for t > 0. Inorder to allow for the growthof the viscous region, the height of the domain was allowedto increasewith time: from O.lc,at t = 0, to 0.2~~ tat = 5c/U0, after whichthe height was heldconstant. This scheme provedto be satisfac- tory,in the light of the solutions obtained, but it must beremembered thatthe solutions are not independent of the extent of the domainbecause theNavier-Stokes equations permit pressure gradients to develop normal tothe wall. The specifiedrate of increase of height must, therefore, be included among thesignificant boundaryconditions imposedon the flow.

The prescribedvariation of U, alongthe upper edge of the domain -- accordingto the definition of the frozen flow considered -- represents a secondboundary condition, and the variation of V represents a third.

It emerged thatthe proper variation of V, alongthe upper edge, had to be chosen with some care. If V istaken to be zero,there, continuity demands thatthe retardation of U, atthe boundary, has to be offset by an accelerationwithin the domain.Consequently, the flow near the wall feelsless retardation than the flow at the upper boundary, and thealle- viationof the longitudinal pressure gradients, due tothe effects of thenormal ones, increases with the height of the domain. A retardation imposed at a largedistance from the wall has 1 ittle or no effect onthe

27 viscousflow, and may notbe large enough toproduce reversal. In an attemptto reduce the extent of the accelerations within the domain, V was given a positivevalue, on the upper boundary, for 0 < x < x and - 0’ a negative va 1 ue for x < x < c.Specifically, V was putequal to 0 saU/ax,where s isthe height of the domain,which extracted enough mass to offset the decrease in U atthe boundary. It was notlarge enough to preventaccelerations entirely, within the domain, butexperimentation showed that.suchlarge values of V wereneeded toachieve this that the cell Reynolds-numberproblem was seriouslyaggravated. It sufficesto say thatthe performance of the present laminar calculations was notas straightforward as it might seem, atfirst sight. Three seemingly sepa- rateproblems: the choice of the height of thedomain, the choice of theV-distribution along the upper boundary, and the artificial-viscosity problem,are actually closely related (computer time is a fourthrelated problem),and a suitable compromisehad to befound that attempted to satisfyall of them.

28 11.2 COMPUTATIONALEXPERIMENTS

The calculations for the frozen flow (in which U t /c = 2.0) werecon- of tinuedup to a timelevel: Uot/c = 8.0. Typicalvelocity profiles

throughthe viscous-flow region are shown inFigures 43 through 45, and

thelongitudinal distributions of wallshear stress anddisplacement

thicknessare plotted in Figures 46,47. Several of thevelocity profiles

exhibit anovershoot which reflects the acceleration within the domain,

mentionedabove, which tends toalleviate the applied retardation. Never-

theless,in spite of this degree of alleviation, reversal occurs at the

wall,and the region of reversalextends over most of theplate at the

highesttime level reached.

Figures 46,47 may becompared withFigures 6,7 whichrelate to the corre-

spondingturbulent boundary layer. A number ofsignificant differences

may benoted between the two sets of results.First, reversal takes

placebefore t = tf, inthe laminar flow, whereas it didnot occur until

some timelater in the turbulent flow. Second, theextent of reversed

flow issubstantially larger in the laminar case.Third, the variation

ofwall shear stress is characteristically different in the twocases:

it isnot only more gradual,in the laminar flow, but it exhibits a double

minimum atthe higher time levels: a double minimum didnot occur in the

presentturbulent flows, andindeed has never been observed in anyof

theturbulent calculations performed by us so far.Finally, andperhaps mostimportant. The abrupt changes ofdisplacement thickness, associated withsingularity onset in the turbulent boundary layer, are completely absentin the laminar flow. Figure 47 indicatesthat the displacement thicknessis tending towards a stable,steady-state solution; it mustbe

remembered thatthe Navier-Stokes equations admit solutions corresponding

tosteady-state separation, whereas the first-order boundary-layer equa-

tions do not.The trend towards an asymptotic solution is emphasized in

Figures 48,49 which show thevariation of the wall shear stress and dis-

placementthickness with time, for one point onthe plate. It is inter-

estingto note, from Figure 48, thatthe wall shear stress in the laminar

flowis of substantiallylarger magnitude than.that in the turbulent case.

