Large-Scale Structure of the Turbulent Wake Behind a Flat Plate

TURBULENT WAKE BEHIND A FLAT PLATE

hy

S.

Jovic

and B. R. Ramaprian

Sponsored by

National Aeronautics and Space Administration Grant No. NSG 2300

IIHR Report No. 298

Iowa Institute of Hydraulic Research The University of Iowa Iowa City, Iowa 52242

TURBULENT

WAKE BEHIND A FLAT PLATE

by

S. Jovic and B. R. Ramaprian

Sponsored by

National Aeronautics and Space Administration Grant No. NSG 2300

with Dr.

J

.

G. Marvin as Technical Monitor

IIHR Report No. 298

Iowa Institute of Hydraulic Research The University of Iowa Iowa City, Iowa 52242

Partial

support

for

this

work

was

received

from

the

Department

of

Mechanical Engineering and the Institute of Hydraulic Research, University of

Iowa.

This support is gratefu1ly

acknowledged.

The authors also wish

to

thank Mr. James Cramer and his staff at the electronic shop for their help in

instrumentation.

This report also forma the Ph.D. thesis of S. Jovic.

An experimental study of the developing two-dimensional turbulent wake behind a streamlined flat-plate is reported. The purpose of this study wàs to understand the evolution of the wake structure from the one characteristic of the boundary layer to that characteristic of the far-wake, and to LnvestIgate whether organized motions are present in the wake flow just as in other free shear flows.

The evolution of the flow and the mixing process in the developing wake was s~udied by using the well-known heat-tagging technique. This consisted of heating very slightly the boundary layer on one side of the plate, so that heat could be used as a pass Ive tracer. The experiments were performed in a wind tunnel at a Reynolds number (based on the trailing-edge momentum thickness) of 1100. Simultaneous measurements of two components of ve locity and local temperaturë were made. The data were obtained at eight stations over the distance range of 0, starting at the trailing edge, to 260 wake-momentum thicknesses and hence practically upto the far'-wake. In addition to obtaining the usual conventionally averaged turbulence properties, special procedures such as conditional averaging and selective ensemble averaging we re employed to analyse the data.

The conditional averaging technique has been used to gain better insight into the dynamics of the interface separating the mixed and

unmixed fluid in the deve Lop Ing wake. Finally, a se

lectIve ensembIe

averaging

technique

was

successfully

used

to

.

identify

and

educe

organized large-scale structures present in

the wake

.

flow.

Two types of

structures were identified on one side of the wake, one originating from

the same side of the wake centerline and the other one from the opposite

side.

The frequency of their occurrence and their dimensions, as well

as all the properties associated with their organized and randoo;l

motiotls

are determined.

LIST OF FIGURES CHAPTER

I. INTRODUCTION

1.1 Problem Introduced

1.2 Objectives and Approach of Present Study 1.3 Organization of the Thesis .' Il. EXPERIHENTAL PROGRAM.

2.1 Apparatus and Hodel .

2.2 Heating Tnchnique . 2.3 Instrumentation .. 2.3.1 Velocity Heasurements 2.3.2 Temperature Measurement 2.4 Calibration of Instrumentation 2.4.1 Hot-Wire ... 2.4.2 Temperature Probe

2.5 Experimental Conditions and Procedure 2.6 Data Acquisition and Processing

2.6.1 Data Acquisition .

2.6.2 Data Processing .

2.7 Conditional-Sampling and Intermittency 111. EXPERIMENTAL RESULTS - CONVENTIONAL AVERAGES

3.1 Longitudinal Mean Velocity 3.2 Turbulence Structure

3.2.1 Reynolds Stress es 3.2.2 An1sotropy Parameter

3.2.3 Diffusion Fluxes and Transport Velocities 3.3 Summaty of the Conventionally Averaged Results IV. EXPERIMENTAL RESULTS - CONDITIONAL AVERAGES

4.1 General . . . . 4.2 Intermittency .... 4.3 Conditional Averages 4.3.1 Reynolds Stresses 4.3.2 Triple Correlations ii Page . iv 1 1 4 6 7 7 7 9 9 10 11 11 12 13 14 14 15 16 29 29 30 30 31 32 36 51 51 52 57 57 60

Page 4.3.3 Conceptual Picture of the Developing

Intermemdiate Wake . . . 61 4.4 General Statistical Properties of Cold/not-cold

Bursts . . . 63

V. EXPERIMENTAL RESULTS- ENSEMBLE AVERAGE 96

5.1 5.2 5.3 5.4 5.5 5.6 5.7 General . . . 96 97 98 102 103 105 Selective Conditional Sampling

Ensemble Averaging .... Coherent S~ructure Statisti~s Coherent Structure Properties Reynolds Stresses . . . . .. ..

Distribution.of the Coherent Structure Properties across the Wake . . . . .

5.7.1 Streamwise velocity U . . . . 5.7.2 Reynolds Shear Stress . 5.7.3 Correlation Coefficient and Anisotropy

Parameter . . . ~ . . . . Spatial Configuration of Educed Organized

Structure . . . ₁₁₂ 107 107 108 110 5.8

VI. CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK

REFERENCES

156

6.1 Conclusions .

6.2 Suggestions fot Future Work

156 157 158

LIST OF FIGURES

Figure Page

2.1. Experimental arrangement . . . 23

2.2. Flow configuration and nomenclature 24

2.3. Schematic diagram of (a) experimental setup (b)

Cold-wire arrangement 25

2.4. Zero bias of the resistance thermometer 26 2.5. Schematic represantation (a) of the flow dynamics in the

intermittent region, (b)Threshold setting for

cold/not-cold discrimination and conditional sampling . . . 27 2.6. variation of (a) intermittency and Cb) crossing frequency

distribution for various threshold settings. . . 28 3.1. Velocity distribution (a) definition sketch, (b) W/W

o

across the wake. · ... 38

3.2. Variation of the wake characteristic scales along the

flow: (a) half-width, (b) maximum wake defect. . ... 39 3.3. Distribution of u2/W~ across the wake at stations: (a)

x/a= 5-35, (b) 70-260. 40

3.4. Distribution of v2/W; across the wake at stations: (a)

x/8= 5-35, (b) 70-260. 41

3.5. Variation of (a) -uv/Wo- 2 across the wake at stations x/8=

- 2

5-260, (b) maximum shear stress luvl /W _along the m 0

wake... · ... 42

3.6. Evolution of (a) normalized eddy viscosity vt/Ue8 and

(b)anisotropy parameter u'Iv' along the wake.. · ... 43 3.7. Distri~ution of ~/W~ across the wake at stations: (a)

x/8- 5-35, (b) 70-260. . ...•... 44 3.8. Distribution of ~/w~ across the wake at stations: (a)

x/8= 5-35, (b) 70-260. . 45

3.9. Distribution of ~/W~ across the wake ··atstations: (a)

x/a= 5-35, (b) 70-260. . 46

3.10. Distribution of ~/W~ across the wake at stations: (a)

x/8= ~-35, (b) 70-260. . .. 47

3.11. Distribution of Vk/WO across the wake at stations: (a)

x/8= 5-35, (b) 70-260. . . .. . ... 48

3.12. Distribution of Vt/WO across the wake at stations: (a)

x/8= 5-35, (b) 70-260. . . 49

3.13. Evolution of transport velocity scale of turbulent kinetic

energy along the wake. . 50

4.1. Distribution of intermittency,crossing frequency and mean

temperature across the wake: (a) x/8=20, (b) 70. . . 66

4.2. Distribution of intermittency,crossing frequency and mean

temperature across the wake: (a)/8=140, (b) 260. . . 67

4.3. Comparison of Q and turbulent-nonturbulent intermittency nc

estimated from Phillips s theory at (a) x/8=S, (b) 10. .. 68

4.4. Comparison of Q and turbulent-nonturbulent intermittency nc

estimated from Phillips s theory at (a) x/8=20, (b) 35 ... 69

4.5.· Comparison of Q and turbulent-nonturbulent intermittency nc

estimated from Phillips s theory at (a) x/8=70, (b)

140" . . . .. . ... . . 70

4.6. Comparison of Q and turbulent-nonturbulent intermittency nc

estimated from Phillips s theory at (a) x/8=210, eb)

260. . . .. 71

4.7. Conditional and c')nventionalaverages of u2 across the

wake at (a) x/8=20, (b) 70. 72

4.8. Conditional and conventionc.laverages of u2 across the

wake at (a) x/8=140, eb) 260. 73

4.9. Conditional and conventional averages of v2 across the

wake at (a) x/8=20, (b) 70. 74

4.10. Conditional and conventional averages of v2 across the

wake at (a) x/8=140, eb) 260.