Thisdifference is partly due tothe difference in the Reynoldsnumber: 4 laminar,flat-plate skin friction at R = 10 isroughly twice the turbu- e lent,flat-plate skin friction at R = 10 7 . The differenceis accentuated, e here,because ofthe larger extent of the reversed-flow region, in the

laminarcase, and because ofthe nature of the velocity gradients within 4 theregion. The displacementthickness in a laminarflow at R = 10 is e aboutthe same as thatin a turbulentboundary layer at R = lo7, and e Figure 49 shows that 6" is of comparablemagnitude, inthe twocases

consideredhere, over the initial time steps. The majordifference in

behaviorfor larger times, lies in the abrupt increase of 6'' withtime,

inthe turbulent flow, compared withthe progressive increase towards an asymptotein the laminar flow.

It mighthave been expected that the laminar calculations would exhibit

some developmentwhich, although not of the proportions of a singularity, would in someway resemblethe steepening gradients characteristic of the singularityobserved in the turbulent boundary-layer calculations. How- ever, the veloci ty prof i l es in Figures 43 through 45 show no suchdevel - opment: theabrupt changes inthe x-direction -- typified by Figure 40,

30 for example -- seem to be completelyabsent in the present laminar

results.Inclusion of the second derivativesin x appear to havehad the

effectof inhibiting the formation of anyregion of unusually large gra-

dients. The onlyfear is that artificial-viscosity effects may havehad

toolarge an inhibitingeffect on those derivatives, and that a more accuratesolution of the Navier-Stokes equations, for a Reynolds number of order lo4, mightindeed exhibit features somewhat intermediatebetween

thoseof the laminar and turbulentresults obtained here. Further

studies,perhaps in which a large number of node pointscould be concen-

trated with I na limitedregion of the flow field, might serveto resolve

thesefears

Be that as I t may, anotherarea of interest, in the lam inarcalculations,

involvesthe distribution of vorticity throughout the flow, and its variationwith time. Figures 50 through 52 show contoursof constant vorticityat three time levels. In the initial phases of themotion,

thevorticity contours lie essentially parallel to thewall. Subse- quently,with increase of time,the contours become distorted, and a

tongueof high-vorticity fluid moves outinto the body ofthe flow,

leaving a regionof low or negative vorticity between it and thewall.

Figure 52 may demonstrate,on a smal I scale, how vorticity is shed from a boundarylayer, near separation, and movesaway fromthe bodysurface.

Thereis, of course, no separation"point" as.such; separation is a progressive phenomenon, associatedwith steadily thickening regions of viscousflow, and corresponding modification of the outer inviscid flow. CONCLUD ING REMARKS

The calculationsreported herein: on the one hand, usingthe turbulent

boundary-layerequations, and, on the other, using the laminar Navier-

Stokesequations, were intended to shed some lighton the phenomenon of

unsteadyboundary-layer separation.

Earlier work hadshown thatflow reversal, in time-dependent boundary

layers,is not a singularevent, and is notassociated with the break-

down ofthe boundary-layer approximations. However, it was also demon-

stratedthat a singularity coulddevelop some timelater: associated with

steepeningstreamwise gradi ents,gross thickening of the boundary layer,

and the breakdown of those approximations,and that the onset of the

singularitycould be identified with the physical separation of theflow

The presentcalculations of unsteadyturbulent boundary layers, which

includedflows relaxing in time and oscillatoryflows, confirm the exis-

tenceof a flowregime in which reversal, but not singular conditions,

prevails. However,they also suggest that singularity onset may not alwayscorrelate with the immediate gross thickening of theboundary layer characteristicof separation. In some ofthe cases examined, it appeared thatsingularity onset was followed by a phase of onlyprogressive boundary-layerthickening; indeed the only unusual gross feature of the solution wasa lackof responsiveness to changes of thepressure gradient.In this situation, alleviation of the adverse pressure gradients didnot bring about a contractionof the reversed-flow region and the

reestablishment of forward flow overthe surface; in some casesthe extent of thereversed-flow region actually increased. The phenomenon described was most noticeablein oscillatory flows, when reversal was fbund topersist throughout the cycle, despite the presence of instan- taneouslyfavorable pressure gradients.

It is,of course, of dubious validity to draw conclusions from the behavior of a solution after a singularity-hasdeveloped, and there is no directevidence that the relatively slow increase of boundary-layer thickness,accompanied by the insensitivity to changes of thepressure gradient,has any analog in a real flow. On theother hand, thereis aninteresting parallel that can be drawn withthe histeresis effect observedon airfoilsundergoing dynamic stall.

The laminar-flowcalculations, basedon the Navier-Stokes equations, were performed so as tostudy separation solutions which would not be contami- natedby singularity onset and the breakdown ofthe boundary-layer equations. The solutionsobtained certainly had thisproperty, but, sur- prisingly,they did not exhibit any of the features associated with the earlystages of singularity onset: steepening gradients or rapidthicken- ingof the viscous region. The calculationswere performed for a flow relaxingtowards a steady-stateseparation, and thedevelopment of the solution was almostdisappointingly uneventful. The reversed-flowregion was found to increasesteadily in extent, the displacement thickness

increasedprogressively, approaching an asymptote, and vorticalfluid moved away fromthe wall into the interior of the flow.

The major enigma posed by these resu 1 ts concerns the phys i ca 1 sign i f i cance of theevents leading up tosingularity onset in the unsteady boundary-

33 layerequations. It was to beexpected that the singularity itself was simply a featureof the boundary-layer approximations. It was somewhat

lessexpected that the formation of a regionof locally steep gradients, and rapidthickening of the layer, might also prove to beno more than a figmentof the boundary-layer imagination.

The matteris not yet resolved, however, and furtherwork needs to address anumber ofspecific questions. First, what is thesignificance of a singularity whose onlyobservable effect is to dissociate the subse- quentboundary-layer development from the imposed pressuregradients?

Second, to whatextent was theuneventful development of the laminar solutions,obtained here, a resultof excessive artificial viscosity, and would it genuinely be thatuneventful in a high Reynolds-numberviscous flow?Perhaps it wouldbe less uneventful in a flowof larger extent thanthe one considered here. Third, at what point, in the introduction ofhigher-order terms into the boundary-layer equations, doesthe transi- tionfrom a singularity-dominatedseparation to a nonsingular one occur, and isthat transition the same for a laminarflow as it isfor a turbulentflow?

34 APPENDIX A

CALCULATION METHOD FOR T IME-DEPENDENT

TURBULENT BOUNDARY LAYERS

1. GoverningEquations

The velocity components inthe x -, y -, z - directionsare expressed inthe form U + u, V + v, W + w, respectively,where U,V,W are defined as ensembleaverage velocities, with W E 0 for a two-dimensional flow, and u,v,w arethe residual fluctuating componentsabout those ensembleaverages. The governing equations are similar to those used

inearlier studies [1,7,9,11], namely,the momentum equation:

the con t i nu i ty equa t ion :

and theempirically-modified turbulent kinetic-energy equation:

InEquations (Al,A3) theconvective derivative is defined by

InEquation (A3) theempirical convective constant, al, and theempirical functionsof position through the boundary layer, a and aretaken to 2 L,

35 bethe same asthose used in Reference [l]; accordingly,the same caution; arystatements apply to their continued use. Experimental substantiat,ion ofthe appropriateness of Equation (A3) and ofthe empirical parameters

in it - or guidanceas to how theyshould be modified or improved - are stillurgently neededand awaited.The quantity CP inEquation (A3) is thetwo-dimensional equivalent of the functions Q Qz in Reference[12], X' and isdefined by

o = r{ 1' where r is some 1 arge number. The inclusionof this term has no effect onthe resultant shear stress, butserves to maintain directionality of theshear-stress vector according to

- whereq2 isthe resultant mean-square velocityfluctuation. Equation

(A6) expresses an assumptionwhich is implied in all the two-dimensional applicationsof Townsend's structuralsimilarity hypothesis (see Ref.

[12]),on which the present model is based.

2. Solutionof the Equations

The solutionof the governing equations follows the approach of

References [1,7,9]. The flowis divided into an inner andan outerlayer, withthe matching station between them lyingat about y = 0.05 . In theouter layer the equations are integrated byan explicit,staggered- mesh finite-difference scheme, advancing in thepositive x-direction.