.

75

4.12.

4.13.

Conditional and conventional averages of -uv wake at (a) x/8=140, (b) 260...

across the

'.

. . . 77 3

Conditional and conventional averages of u across the

wake at (a) x/8=20, (b) 70. . . ... 78 4.14. Conditional and 'conventional averages of u3 across the

wake at (a) x/8=140, (b) 260. 79 4.15. Conditional and convehtional averages of u v across2 the

wake at (a) x/8=20, (b) 70. 80

4.16. Conditional and conventional averages of u v across2 the

wake at (a) x/8=140, (b) 260. 81 4.17. 4.18. 4.19. 4.20. 3

Conditional and conventional averages of v across the

wake at (a) x/8=20, (b) 70. . . .. . ... 82 3

Conditional and conventional averages of v across the

wake at (a) xj8=140, (b ) 260. . . 83 2

Conditional and conventional averages of uvacross the

wake at (a) x/8=20, (b) 70. . . . .. . 84 2

Conditional and conventiona1 averages of uvacross the

wake at (a) x/8=140, (b) 260. . . .. ....•. .. 85 4.21. Conditionally averaged transport velocity of turbulent

kinetic energy at (a) x/8=70, (b) 140. . . 86

. .

4.22. Conditionally averaged transport velocity of turbulent

kinetic energy at (a) x/8=210, (b) 260. . . 87

4.23. Conceptual representation of developing wake, (a) zones in streamwise and transverse direction, (b) variation of

length scales along the wake. . . 88 4.24. Evolution of the eddy viscosity normalized with the

wakelike region lenght scale L . . . 89 w

4.25. Variat~on of (a) mean tb and stàndard deviation ob of cold and not-cold burst durations and (b) ob/tb across the

wake at x/8=70.. " . . 90

4.26. Variation of (a) mean tb and standard deviation ob of cold and not-cold burst durations and (b) Ob/tb acro~s the

wake at x/8=140. . .. 91

4.27. Variation of (a) mean tb and standard deviation ob of cold and not-cold burst durations and (b) ab/tb across the

wake at x/ 6=260. 92

4.28. Probability density distributuion of (a) cold time

intervals,_(b) not-cold time intervals at x/6=70 and at

y/6 where 1 has a given value. . 93

nc

4.29. Probability density distributuion of (a) cold time

intervals, (b) not-cold time intervals at x/6=140. . ... 94 4.30. Probability density distributuion of (a) cold time

intervals, (b) not-cold time intervals at x/6=260. . ... 95

5.1. Definition sketch of (a) structure and thresh6ld setting for the selection duration tand selective sampling

o .

(b)

of a structure of

118 5.2. Distribution of <u>/W

O

and <V>/W

O

for (a) cold and (b)

not-cold structures of different durations. '

..

119 5.3. Distribution of -<uv~s/W~ along the structures of

different durations for (a) not-cold and (b) cold

5.4. 5.5. 5.6. 5.7. 5.8. 5.9. structure. 120

Distribution of the (a) total time share and (b) total

number of the selected structures. 121

Peak frequency of structure occurrence and its evolution

along the wake.D,frequency of ... 122

Distribution of <u>/WO along the structure and across the intermittent region for (a) cold and (b) not-cold

structure at x/6=35. 123

Distribution of <v>/W0 along the structure and across the

.intermittent region for (a) cold arideb) not-cold

structure at x/6=35. 124

Distribution of <u>/WO along the structure and across the intermittent region for (a) cold and (b) not-cold

structure at x/6=70, _" 125

Distribution of <v>/WO along the structure and acros s the intermittent region for Ca) cold and (b) not-cold

structure at x/6=70, 126

intermittent region for (a) cold and Cb) not-cold

structure at xj8=140. .. ..... 127 5.11. Distribution of <v>jWo along the structure and across the

intermittent region for (a) cold and Cb) not-cold

structure at xj9=140. 128

5.12. Distribution of <u>jW

o

along the structure and across the interrnittent region for Ca) cold and (b) not-cold.

structure at xj8=260.

.

129

5.13. Distribution of <v>/WO along the structure and across the interrnittent region for (a) cold and Cb) not-cold

structure for x/9=260. . . 130

5.14. Distribution of the shear stress along the cold structure

(a) <uv>l/üV and (b) <uv>s/üV at xj6=70. .... . 131

5.15. Distribution of the shear stress along the not-cold

structure (a) <uv>l/üV and Cb) <uv>sjüV at xj9=70. . 132

5.16. Distribution of the shear stress along the cold structure

(a) <uv>ljüV and (b ) <uv>sjüV atx/9=140. . . 133

5.17. Distribution of the she~r stress along the not-cold

structure (a) <uv>l/üV and (b) <uV>/Uv at xj8=140. . 134

5.18. Distribution of the shear stress along the cold structure

(a) <uv>l/üV snd (b) <Uv>/üV at xj6=260. ; . . . 135

5.19. Distribution of the shear stress slang the not-cold

structure (a) <uv>lj~ end (b) <uv>s/üV at x/6=260. 136 5.20. Total contribution of shear stress by (a) cold and Cb)

not-cold organized structure at x/8=35. 137 5.21. Total contribution of shear stress by (a) cold and (b)

not-cold orgacized structure at x/8=70; 138 5.22. Total contrihution of shear stress by Ca) cold and (b)

not-cold organized structure at x/8=140. 139 5.23. Total contrfbuti.onof shear stress by (a) cold and eb)

not-cold organized structure at xj8=260. 140

5.24. Evolution of -<uv>/W~ along the wake for eb) not-cold structure.

(a) cold and

141 5.25. Distribution of mean streamwise velocity U/U across cold

e

and not-cold structure at x/8=35. 14~

5.26. Distribution of mean streamwise velocity U/U across cold e

and not+co ld structure at x/8=70 ... 143

5.27. Distribution of mean streamwise velocity U/U across cold e

and not-cold structure at x/8=140. 144

5.28. Distribution of mean streamwise velocity U/U across cold e

and not-cold structure at x/8=260. 145

5.29. Distribution of correlation ëoefficient across (a) cold

and (b) not-cold structure at x/8=35. 146 5.30. Distribution of correlation coefficient across (a) cold

and (b) not-cold structure at x/8=70. 141 5.31. Distribution of correlation coefficient across (a) cold

and (b) not-cold structure at x/8=140. 148 5.32. Distribution of correlation coefficient across (a) cold

anq (b) not-cold structure at x/8=260. 149 5.33. Distribution of anisotrpy parameter across

(b) not-cold structure at x/8=35.

(a) coLd and

150 5.34. Distribution of anisotrpy parameter across

(b) not-cold structure at x/8=70.

(a)'cold and

151 5.35. Distribution of anisotrpy parameter across

(b) not-cold structure at x/8=140.

(a) coLd and

152 5.36. Distribution of anisotrpy parameter across

(b) not-cold structure at x/8=260.

(a) coId and

153 5.37. Spatial representation of (a) cold and

structure at x/8=140

(b) not-cold

154 5.38. Spatial representatioriof (a) cold and

structure at x/8=260

eb) not-cold

155

CHAPTERI INTRODUCTION

1.1 Problem Introduced

The turbulent wake of a flat plate has been studied in recent years both because of the fundamental nature of the problem and because of its practical applications in aerodynamics, ship hydrodynamics and chemical engineering, e;g. airfoil drag cal cu l at i.on, noise control , propeller design, chemical mixers, etc. This research is the continuation of an earlier stndy at the lnstitue of Hydraulic Research on developing turbulent wakes behind streamlined bodies (Sastry , 1981 and Ramaprian, Patel and Sastry , 1982). The main finding of that study was that the developing symmetrie wake behind a flat-plate can be divided into the following three regions in the streamvise direction ex) based on the evolution of its turbulent structure.

(i)

A

'Near-Wake' region extending trom the trailing edge to about x/a

= 25, where a is the momentum thiekness of thè wake. It is eharacterised by the presence. of an inner wake, whose properties are governed by the details of the near-wall region of the boundary layer upstream of the trailing edge. This region is also eharacterized by

fine scale· mixing between the boundary layers emerging trom the two sides of the plate.

2

(H) An 'Intermediate Wake'·extending over the region 25 < x/a < 350. Direct influence of the wall reg ion of the upstream boundary layer is insignificant in this region since the inner wake has consumed the log-region of the boundary layers. The wake,thus,evolves essentially as a free turbulent flow in this region. It should be pointed out that two turbulent structures are present in this region, namely one associated with the inner wake evoIv Lng from the small-scale mixing and other

associated with the outer boundary layer characterized by large-scale structures, approaching eventually the final structure characteristic of the far-wake. It is, therefore, anticipated that the interaction between these two zones in the intermediate wake governs the evolution of the flow.

(iii) The 'far-wake' region extending beyond x/a

=

350. The turbulent structure iJl this region, having lost memory of its origin would attain a universal self-preserving form. This is the reg ion which has been extensively studied both analytically and experimentally.

Wakes behind cylinders and usually in t.he self-preserving region have been studied in detail by Townsend (1946), Grant (1958), Thomas

(1973), LaRue and Libby (1974, 1976) and Fabris (1976). In addition to documenting the usual mean and turbulent properties of the flow, they studied the intermittent interaction between turbulent and nonturbulent flow in the outer regions of the wake. Chevray and Kovasznay (1969), Pot (1979), Andreopoulos and Bradshaw (1980), and Sastry (1981) studied the developing flat plate wake and reported the mean and turbulent flow properties in the wake. Pot' s measurements extend upto x/a = 950 and

are the only available data extending from the trailing edge well into the far-wake, even though they do not contain information on diffusion (triple products of velocity fluctuations). More detailed studies of the turbulence structure of flat-plate wakes have been reported by Kovasznay and Ali (1974) and more recently by Bonnet, Jayaraman and Roquefort (1984). The former studied the wake of a heated flat plate using heat as a 'tag' to investigate the properties of the turbulent-nonturbulent interface between the wake and the outer flow. Bonnet et

al.

obtained extensive data on space-time correlations in the compressible wake of a flat plate.

All the above investigators considered wake flow essentially as a single flow, with the exception of Andreopoulos and Bradshaw who studied the near-wake as a flow characterized by the interaction of two boundary layers~ They also use.I the heat-tagging technique. For this purpose , they heated the boundary layer on one side of the p late. Using temperature as a detector, they emp10yed conditiona1 averaging techniques to study the fine-scale mixing between the two sides of the inner wake, in the region 0 < x/a < 30. They measured the contributions to various turbulent properties from each of the two sides which time-share the inner-wake region. Their experiments did not extend over the outer part of the near-wake and the very important intermediate wake. These regions are cbaractrized by large scale mixing and large structures whose evolution requires careful study.

The subject of large structures has received extensive attention during tbe last several years, especially after the discovery tbat in

many flows these structureS are h ighl y organized, th~tàiled studies öf these coherent structuresj including their eductioh ártd interpretation; have been reported for boundary layers, jets and inixing iayers. Fiom flow visualisation studies and muitiple-probe measurements, of gan Lzed structures have been successful1y educed especia11y in jets and mixing 1ayers. Crow and Champagne (1971) applied flow visuá1isatioh technique to show the presence of organised motion in the initial region of the axisymmetric jet. Brown and Roshko (1974) obtained the same for mixing 1ayers. Hussain and coworkers (e.g. Hussain

and

Zaman 198Ö)j among

others, applied multip1e-probe measurements, conditional s ampl i.ng and ensemble averaging techniques to educe orgánized structures and s t udy their dynamics in the case of undisturbed circular jets and circular jets under controlled excitation. Flow visua lis at Lon studies of boundary 1ayers made by Falco (1977) and Head and Bandyöpadhyay (1981) provided evidence about the existence of large organized structutes ih these flows also.

The questLons : "Are there coherent structures in wákes?i' and

"rr

so, what is their nature?" have not been answer ed .. A related questLon is whether all wakes reach the same asymptotic state as has been usually assumed. These aspects need detailed and extensive study.

1.2 Objectives and Approach of Present Study

The objective of the present study is to investigate in detail the mixing process in the developing flat-plate wake. Since both sides of the wake are turbulent, heat-tagging is used to mark one side. UsLng

heat as a passive tracer and employing the weIl known conditional- and ensemble-averaging techniques, the details of mixing are studied.

Experiments were conducted in a wind tunnel on a flat-plate wake at

a trailing-edge Reynolds number U 6/\)=1100 in the range of x/6=0-260

e

which covers most of the developing wake. Velocity fluctuations were

measured by two-sensor hot-wire anemometry and temperature by 'cold

wire anemometry' of adequate dynamic response.

The initial objectives of the study were to investigate the manner

in which fluid from one side of the wake penetrates to the other side

and how it contributes to the evolution of the turbulent properties of

the flow. These studies required the use of conditional averaging

techniques. As the work progressed the results suggested astrong

possibility of the existence·of organized structures in the mixing

region. Hence, ensembie averaging technique of the type used by Hussain

and Reynolds (1972) and Sreenivasan et al. (1979), was used to educe

these structures and study the details within the ~tructures.

Single-point measurements and the absence of externally triggered reference

phase made this task quite challenging. Fortunately, however, it turned

out that the structures were very strong and hence could be educed and

reconstructed even with these relatively simple measurements. The major

contribution of the present study is the identification and

quantification of very strong coherent structures that are present all

along the developing wake and continuing into the asymptotic wake. The

origin of these structures and their dependence on the trailing edge

Reynolds number are still unknown but their presence even in the

6

As the present work was nearing completion, information become available on a similar study under progress at the California Institute of Technology by Roberts and Roshko (1985). They studied a flat-plate wake at a trailing-edge Reynolds number of 160. Their experiments

included cases in which the upstr eam flow on one side was artificially excited as well as the case in which no external excitation was applied.

In these experiments, which were conducted in ,water, laser f Iuores cence technique was used to distinguish fluid from the two sides of the wake. The flow visualization pictures from their study are remarkably similar

to .the large scale structure reconstructed from the present measurements.

1.3 Organization of the Report

The experimental appar at.us , instrumentation and the heat-tagging technique are described in Chapter 2. Chapter 3 describes the results of conventional averaging for the developing wake. Chapter 4 deals with the conditionally averaged results. The results of ensemble averaging pertaining to the properties of the organized structures, and their

spatial configuration are described in Chapter 5. The main conclusions from the study and some suggestions for future work are presented in Chapter 6.