The only aspect of the finite-difference scheme, worthmentioning here concernsthe method of obtainingx-derivatives. In order to avoid

36 violation of therelevant zonesof dependence, derivativeswith respect to x areformed using two-point backward differences in region of positive U, andtwo-point forward differences in region of reversed flow

(U < 0). At stationswhere the local flow direction is ambiguous:i.e. wherethe sign of U at some node point(x,y,t) is different from that at theadjacent points: (x - Ax,y,t) and (x + Ax,y,t),the x-derivatives areset equal to zero.Thi s refinementleads to improved smoothness of thesolution at points of i ncipientflow reversal. Little loss of accu- racy resul ts because U, whi chis multiplied bythe x-derivative in the momentum equation, i s i nevi tablyclose to zero at such points. It should be stressed that aU/ax is -not set equal to zero throughout the reversed- flowregion, as hasbeen done incertain other analyses of flowswith reversal;such a procedureclearly leads to an invalid solution because typicalvalues of UaU/ax are byno means numerically small compared to theother terms in the momentum equation.

Furtherdetails of thenumerical scheme, inthe outer layer, can be found in References[1,9,12,13,14], and the reader is referredthereto.

Inthe inner layer, near the surface of theplate, the numerical solut ionis again matched to an approximate solution basedon the Law of theWall. The detailsof this inner solution havebeen modified, as compared withthe earlier work, in order to handle the transition from positive to negativevalues of thewall shear stress. In the present wo rk,the turbulent kinetic-energy equation, for the inner layer, is wr i tten

37 wherethe dissipation length, L, hasbeen equated, inthe usual way, to

KY. The function A, where

representsthe residual imbalance between production and dissipation of

turbulentkinetic energy which, near the wall, corresponds chiefly to con- vectivetransport. In the inner layer A << I aU/ayl , and is,replaced by

itsvalue, A atthe matching point with the outer-solution domain. With m now independent of y, we have,upon forma 1 integration Equation (A7) : A m of

The integralis evaluated, following Townsend [Is], by prescribing a

1 inear stress relationship

where a is independent of y, and typicallyof the same order asap/ax.

It isnot difficult to perform the integration analytically; however theresulting forms: one, if isof the same signthroughout the inner layer, andtwo, if it changes sign, do notlend themselves readily to programming forthe computer. To avoidthese problems, Equation (A9) is integrated by a simpleiterative numerical scheme. With U andknown atthe matching point, an approximate value of T~ is estimatedfrom - which ct is determined;uv and U canthen be found. as functionsof y, for 0 < y < y,. The valueof T is adjusted, by means of a simplepre- W dictor-corrector method, untilthe associated values of U mergesmoothly into those for the outer domain.

I 3. Integration Domain

The integration domainextended from x = 0 to x = c, andfrom y = 0 to y = s(x)where s = 1.25b,.approximately.The collocation pointsin the y-direction, 20 in number,were distributed,as in Refer- ence [131, togive increased density near the wall. The collocation pointsin the x-direction were also distributed nonuniformly, so asto giveincreased density in the center portion of the plate. Specifically, the 24 pointswere distributed according to

(A1 1) with n = 24 and 0 -< j 5 n.

4. Boundary Cond i t ions

For times t -< 0, the flow corresponds to steady, constant-pressure flow inthe x-direction. The boundarylayer isin constant-pressure equilibrium,with a thickness, 6, at x = 0, of 0.00444c. The Reynolds number: Uoc/v, is taken to be 107 .

At x = 0, thevelocity and shear-stressprofiles are maintained, forall time, in the same formas at t = 0; i.e.,in steady constant- pressureequilibrium with an externalvelocity of U and theReynolds- 0 number conditionsspecified above.

39 APPENDIX B

SOLUTIONMETHOD FOR THETIME-DEPENDENT NAVIER-STOKES EQUATIONS

1 . Govern i ng Equat ions

TheNavier Stokes equations, for incompressible, time-dependent flow,can be written in vorticity-transport form as

inwhich the single non-zero component of vorticity is e, where

av au e=”-ax ay ’ and

v 2 =-+-a2 a2 2 2. ax ay

From Equation (B2), togetherwith the continuity equation:

a Poissonequation can be derived for the velocity components, in terms of derivatives of the vorticity:

vv=-2 ar; ax

Equations (B1 ,B5,B6) form a parabolicset, and representthe governing equations,in three unknowns: c,U,V, whichare integrated in a three-

40 dimensionaldomain consisting of two spacedimensions and time. The method of solutionis closely related to that of References [16,17] for solvingthe parabolicized Navier-Stokes equations in three-dimensional steady f 1ow.