CHAPTER 11

E

X

PERI

M

ENTAL PROGRAM

2.1 Apparatus and Model

The experiments were conducted in a closed circuit, open-jet, low-speed wind tunnel. The tunnel freestream velocity was maintained at about 13 mis. The working section has an octogonal cross-section of 0.5 m side and is about 1 m long. There is no measurable longitudinal pressure gradient in the test section. The freestream turbulence is about 1~~at the tunnel speed studied. This is large but, as will bo seen later, irrelevant to the flow under study.

The details and dimensions of the wake generator (referred to as an 'airfoil' hereafter) used for the experiment are shown in Fig. 2.1. It is a flat plate 65 cm (chord) x 86 cm (span) x 1.5 cm (thickness) with the last 15 cm of its length tapered to end in a pointed trailing edge. The airfoil was placed vertically in the midplane of the tunnel working section .. The boundary layer on each side of the airfoil was tripped by means of a 2 mm diameter wire glued to the surface at 12 cm from the leading edge.

2.2 Heating Technique

Conditional sampling, in the present case, requires that one of the boundary layers be tagged by heating it. At first, this heating was

8

accomplished by a method similar to that used

Dy

Andr-eopoulos and Bradshaw (1980); namely by using heating wires resting on the surface sf the airfoil. This method has been found to be satLsfactory for the study of the fine mixing processes in the near-wake. However, this heating arrangement was found to be unsatisfactory for the study of the outer part of the neax-wake and the intermediate wake. SpecLfLcalLy , this method of heating wou Id not produce a region of Uniform high temperature in the free stream beyond the shear layer. A modi fLed heating technique which would provide a reasonably 'uniform high temperature' region on the heated side of the airfoil was therefore to be developed. After several trials, the details of which are described in Kalale (1982), the heating confLgurnti.cn shown in Fig. 2.1, was arrived at. Twelve nichrome resistance wires of 0.8 mm diameter, extending over the span of the airfoil and spaced 6 mm apart were pla6ed on each side of the airfoil at about 45 cm trom the trai1ing edge in a plane perpendicular to its surface. All the wires were spring loaded so that they would remain straight when heated. The set of twelve wites on one side of the airfoil were heated and the wires on the other side merely served to provide flow symmetry. The heating wires were connec'ted in 3 parallel branches with each branch having 4 wires in series. Each wire had a resistance of approximately 5 ohms. Thus, this arrangement would draw a power of about 1.7 kWat 110 volts. With this arrangemeht, a temperature differènce of about 1.2-1.5° C was obtained between the heated and unheated str~ams at the trailing edge at a tunnel velocity of

provide the required discrimination between 'cold' and 'hot' fluid, yet small enough not to introduce buoyancy effects . The drag on the wires resulted in a 'wake' behind each wire. However, these individual wakes merged with one another and the flow at the trailing edge (about 900 wire diameters downstream) was found to be 'weIl mixed' with regard to temperature distribution. Fig. 2.2, shows schematically the flow conditions at any station in the wake and explains the wake nomenclature used henceforth. Note that the 'freestream' for the airfoil wake has a slightly smaller velocity than the undisturbed flow in the tunnel namely, U

=

12.5 mis. It was found that even at the last measurement

e

station in the wake, there was a 'free stream' of about 3 cm in width in which the velocity

U

was constant.

U

remained practically constant in

e e

the x-direction. The turbulence level in th Ls 'free stream' was about

1.2%. This is rather high and unfortunately, is one of the

shortcomings of the present design. This aspect will be discussed later in Chapter 5.

2.3 Instrumentation 2.3.1 Velocity Measurements

Mean velocity in the flow was measured with a total head tube of 0.8 mm.diameter. Turbulence measurements were made with two DISA 55M01

constant-temperature anemometers using a DISA cross-wire probe (type 55P51) with 5~m gold-plated platinum-wire sensors. The outputs from the two channels were first low-pass filtered at 1000 Hz using sharp cut-off (32 dbjoctave) analog filters before being read by the data-acquisition

10

system. The low-pass filtering was deliberately introduced in order to facilitate the efficient sampling and processing of the large eddies in the flow. The overall loss of information due to this filtering was

2

found to be less than 5% in the case of u .and even smaller in the case of uv. This is acceptable since the interest of present research is

primarily in the large eddies.

2.3.2 Temperature Measurement

A thermistor probe conected to a .calibrated digital thermometer o

with an accuracy of 0.01 C was used to measure the ambient temperature

in the tunnel. The mean and fluctuating temperature across the wake were

measured differentialy, with respect to the 'freestream' temperature,

using a pair of DISA temperature probes ( type 55P31) each with a 1 ~m

platinum wire as sensor. The probes w~re operated togetner as a

differential microresistance thermometer by using them in a const

ant-current Wheatstone bridge circuit as shown in Fig. 2.3(b). The bridge

was powered by a 12V battery with a current of 0.6 mA so as to obtain an appreciably large out-of-balance voltage signal. The sensors were thus

operated in the so called 'cold-wire' mode. One of the 'cold wires' was

located close to the fixed thermistor probe in the free stream of the

cold side of the wake, while the other wire was located near the

cross-wire probe and traversed with it. The center of the temperature sensor

was within about 1 mm from the center of the cross-wire probe. The

differential voltage from the Wheatston bridge was amplified by a

was filtered by a 1000 Hz Iow- pass filter to remove any undesirable high-frequency noise. The electronic noise in the temperature circuit was of the order of 5 millivolts whereas the signal was about 80 millivolts for a temperature difference of 1° C. The frequency response of the temperature system ( estimated to be nearly flat in the range 0-1000 Hz) is adequate for the study of large scale structures.

2.4 Calibration of Instrumentation 2.4.1 Hot-Wire

The hot-wire probe was calibrated in the undisturbed stream in the tunnel at six different velocities. Calibrations we re made before the start and the end of each experiment. The overheat ratios in the two anemometer channels we re so adjusted that the calibration curves for the two wires were as closely matched as possible. The overheat ratio used was about 1.6. All the calibration curves were found to obey the

following relation:

E2

- A

+

B

IU

T

T

-w (2.1)

Note that eq. (2.1) is a more general form of King's law, than the

expression usually used, namely

12

The temprature terms are explicitly retained in eq. (2.1) beeause of the faet that the Local air temperature T is also a variabie in the experiment.

2.4.2 Temperature Probe

The temperature bridge has a ealibration re1ation as follows ( see Kalale 1982):

E

a

C

+ D (T-T )

(2.3)

r

where E is the instantaneous output voltage, T-T is the instantaneous r

temperature differenee between the measuring and referenee probes. C and

D

ean. in general, be dependent on the referenee ( ambient) temperature T .

r D was found to be almost independent of the loeal temperature T, while evaried somewhat with

T .

Crepresents the zero

r

bias of the instrument, i.e. the voltage output for zero temperature differenee between the measuring and referenee probes. Dependenee of C on

T

was determined from an experiment by bringing the two temperature

r

probes together ( so that

T=T )

and measuring the bridge output voltage r

E at several ambient temperatures T

r ( measured by the thermistor termometer). This experiment was done before eaeh detailed experiment. Fig. 2.4 shows one typical result of temperature ealibration. It is seen that the data ean be fitted by a straight line, from whieh C can be obtained as a funetion of T .

2.5 Experimental Conditions and Procedure

The experiment was conducted at a constant freestream velocity of 12.5 mis corresponding to a Reynolds number Re= 1100 based on the momentum thickness of 1.5 mm at the trailing edge of the airfoil. It took usually about two hours of continuous operation of the tunnel to reach steady state. Detailed measurements of mean and turbulent flow characteristics were made at eight streamwise locations, namely at x/a= 10, 20, 35, 70, 140, 210 and 260. Even though the purpose of this experiment was to study the large-scale mixing process in the developing-wake region with the aid of heat tagging, it was feIt des;i.rableto obtai.n a complete set of· baseline data of all the flow properties in the wake in the absence of any heating. There were three reasons for performing this additional experiment:

ei) to check the quality of the flow in the presence of the wires, (ii) to provide a reference for comparison with the data obtained with one half of the wake being heated, and

(iii) to obtain a set of detailed data for the developing wake in the range x/8= 0-260. It might be noted that with the exception of Pot's (1979) data, such information is not available in the literature over this range of the developing wake.

The measurements of the heated wake were carried out under the same conditions as the isothermal wake, except that the flow on one side of

o

the airfoil was heated by about 1.2-1.5 C, relative to the flow on the other side, as described in section 2.2.

14

During the actual experiment, the temperature T of the tunnel air r

increased slowly but steadily because of the constant heat input. Therefore, the reference temperature was monitored and the bias value C was accordingly updated continuously during the experiment (usually

after every 3rd point in the traverse) . Finally, instantaneous

temperature at a point was obtained from eq. (2.3) as

T ~ T

+

(E-C)/D

r (2.4)

The traverse cönsisted of a set of 30 points at each x-Iocation.

2.6 Data Acquisition and Processing

2.6.1 Data Acquisition

A HP 1000 minicomputer with a Preston analog to digital converter

was used for data acquisition and proces;ing. All the components of the

instrumentation are schematicaly shown in Fig. 2.3(a). The

instantaneous outputs of the two channels of the cross-wire and the

temperature system were simultaneously sampled, digitized, stored first

on disc and transfered subsequently to magnetic tape ( for permanent

archival) after each traverse was completed. The instantaneous output

voltages from the three channels of the final set of experiments were

sampled simultaneously at the rate of 2000 samples per second to obtain

a total of 40960 samples per channel. These samples were acquired in

eight continuous batches of 5120 samples each with a short gap of about

five seconds between successive batches. The total record length was

thus about twenty seconds, while the total time over which these batches

2.6.2 Data Processing

The analysis of the data was carried out essentially in three phases:

(i) conventional averaging (ii) conditional averaging (iii) ensemble averaging

The instantaneous voltage fluctuations for the cross-wire and co

ld-wire sensor~ are given, respectively, by

s

u

+

S lt

u I el t (2.5)

(2.6)

(2.7)

using linearized form of (2.2) and (2.3) under the assumptions u«U and

t«6T (see Kalale,1982) where Su and St represent the sensitivities of

the sensor to velocity and temperature respectively. It is seen from

eq. (2.5) and (2.6) that the velocity signals from the cross-wire sensors, are contaminated by the nonconstant local temperature. This is removed simply by subtracting St1t and St2t from eland e 2. Likewise, the fluctuating temperature signal is, in principle, contaminated by the

velocity fluctuations. _{However, it can be shown that at a sensor}

current of 0.6 mA, and at the tunnel velocity studied, the velocity

16

(Wyngaard,1971). The fluctuating components can be recovered from (2.5) and (2.6), as

u .. (u +u2)/{ï

el e (2.8)

(2

.

9)

and the temperature fluctuation from eq. (2.7) as

(2.10)

In the case of the conditional sampling and averaging, velocity and

temperature sensitivities were obtained, using the conventionally

averaged velocities U and V and temperature AT in (2.5)-(2.7) and the

zonal fluctuations (namely, u , V ,u , V ) were measured with respect

c c nc nc

to the overall averaged quantities U and V. From the instantaneous

fluctuating quantities, the appropriate statistical properties were

determined in each of the three phases of the analysis. These include

h .. 22 2 233

t e quant1.tles·u , v , uv, u v, uv , u , v .

2.7 Condltional-Sampling and Intermittency

The outer region of a boundary layer, or any flow which interfaces

with an outer inviscid (nonturbulent) flow, is not turbulent all the

time, i.e. flow is intermittently turbulent and nonturbulent. The flow

Bandyopadhay (1981) and Fabris (1977) on boundary layers, Brown and Roshko (1974) on mixing layers, Grant (1958) and Roberts and Roshko (1985) on wakes and Crow and Champagne (1971) on jets have confirmed the intermittent nature of turbulence in the outer regions of these flows. Similarly, when two turbulent flows, which are originally spatially separated, merge together, there would be an interface region in which mixing would be intermittent, i.e. the region would be occupied

intermittently either by the fully mixed fluid or fluid from either of the unmixed streams. In this case, however, the fluid is always turbulent. In either of the above two cases the interaction of the two flows can be studied by subjccting the output signal, corresponding to

some flow property,· to an appropriate criterion (which can determine

whether the fluid particles present at the probe at the instant of

s smpl i.ng be long to one .or the other flow) .and .obtaining therefrom

statistical properties of the flow only when the criterion is satisfied.

Such conditi.onal: sampling and averaging techniques have proved to be

very useful in studying the interaction between two adjacent flows.

Many investigators have studied the first kind of intermittency

(turbulent-nonturbulent) by using such criteria to distinguish between

turbulent and nonturbulent fluid. The criterion originally used by

Townsend (1949) was that thc magnitude of the derivative lau/atl of the

longitudinal velocity fluctuations must be above a certain pres et

threshold level in order that the flow could be considered to be

turbulent. This initial idea was used by him in the study of the wake

18

years and has been modified by many investigators (see Hedly and Keffer 1974), who used other flow variables . Kovasznay et al. (1970) used

Ia.u/ayatl, Antonia (1972) used (duv/at)2, and most recently Bradshaw and Murlis (1973) and Murlis et al. (1982) used lauv/atl in combination with (duv/at)2. But all these methods, to· different degrees, suffer

from the· same problems, namely the selection of a .' right' threshold level and sharp discrimination between velocity fluctuatiens due to turbulence and irrotational velocity fluctuations due to pressure fluctuations. Murlis et al. (1982) showed that heating the flow slightly and using the temperature level relative te an appropriate threshold level as the detection criterion is a more reliable technique. Bradshaw (1974) used such a 'heat tagging , technique in a series of studies on 'complex'· turbulent flows for distingushing between two turbulent flows of different origins. This technique was used, as already mentioned, .by Andreopoulos and Bradshaw (1980) for the study of the fine grain mixing in the inner part of the near-wake of a flat. plate. This technique has been adopted for the present study of

internal large scale mixing in the intermediate wake and its evolution into the asymptotic far-wake.

One side of the plate was heated as described in section 2.4. All fluid part ic les were labelled 'cold '. on one iside and 'hot' on another. In the interaction reg ion beyond the trailing edge the labelled fluid particles are convected by the eddies of different sizes essentially in

all three directions , and consequently the probe detects occasional "coId ' , 'hot' or

,

warm

,

f Iuid . This is schematicaly shown in Fig. 2.5(a).

1'he time history of velocity and temperature signal in the washown

qualitatively in Fig. 2.5(b). In order to label the fluid as 'cold' or

'hot' _{it is necessary to specify a lower (Tl) and an upper}

threshold level, defined as follows

(T ) u T~.= Tc

+

r 1

(2.11)

T u

(2.12)

where rl and r2 are the values selected so as to compensate for the

electronic noise and/of any thermal fluctuations present in the free

stream. Note that LdeaIly (i.e. in the absense of these extraneous

influences) rl and r2 should simply be equal to zero. Depending on

whether 1'<Tlof T>Th the fluid is 'cold' or 'hot', respectively, refered to also as 'unmixed' and if Tl<T<Th the fluid is designated to be 'warm'

refered to also as 'mixed'. Note that conditional sampling can be made

either by discriminating between 'cold' and 'not-cold' or by

discriminating between 'hot' and 'not-hot'. However, since the cold

freestream of the fluid has neither thermal fluctuation nor any ripples

in the temperature distribution it is easier to set the Tl .

. threshold

level. Here, in the present studies, only the lower threshold level Tl

was used for discrimination. _{Hence, only two different flows were}

distiguished, namely, 'cold'when T<Tl and 'not-cold' when T>Tl. Antor funetion I ean hence, be defined as

I

20

indicating 'not-cold' flow at an instant. The intermittency factor, or

simply intermittency, is defined as

n

_nc

- I

_nc := 1

N

(

2

.