2. Solutionof the Equations

Thegoverning equations are integrated by means of an implicit,alterna- ting-direction (ADI) scheme, advancing inthe positive time direction.

A rectangular mesh:x = mAx, y = nAy, iserected on the plate, permitting discretizationof the derivatives at eachtime level, R:

-=a2F 2 ax

a2F "- 3Y2

-=aF at where F is anyvariable. Any one of thegoverning equations can then be

41 written in difference form as

where A throu'gh E arecoefficients which, in general, dependon thesolu-

tion,but which are regarded as known at each iterationlevel reflecting

thecustomary linearization procedure. R, inEquation (B14) involves

thesolution at time level a-1, whichhas already been calculated. The

field, m,n, is scanned alternatelyin the m-, and n-directions,convert-

ingEquation (B14) intothe successive forms:

(B16) where R1 and R have absorbedthe passive terms on the left-hand side 2 ofEquation (B14). Equations (B15,B16) aresolved by theextended

Choleskimethod [18], forwhich efficient solution algorithms are available.

The procedureduring any iteration cycle is to solvethe vorticity equa- tionto provide updated values of 5, andthen, in a secondstep, to solve the twoPoisson equations to provideupdated values of the velocities

U,V. ThePoisson equations were solved simultaneously, by regarding F and R, inequation (B14) astwo-dimensional vectors, and A through E as square matrices.

3. BoundaryConditions

It is assumed thatthe upper surface of the integration domain(y = s) isoutside the viscous region, so that 5 = 0. The velocity components

42 are assumed to be prescribed. At the upstream boundary (x = 0), the vorticity is assumed to be zero exceptat the wall where a delta function is imposed, corresponding to the no-.,lip condition.At the downstream boundary (x = c), the vorticity is assumed to be zero outside the boundary layer, while conditions inside the boundary layer are assumed to conform to the first-order boundary-layer approximations: 2 a2/ax = 0. At the wall: y = 0, the boundary conditions are U = V = 0; the vorticity at the wall is determined as part of the solution,and is proportional to the wall shear stress.

"" __ "" - "".."__"""-__I" REFERENCES

1. J.'F. Nash, L. W. Carr and R. E. Singleton, "Unsteady Turbulent Boundary Layers in Two-DimensionalIncompressible Flow," AlAA J. -13, No. 2, February 1975. 2. M.R. Scruggs, J. F. Nash and R. E. Singleton,"Analysis of Flow- ReversalDelay for a Pitching Foil ,I' A. I.A.A. 12thAerospace SciencesMeeting, Paper No. 74-183, Feb. 1974.

3. Sears, W. R. and Telionis, D. P., "UnsteadyBoundary-Layer Separa- tion,"Recent Research Boundary Layers(Proc. I .U.T.A.M. Symposium, Quebec 19711, E. A. Eichelbrenner, ed.,Presses de L'Universit6 Laval, Quebec 1972.

4. D. P. Telionis and M. J. Werle,"Boundary-Layer Separation from Downstream MovingBoundaries," J. Appl. Mech., p. 369, June 1973.

5. D. P. Telionis and 0. Th. Tsahalis, "The Response of Unsteady Bound- ary-LayerSeparation to Impulsive Changes of Outer Flow,'' AIAA 6th Fluid and Plasma Dynamic Conf.,Paper No. 73-684, July 1973.

6. D. P. Telionis, "Calculations of Time-DependentBoundary Layers," UnsteadyAerodynamics, Uol. 1, Univ. of Arizona (R. B. Kinney,Ed.), 1975.

7. J. F. Nash and V. C. Patel,"Calculations of UnsteadyTurbulent Boundary Layerswith FlowReversal," NASA CR-2546, May 1975.

8. V. C. Patel and J. F. Nash, "Unsteady Turbulent Boundary Layerswith FlowReversal," Unsteady Aerodynamics, Vol. 1, Univ. of Arizona (R. B. Kinney,Ed.), 1975.