1

4)

representing the fraction of time when fluid is not-cold.

n

=1 means

nc

that the probe is occupied by the not-cold fluid all the time, and

n

=0

nc

means that the probe is within. the cold fluid all the time. A

complimentary intermittency )2=1-)2 is the fraction of time for which

c nc

the fluid is cold. Another parameter that has been used for describing

interface dynamics is the 'crossing frequency'. It represents the rate

at which the interface crosses a given location.

Problems often encountered in setting a temperature threshold are

the slow variation of the level T either due to electronic drift or due c

to a gradual change in the ambient temperature and certain noise level.

Dean and Bradshaw (1976) used two criteria, namely temperature level as

the first criterion and the time derivative of the temperature signal as

the second criterion for the LnderrtLficetion of the labelled fluid.

This was because there was a slow variation of the ambient temperature

T in their experiment.

c Andraeopoulos and Bradshaw (1980) used the

derivative only as a back up criterion. In the present experiment only

the level criterion was used. The variation in T was slow enough to

c

justify this procedure. The threshold level Tl was updated for each

Ideally, the results for intermittency, and consequently, for all conditional averages should be independent of the threshold level setting. However, it is more difficult to achieve this or determine the appropriate value for r in the case of a turbulent-turbulent interface than in the case of a turbulent-nonturbulent interface. Tl should be close to T provided that signal/nojse ratio is suficiantly high.

c

The effect of vàrying the threshold level on the intermittency and conditional averages was studied in detail. Some typical crossing frequency and intermittency distributions for different threshold level settings (Tl=Tc+r, r is a parameter to compensate for above mentioned problems) are shown in Fig. 2.6. After a careful examination of intermittency distribut i.cnfor different va lues of r in eq. (2.11) the value of r was chosen to he 3.5 times the rms noise level of the

. 1· h "co l d ' f 21/2

temperature s1gna 1n t e co ree stream te . This chJice of the threshold level was found to be adequate for the present study. No significant changes in the conclusi.ons would result from choosing a slightly different threshold level.

Further insight into the structure of the mixing process can be obtained by studying the corid itionally averaged properties defined as follows Q ne 1 (2.15) 1 ne 1 _(2.16)

(I-I )

ne

22

where Q is any property being studied. the quantities Q Q are nc' c usually known as the zonal averages ( Kovasznay et al. 1970). The conventional averages are related to the zonal-average as

Q

=

I

Q

+ (1-1 )

Q

nc nc nc c

(2.17)

Fluctuations in velocity were measured with respect to the conventional

mean velocity for the purpose of conditional averaging. This is appropriate if one wants to determine the contribution from the labelled fluid ('cold' or 'not-cold') to the conventionally averaged flow

property .. On the other hand, if the fluctuations are measured with respect to the zone-mean ve I'ocity , one can obtain information on the structure of turbulence within the cold flu:id or the mixed fluid ( Fabris 1979). Such an analysis also was performed in the present study, in connection with the determination of the ensemble averaged flow properties. This will be discussed in Chapter 5.

Finally, it should be pointed out that the analysis of the large scale mixing process using the ' heat tagging' technique is based on the assumption that temperature and velocity ( momentum) interfaces coincide with each other. This assumption is valid only for Prandtl number close to unity. This condition is.reasonably well satisfied in the present case since the flow is fully turbulent in the interfacial region.

FlOW HEATED SIDE NICHROME HEAT G WIRES AT 8mmSPACING TENSIONING SPRING

Figure 2.1. Experimental arrangement

MEASURING PROBE8

:Z:::ERENCE TEMPERATURE

PROBE

~--UC%,J-- -UNOISTURBEO FLOW T, SHEAR LAYER -y +y 'HOT FREESTREAM' Tmox SHEAR LAYER UNDISTURBED FLOW T, 24 I I \ \

,

"

" "'~,,, ----

.

_

-

--t:;_T-~

_,

"

,

_,

,

I I I 05Wo : ----l:ITma.--~ I t I ------_._------1 _'

_--.:»>

VARIAC

,----j AUTOTRANSFORMER

20 A. 140 V

PITOT TUBE MEASUREMENT STATION

TEMPERATURE PROBE CROSS WIRE PROBE

2 REFERENCE THERMISTOR REFERENCE TEMPERATURE~----~ PROBE FREE STREAM DlGITAL THERMOMETER ( a) (b)

Figure 2.3. Schematic diagram of Ca) experimental setup, wire arrangement

Cold-~.~---~

V'l I- _4.11 ~ > (J) <

....

al 4.8

t

•

v

T

1

t

Figure 2.5. Schematic represantation (a) of the flow dynamics in the intermittent region, (b)Threshold setting for cold/not-cold discrimination and conditional sampling

o

ltS

28 ( a)