9. R. E. Singleton and J. F. Nash, "A Method forCalculating Unsteady Turbulent Boundary Layers in Two- and Three-DimensionalFlows," AlAA J. -12, No. 5, May 1974.

10. W. J. McCroskey:"Recent Developments in Dynamic Stall ,I1 Unsteady Aerodynamics,Vol. 1, Univ. ofArizona (R. 13. Kinney,Ed.), 1975.

11. V. C. Patel and J. F. Nash. "Some Solutionsof the UnsteadyTurbulent Boundary LayerEquations ,'"RecentResearch on Unsteady Boundary Layers(Proc. I.U.T.A.M.Symposium, Quebec 1971), A. E. Eichelbrenner, ed., Presses de l'Universit6Laval, Quebec 1972.

12. J. F. Nash and V. C. Patel,"Three-Dimensional Turbulent Boundary Layer,'' SBC Technical Books,1972.

13. J. F.Nash, "An Explicit Scheme forthe Calculation of Three-Dimen- sionalTurbulent BoundaryLayers," J. Basic. Eng.., 94D, p. 131, March 1972.

44 14. J. F. Nash and V. C. Patel, "A Generalized Method for the Calcula- tion of Three-Dimensional Turbulent Boundary Layers,'' Proc. Project SQUID Workshop, Ga. Inst. Tech., (Ed. J. F. Marshall) , June 1971.

15. A. A. Townsend, "Equilibrium Layers and Wall Turbulence," J. Fluid -Mech., p. 97, 1961 16. R. M. Scruggs and C. J. Dixon, "Vortex/Jet/Wing Viscous Interaction Theory and Analysis," Final Report, O.N.R. Contract N00014-74-C-0151 , Feb. 1976.

17. R. M. Scruggs, C. J. Dixon and J. F. Nash, "Vortex/Jet/Wing Inter- action Loads by Viscous Numerical Analysis," Proc. A.G.A.R.D. Symposium: Prediction of Aerodynamic Loading, Sept. 1976.

Richtmyer Morton, Difference Methods for Initial- .- 18. R. D. and K. W. Value Problems, Wiley, 1967.

45 t=O

0 I 0.5 1.0 x/c

I I I I 1 I 1 OO0 1I T tf t-I time, t PPI

Figure 1 Definition of the ExternalVelocity Distributions.

46 Uot/c = 2.0

-3 i

& U Figure 2 Wall ShearStress Distributions for Increasing Time; Frozen Flow, U t /c = 0.2, ff = 0. of 0.2

6*/c

0 .:

Figure 3 DisplacementThickness Distributions for Increasing Time; FrozenFlow, U t,/c = 0.2, f = 0. 0 f Figure 4 Wall ShearStress Distributions for Increasing Time; Frozen Flow, Uotf/c = 2.'0, ff = 0. vl 0

0.:

6*/C

1 Uot/c = 3.3 1

Figure 5 DisplacementThickness Distributions for Increasing Time,Frozen Flow, tf/c = 2.0, = 0. U 0 ff Figure 6 Wall ShearStress Distributions for Increasing Time; Frozen Flow, tf = 2.0, ff = 0.5. VI N

Figure 7 DisplacementThickness Distributions for Increasing Time; Frozen Flow, tf = 2.0, ff = 0.5. Figure 8 Variation of Wall ShearStress with Time; Frozen Flow. Figure 9 Variation of Displacement Thickness with Time;Frozen Flow. I I

"0" 1st. onsetof reversal

0 tf = 0.2, f = 0 approximate point f of singularity 0 tf = 2.0, ff = 0 onset A tf = 2.0, ff '= 0.5

1.o 2.0 3.0 4.0 U,t/c 5.0

Figure 10 Movement ofthe Reversal Point with Time; Frozen Flow. LJl tf = 2.0 m O.( 5- ff = 0 Max. reversedflow - - -Tonvection velocity" tf = 0.2 -u/uo velocity" ff= 0 0 0. 4-

tf = 2.0 f6 = 0.5

0.: 2-

lgularity onset-"

0- CI 1.0 2.0 3.0 4.0 5.0 Uot/c

Figure 11 Prediction of SingularityOnset, Using the Criterion of Reference -7/; Frozen Flow. Figure12 The Function f(t) for Oscillatory Flows of VariousPeriods; A = 0.5. 2.0

0.5

1 (

-0.5

-1.c

Figure 13 Variation of Pressure Gradient at x = x with Time; Oscillatory Flow, A = 0.5. 0 -~ x- . ~. -

u1 W Figure 14 Wall Shear Stress Distributions for Increasing Time;U t /c = 2.0, A = 0.5. Oscillatory Fiow OP Q\ 0 I I ;I 0.2 I I I 6*/C I I I I I I I

I I I I / I I I I I 0.1 I \ I

0.2 0.4 0.6 1.0 x/ c x = x.