o.o~

~

~

~

~

~~

~~~

o.

2. 4.

B.

y/8

B.

10.

12.

3

0

0.

... 200.

(b)

O~~ __

~_% __ ~ ~ __ ~ __ ~~~~

o.

B.

y/8

Figure 2.6. Variation of (a) intermittency and (b) crossing frequency distribution for various threshold settings. T1=Tc+r; O,r=r

o

;V,r=O.9r

o;

<>

,r=O.8rO;6,r=1.lrO; o,r=1.2rO ( rO=3.S

21/2 21/2 . 1 1· h f )

te ,t e temperature noise eve an t e ree stream

CHAPTER 111

EXPERIMENTAL RESULTS - CONVENTIONAL AVERAGES

3.1 Longitudinal Mean Velocity

The distribution of the mean longitudinal defect velocity in the wake is shown in Fig. 3.1 (b) for x/a from 5 to 260. These results were obtained in the absence of any heating of the wires. The defect velocity profiles at different stations have been normalised by

Wo

and b, and it can be observed that they exibit near-selfsimilarity for x/9>25. This distance coincides with the termination of the near-wake as defined by Ramaprian, Patel and Sastry (1982). It is seen that the self-similar velocity profile is desciibed reasonably well by a Gaussian distribution (see Sastry 1981).

The growth of the wake width and decay of the maximum defect velocity with downstream distance, representing developments of the characteristic length and velocity scales, are shown in Fig. 3.2. The wake width is measured in terms of the half width b, defined as the distance between the two points in the wake where the defect velocity is equal to half the local maximum defect. This is illustrated in Fig. 3.1 (a). The growth of the wake width and decay of maximum velocity defect in the asymptotic far-wake are given respectively by (see Sastry 1981)

2 (.Q.) x 6

0.355

e

U 2 (~) =

_0.402

e

x W 0 (3.1) (3.2)

The measured rates are seen to approach the asymptotic behav ior at rhe downstream stations. It is also seen that these races are in g,ener[Jl qualitative agreement with the measurement·s of Ch.evray and Kovaznsy

(1969), Pot (1979) and Sastry (1981).

3.2 Turbulence Structure'

3.2.1 Reynolds Stresses

The distribution of Reynolds stress components u • v ,2 2 and -uv across the wake are shown in the usual similarity coordinates in Fig. 3.3 - 3.5. It is seen that the turbulent stresses u2• v2 continue to evolve even upto the last measurement station xj9=260. A closer approach to asymptotic state can be observed from the data for uv at the

last two stations xj8=210 and xj8=260. This is much further downstream of the point (xj8=25) at which the mean defect velocity distribution attains near-selfsimilar distribution. The evolution of the turbulent stresses along the developing wake is similar to that observed by Chevray and Kova5znay (1969), Pot (1979) and Sastry (1981). The present data seem to support the earlier conclusion of Ramaprian et al. (1982), that it takes about 350 momentumthicknesses for the wake to reach practically the fully developed asymptotic state. This slow evolution of the turbulent flow properties is to be expected since the turbulent structure, especially the large scale structure, evolves more slowly than the mean flow structure.

The variation of the normalized maximum she ar stress along the developing wake [uv]_mjWo2 is shown in_. Fig 3.5 (b). Even though it has

not quite reached an asymptotic value at the last (x/e=260) station, the value of 0.041 at this station is seen to be close to 0.048, which is the value suggested by Pot's (1979) data (or the value of 0.045 observed by Narasimha and Prabhu ,1972) as wel! as by the linear-wake theory

(Schlichting, 1968).

Another important quantity associated with the asymptotic turbulent wake is the nondimensional eddy viscosity "t/U ee, which (under

assumption of constant eddy viscosity across the wake) is independent of the Reynolds number at the trailing edge and has a universal constant value of 0.032 (see Narasimha and Prabhu,1972) or 0.035 suggested by the data of Pot. The present data shown in Fig. 3.6 (a) approach a value of 0.032 at the last measurement station.

Based on the above mean flow and turbulence data, it is therefore reasonable to assume that the wake being studied is indeed developing 'normally' _{into the asymptotic far-wake at least in respect of} conventionally averaged flow properties. Freestream turbulence generated by the heating wires does not seem to have affected the overall behavior of the wake in any significant manner.

3.2.2 Anisotropy Parameter

The present measurements can be used to obtain some additional information on the turbulent structure of the developing wake. One of these is the so called 'anisotropy parameter' u'/v'. This parameter is a measure of the degree of anisotropy in the turbulence structure in the flow. It is equal to unity in isotropic turbulence. It has a value

greater than unity and can vary from point to poi.nt;in a shear flow. lts average value in a turbulent boundary layer is about 1.7, as seen

from data of Klebaroff, reported in Hinze (1975). The large value is due to the fact that t.urbu Lerrt energy produced by the action of the Reynolds shear stress on the mean velocity gradient passes

preferentially into the u-fluctuations from which it gets redistributed

over the v and w modes by pressure fluctuat ions . Figure 3.6 (b) shows

the variation of the average value u' /v' (averaged across the wake)

along the wake in t.hedifferent flat-plate wake experiments. It is seen that all the experiments indicate the same trend, namely a decrease in

the value of u' /v'.along the developing wake. The data suggest that all

the wakes would eventually reach near-isotropic asymptotic state. The

data of Townsend at 500-950 diameters downstream in the wake of a cylinder indicate a value of about 1.06 and data of Pot 1.07 at distance

x/a = 950. The flat-plate wake thus evolves from a boundary-Iayer like structure to a 'universal' near-isotropic far-wake structure.

TIlenear-isotropy in the far -wake is essentially due to the absence of any significant product ion of turbulent energy. The turbulence in the wake

is largely characterized by redistribution of the turbulent energy among

the u, v and w modes and viscous dissipation.

3.2.3 Diffusion Fluxes and Transport Velocities

. 3

The triple correlation u , shown in Fig. 3.7, represents the transport

of u2 in the streamwise direction by u-fluctuations. Positive va lues of

3

mean velocity U are confined to the vicinity of the centerline, while negative values in the outer region imply prevailing slow moving fluid.

The diffusion fluxes of u2 and v2 by the turbulent fluctuation v

2 3

u v and v . across the wake are represented by the triple produets

Their evolution along the wake is shown in Fig. 3.8- 3.9. The term uv

2

shown in Fig. 3.10 represents, likewise, the turbulent transport of uv by the v fluctliations. These figures suggest that these fluxes also approach self-similar distributions at large distances downstream, even though they have not yet actually attained such distributions at the last measuring station, name1y xj9=260.

A

comparison with the cylinder-far-wake data of Townsend (see Hinze,1975) shows that these distributions are in reasonable agreement with each other. This level of agreement (within 20 percent) is considered to be within the experimental uncertanties in the triple correlation measurements in the far-wake.

From the measured diffusion fluxes, one can estimate the so called transport velocities associated with the diffusion of turbulent kinetic energy and Reynolds shear stress in the lateral direction. These veloeities are usually defined as (Bradshaw,1967)

2

3"

-2-u v + v + w v -2 -2 -2 u

+

v

+

w

(3.3)

-2 uv (3.4)

34

However, in the present experiment the w component was not measured. Hence, the following alternative definition will be used in the present

study -2-

3"

u v

+

v u2

+

v2 (3.5) V t = ---2 uv (3.6) uv

It is assumed that equation (3.3) can be used as a measure of the diffusion velocity for the turbulent kinetic energy in the lateral

direction. The distributions of Vk and Vt are shown in Fig. 3.13 The

estimation of V,!:near the centerline is difficult since uv is zero at

the centerline. Vk, however, is wéll behaved everywhere. The figures

show that these velocities exibit definite trends in their distribution.

Considering the right half of the wake, it is seen that Vk and V,!:are

negative in the central region ( y/b<O.2) and positive in the outer

region of the wake. Thus the wake is characterized by outward diffusion

in the outer part and inward diffusion in the inner part. This is consistent with the gradient diffusion model, if it is recognized that

the crossover point corresponds roughly to the point of maximum shear

stress and maximum turbulent energy in the wake.

In order to study the evolution of the diffusion process along the

the inner and outer regions respectively in the following manner (these definitions are for the right half of the wake):

I

Yo (u v2 + v3) dy

Vki

0 (3.7)

I

Yo (u

2"

+ v ) dy

2"

0 00

I

(u v2 + v3) dy '"

_Vko

Yo 00 (3.8)

I

Cu2 + v2) dy Yo

where

Y=Yo

corresponds

evolution of these velocities along the wake are shown in Fig. 3.11 and to the point of maximum shear stress. The

Fig. 3.11. Several observations can be made from the figure. First, it is seen that the transport velocities (normalized with WO) increase in the downstream direction and tend to approach asymptotic values far downstream. The outer region of the wake is characterized by strong outward diffusion velocities. These velocities are of the order of O.lWO indicating significant diffusive turbulent motions in the outward direction. _{The inner region, by contrast, shows relatively small} diffusion veloeities. These velocities are directed towards the center of the wake. The present results also suggest that turbulent diffusion plays a much greater role in determining the flow structure and evolution in the case of the wake than in the case of the boundary layer.

36

3.3 Summary of the Conventionally Averagep Results

The results presented in this chapter have shown that the flow configuration studied is reasonably representative of a standard flat-plate wake developing into an asymptotic far-wake. It should be mentioned that the wake studied, has not yet fully attained the far-wake state at the last measurement station x/8=260. Any subsequent evolution is likely to be completed within x/8!:350 as suggested by Ramaprian et al. (1982) but is unlikely to introduce drastic changes in the detailed structure of the flow. The region 25<x/8<260 can therefore be expected to exibit all the essential features of a developing wake. Lastly, the detailed results of ReynoIds stress and turbulent diffusion fluxes presented in this chapter are useful in their own right as data on the developing symmetrie wake behind a flat plate.

Quantitative differences (between different wake experiments) in the developing region of the wake are the result of different initial conditions in the experiments, as reflected in the different virtual origins for each flow. This fact was also indicated by the results of recent wake calculations by Chen and Patel (1984). These calculations were based on the solution of partially parabolized Navier-Stokes equations. The specific geometry of the trailing edge and the turbulent structure of the boundary layer near the trailing edge have astrong effèct on the interaction in the near-wake region and hence alter the virtual origin. Therefore, asymptotic conditions in the wake may be reached at different distances downstream (x/8>350) in different

experiments on 'flat plate' wakes. It is generally believed that all these flows would eventually attain the same universal asymptotic structure. The findings from the present study, however, have raised some questions regarding the existeance of such a completely universal structure. This aspect will be discussed later.

( a) y < 1••

r---

---:-

a.~

+

a

38

"

o 31: <, 31: v (b)

o

rr

+

o

*

a..

(Y/b)

Figure 3.1. Velocity distribution Ca) definition sketch, eb) W/W

o

across the wake.x/e:*,5;O,10i+,20i9,35iÄ,70;<>,140;~,210i

0,260

o

+

a

q

+

a

o

-It + + +0. a

+

cr

.S 1•• 1.S 2.'

100. ( a) 0 A 80. N """""'_<D <, 60.

.o

'-' 40.

o.

50. 100. 150. 200. 250.

X/8

300. 80. (b) 60. 40.

o

A 20. VA

o

o

50. 250.

o.~

~~

~

~

~

~

~

O. 100. 150. 200. 300.

x/-e-Figure 3.2. Variation of the wake characteristic scales along the flow: Ca) half-width, Cb) maximum wake defect.~,PotjV,Sastry; <>,Chevray

&

Kovasznay; o,Andreopoulos

&

8., N ,(a) 0 3: <,

0°

IS

4.

0

'

0 °

,0 0 00

SCJ

O 0 °0

°

~ _A A

°

,AD OA

°

2. 0:sOA 0

°

0A A

_tP

Aa A A ~ ~ .0. -1.'5 -1..0 -.5 '.0. .0 .5 1..0 1.5

ylb

'1,2. I I I 'I I N

_ie.

(b) 0

r-

V

-3: _Vv V <, V V

<>

Is

_8.

<><>

<>

f-v<>

v<>

<>v

_

<>v

Ol Ol 0 Ol Ol

<>

V V 'Ol 0

_<>

Ol

<>

_~ ..-i _8.

r-

_V" '

..

<><>

..

+ +

v

_

'

+ +

..

₊ + ti

..

~ + ti .. + + ~ 4.

-V + + +

-v

+ + ~ ti(> ₊+ + ,'V + ~ ~ ï V ~ ,

,

v v v<>"<>"~

+" ..

<>~V>'fJ

'

(1<:>.. + + OI"".,.Q

.o.~~*_+_+_~I ~I ~I ~I~ ~I+_+__+~

-1.'5 -1.'.0 -.5 .0•.0 .5 1..0 .1.5

y/b

Figure 3.'3. Distribution of u2jW~ across the wake at stations: (a)

x

/

a=

2.5 ( a) N 0 _2.0 3: <, 0°0 00

I~

080°00 ° ° ° 0 _OA

o

BA A 0 0 .,.; _{A A .} ° A A 0 0 0 ° ₀A Ac° A ~

CE

~ -1.0 -.5 0.0 .5 1.0 1.5

y/b

10. I I I I I (b) N 0 _8. 3:

f-vVVVvVVv

-<,

I~

v

_<><> <>

v

<><><><> 0 6.f- _{<><> <>} -0

_v<>

_<>v

.,.; 4. f-2. I--1.5

v

<> <>v

-..

..

_..

v

..

--1.0 _{1.0 1.5}

y/b

Figure 3.4. Distribution of v2

/w;

across the wake at stations: (a)

x/a=

5-35, (b) 70-260. Symbols as in Fig.3.3

j

Figure 3.5. -'6.L..,_ __ ..l...-__ ..l- __ ...L-_-~----'--_---l -1.5 -1.0 -.5 0.0 .5 1.0

y/b

(b) 0.00~ ~ ~ -4 ~ ~

O.

80. 120. 180. 240.

x/.e-1.5 300. - 2

Variation of Ca) -uv/Wo across the wake at stations x/a=

- 2

5-260, (b) maximum shear stress iuvlm/Wo along the wake. Symbols as in Fig.3.2

> <, :J Figure 3.6. .04r---.---r---r---r---r---~ (a)

O.OO~

~

~

~

~

~ __ ~

O.

100. 200. 300.

x/a-2.0r---r---,---r---r---.- __ ~ (b) o

.

5~

~

~

~

~

~

~

O.

100. 200. 300.

x/.g..

Evolution of (a) normalized eddy viscosity v

IU

e

and

t e

(b)anisotropy parameter u'

Iv'

along the wake. Á,Pot; o,present;-ïse1f-preserving wake

.4 ('J 0 3: <, .2

I~

0 0 "C'"'I _0.0 1. ('J 0 3: <,

I~

o.

0 0 orl -1. -.2 I I I I (a)

,__

-Os

00 ~[J t:.. _t:..t:J[J '&l. v 0

-~~

~ t:.. t:. _t:. liP

'b

t:.. [J t:. t:. t:..cx> ,__ [J [J

-t:.

°

&

[J 0 [J DO

°

I ,.. I [J I I -.4 -1.5 -1.0 -.5 0.0 .5 1.0 1.5

y/b

I I I I (b) ~~~ iN ~' ~ n+ "'tI> ~v Q ~w V + + <> + _~ vN N N + V v+ + "+++M<> <> ++ ~ V N V ,__ NO.V _N

-<><> <>Ne v<> VV _v v I I I I ...,.2. -1.5 -1.0 -.5 0.0 .5 1.0 1.5

y/b

Figure 3.7. Distribution of ~/w~ across the wake at stations: Ca)

x/

a=

5-35, Cb) 70-260. Symbols as in Fig.3.3

.3 (I] 0 .2

::s::

<,

I~

.1

0 0 ..-1 _0.0

-.1

-.2 -.3 1.0 (I] 0

::s::

<, .5

I~

0 0 ..-1 _0.0 -1.0 -1.5 I I I I (a) f- ₀₀

-~°cP

f-~

-08

p

AlJ:D A o.

-èDA...

,... ~AA Dl.lAJ 00

cf

_A

DO f- 0

-0 rIlJA 00 0 f-

-I I I I -1.5 -1.0 -.5 0.0 .5 1.0 1.5

y/b

-.5 I I 'V r V (b) ~<><>

..

..

f-8

-<> ..

..

<>Cf.

_..

+ + + + ~ "t7+ +i <;JI + '!l <>+ + ~...n

·v"'~

+ ~

.

...

.". V +

~+""v

~ 'V

v"

+ + ~

..

+ ..<>

o

f- ~ V

-V ....

v

o ~ V I I I I -1.0 -.5 0.0 .5 1.0 1.5

y/b

Figure 3.8. Distribution of ~/w~ across the wake at stations: (a)

.2~----~,----~,---,----~,r---r-,--~

( a) .1_

-- .1.... I 1.0 I .5 1 -.5 -.2 I -1.5 -1.0 0.0 1.5

y/b

1.2 ('I] 0 3: <, .8

I~

0 0 ..-f 0.0 I. I 1 I (b) Vv v

e

....

~ ~~V

-o

~ V _MMM

J

"

"

$<>"+++++

"?

.6 "+ + + ....,r,u• y~ + ++.4;

,.

...

V .~/++++" V V

<:\t

"~V V ~"" ~

....

_o

~V

-V ~~ vV I I I I -.8 -1.2 -1.5 -1.0 -.5 0.0 .5 1.0 1.5

y/b

Figure 3.9. Distribution of ~/W~ across the wake at stations: Ca)

x/a=

5-35, eb) 70-260. Symbols as in Fig.3.3

.2 (IJ 0 3: <, .1

I~

0 0 ...-1 _0.0 I I I I ( a)

-

-0

_'8

Jt:O °A 0 _n

...

ct!

A 0

't

»r:

OCJA 0

61

0 !:lA

è

A 0 AA 00 t- OA

°

-0 0

°

ól

0 0 0 ..l I ..l I -.1 -.2 -1.5 -1.0 -.5 0.0 .5 1.0 1.5

y/b

1.0r---.---~----,_----ïï----_r----_,_{I I I I} (b)

-ti -ti -.5_

--1.0 -.5 0.0 .5 1.0 1.5

y/b

Figure 3.10. Distribution of

~/W6

across the wake at stations: Ca)