Figure 15 Displacement Thickness Distributions for Increasing Time; U t /c = 2.0, A = 0.5. OP Oscillatory Flow Figure 16 Wall ShearStress Distributions for Increasing Time;

Oscillatory Flow, U t /c = 4.0, A = 0.5. OP 0.2

6*/C

0. : u t/c = 4.0 0

( c 1 0.2

x = x.

Figure 17 DisplacementThickness Distributions for Increasing

Time; Oscillatory Flow, U t /c = 4.0, A = 0.5. OP . . -.

-5c

-6L Figure18 Wall ShearStress Distributions for Increasing Time;

Oscillatory Flow, U t /c = 8.0, A = 0.5. OP 0.;

6*/C

0. I Uot/c = 4.6

x=x 0

Figure19 Displacement Thickness Distributions for Increasing

Time; Oscillatory Flow, U t /c = 8.0, A = 0.5. OP u t /c OP

I 1 2 3 4 5 Uot/c

Figure 20 Movement of theReversal Point with Time;Oscillatory Flow, U t /c = 2.8 OP Figure 21 Variation of Wall ShearStress with Time

1.a

0.:

Figure 2.3 The Function f(t) for Oscillatory Flows of VariousAmplitudes; U t /c = 2.0. OP Figure 24 Variation of PressureGradient at x = x with Time; Oscillatory Flow, U t /c = 2.0. 0 OP amplituderatio: A=O

D 0.1

0 0.3 0 0.5

Figure 25 Wall ShearStress Distributions for Various Amplitudes; Oscillatory Flow, U t /c = 2.0, Uot/c = 4.0. OP amplitude ratio: 0 A=O

D 0.1 D 0.3 0 0.5

Figure 26 Displacement Thickness Distributions for Various Amplitudes; Oscillatory Flow, U t /c = 2.0, U t/c = 4.0. OP 0 -1 -

amplitude ratio: 0 A=O -2 - D 0.1 V 0.3 0 0.5 -3 -

Figure 27 Wall Shear StressDistributions for Various Amplitudes; Oscillatory Flow, U t /c = 2.0, Uot/c = 5.0. OP I

0.2 - 6*/C I

0.1- amplitude ratio : 0 A=O 0 0.1 0 0.3 0 0.5

x=x 0

Figure 28 Displacement Thickness Distributions for Various Amplitudes; Oscillatory Flow, U t /c = 2.0, Uot/c = 5.0. OP

I amplituderatio:

0 A=O

D 0.1

V 0.3

0 0.5 6

-2

Figure 29 Variation of Wall ShearStress with Time; Oscillatory Flow, U t /C = 2.0, x = 0.65~. OP amplitude ratio: gA = 0 0.2 D 0.1 0.3

6*/C 0 0.5

Figure 30 Variation of Displacement Thickness with Time; Oscillatory Flow, U t /c = 2.0, x = 0.65~. OP x/c

0.f

amplitude ratio:

0 A=O A 0.1 D 0.3

0 0.5

1 2 3 4 5 Uot/c

Figure 31 Movement of the Reversal Point with Time; Oscillatory Flow, u t /c = 2.0. OP Figure 32 The Function f(t) for Oscillatory Flows of Various Amplitudes; U t /c = 4.0. OP 1.or amplituderatio:

Figure 33 Variation of PressureGradient at x = x. with Time; Oscillatory Flow, U t /c = 4.0. OP Figure 34 Wall ShearStress Distributions for Various Amplitudes;Oscillatory Flow, U t /c = 4.0, U t/c = 4.0. OP 0 co 0

\(values for U t/c = 5 .O) I\ 0 I\ I\

Figure 35 Displacement Thickness Distributions for Various Amplitudes; Oscillatory Flow, U t /c = 4.0, U t/c = 4.0, OP 0 0.8' x/c I I

0.,6- b\

amplituderatio: 0.,4 - 0 A=O D 0.1 V 0.3 0 0.5

0- 0 1 2 3 4 5 U0t/C

Figure 36 Movement ofthe Reversal Point with Time: Oscillatory Flow, U t /c = 4.0. OP Figure 37 Velocity Profiles at U,t/c = 1.0; Oscillatory Flow, U t /c = 2.0, A = 0.5. OP 1- I

I !

>- 2dge of boundary layer (Y = 6)

3- A I 1 0.3 0.4 0.5

Figure 38 Velocity Profiles at Uot/c = 1.5; Oscillatory Flow, U t /c = 2.0, A = 0.5. OP Y/C

0.0 5

edge of reversed- edge of boundarylayer

reversal

Figure 39 VelocityProfiles at Uot/c = 2.0; Oscillatory Flow, U t /c = 4.0, A = 0.5. OP 0.lOy

0.05 t edge of boundarylayer

c \.J ~ "'

-edge of reversed- region (U =

"". "".

0 I I I I 0.1 0.2 0.4 0.5 I x/ c reversal

Figure 40 Velocity Profiles at U t/c = 3.0; Oscillatory Flow, U t /c = 4.0, A = 0.5. 0 OP 0.6 0. u/uo u/uo

0.4 0.

0.2 0,

-0.2 -0. u t /c = 2.0 0 1'

d- I I I 1 0.30.5 0.4 0.6 " 0.1 0.2 0.3 0.4 0.5 x/ c x/c

Figure 41 Velocities at y = 0.16; Oscillatory Flow, A = 0.5. 1- "

mesh density-r (node points) m 11 x 11 A 21 x 21 o 41 x 41

! 5- "isBlasius

3L c 0.2 0.4 0.6 0.8 1.0 X/C

Figure 42 Computed VelocityProfiles Over a Flat Plate; Laminar Flow, Uoc/u = lo4. -

mescribed external

Figure 43 VelocityProfiles at U t/c = 2.0; Laminar Frozen Flow, U t /c = 2.0, ff = 0.5. 0 of prescribed external velocities

Figure 44 VelocityProfiles at U t/c = 4.0; LaminarFrozen Flow, U t /c = 2.0, f = 0.5. OJ 0 of f u) i

I CD 0

Figure 45 Velocity Profiles at Uot/c = 8.0; Laminar Frozen Flow, u t /C = 2.0, ff = 0.5. of 1

Figure 46 Wall ShearStress Distributions for Increasing Time; Laminar Frozen Flow, Uotf/c = 2.0, f = 0.5. f 0.

6*/c u t/c = 8 "/

0. 1

x=x 0

Figure 47 Displacement Thickness Distributions for Increasing Time; Laminar Frozen Flow, U t /c = 2.0, f = 0.5. of f 20 2 T,/(PUo :

XI o4 la

-IC

-20

Figure 48 Variatioh of Wall ShearStress with Time; Frozen Flow, U t /C = 2.0, f = 0.5, X = 0.65~. of f Figure 49 Variation of DisplacementThickness with Time; FrozenFlow, U t /c = 2.0, ff = 0.5, x = 0.65~. of I

”- 1 \ /” \ cr;/uo =, 1 // ”””-“ ‘1 / 3” -\ \ /’ -\ 1 / / -/ I // ” 4” 10 \. ” .-. ”” -\ “

10 ”-” I - 1 0.2 0.4 0.6 0-1 0.8 1.0 x/ c

Figure 50 VorticityContours at U t/c = 2.0; FrozenLaminar Flow, U t /c = 2.0. 0 of 0.

Y/ C 'i

-0.2,

/"""" 0 0 0.2 0.4 0.6 0.8

Figure 51 VorticityContours at Uot/c = 4.0; FrozenLaminar Flow, U t,/c = 2.0. 0

I 0.;

0.1

0 0.2 0.4 0.6 0.8 1 .o x/c

Figure 52 VorticityContours at U t/c = 8.0; Frozen Laminar Flow, U t /c = 2.0. 0 of