AN INVESTIGATION OF TRANSITION AND TURBULENCEIN OSCILLATORY STOKES LAYERS

by

Rayhaneh Akhavan-Alizadeh

B.S., California Institute of Technology (1980)S.M., Massachusetts Institute of Technology (1982)

Submitted to ths Department of Mechanical Engineeringin Partial Fulfillment of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGYJUNE, 1987

c Massachusetts Institute of Technology, 1987

Signature of Author

Certified by ,

Certified by _.

Dep ttment of Mechanical EngineeringMay 1987

Professor loger D. KammThesis Supervisor

Profe sor Ascher H. ShapiroPfs Thesis Supervisor

Accepted byProfessor Ain A. Sonin ARCHIVES

Chairman, Department Committee on Graduate Studie§:~TE:,NLr.G;:;.T,

JUL 0 2 1987

LIBRARIES

\..... _

- _-

A] INVESTIGATION OF TRANSITION AND TURBULENCEIN OSCILLATORY STOKES LAYERS

by

Rayhaneh Akhavan

Submitted to the Department of Mechanical Engineeringin May, 1987 in partial fulfillment of the requirements

for the Degree of Doctor of Philcsophy inMechanical Engineering

ABSTRACT

Transition to turbulence in bounded oscillatory Stokeslayers is investigated experimentally by LDV measurementsfor flow in a pipe, and numerically by direct simulationsof the Navier-Stokes equations for flow in a channel. Theexperiments indicate a transitional Reynolds number based onStokes boundary layer thickness of Re-500. The resultingturbulent flow is characterized by the sudden appearance ofturbulence with the start of the deceleration phase of thecycle, and reversion to "laminar" flow during theacceleration phase. It is proposed that generation ofturbulence in this flow is due to turbulent "bursts" similarin nature to those that occur for steady wall-bounded shearflows. Furthermore relaminarization of the flow during theacceleration phase is attributed to cessation of "bursting"because of the favorable pressure gradients that are presentduring this period.

Investigation of the stability of oscillatory channelflow to various infinitesimal and finite-amplitude two- andthree-dimensional disturbances reveals that; (a) Aninfinitesimal perturbation introduced in the boundary layerwould, over the course of a few cycles migrate to the centerof the channel where it would decay with viscous time scalesas predicted by a Floquet analysis. However, as long as thedisturbance is still localized within the boundary layer, itwould amplify with inviscid growth rates as predicted byquasi-steady theories. (b) No finite-amplitude two-dimensional equilibrium states were found; however, quasi-equilibria (with viscous decay rates) were observed attransitional Reynolds numbers. (c) The secondary instabilityof these two-dimensional finite-amplitude quasi-equilibriumstates to infinitesimal three-dimensional perturbations cansuccessfully explain the critical Reynolds numbers andturbulent flow structures that were observed in experiments.

Thesis Supervisors: Prof. R.D. KAMMProf. A.H. SHAPIRO

Thesis Committee : Prof. D.J. BENNEYProf. C.F. DEWEYProf. A.T. PATERA

-3-

ACKNOWLIEDGENTS

I would like to extend my sincere gratitude to my

thesis supervisors Professors Roger Kamm and Ascher Shapiro

for their guidance , patience and support throughout the

period of my stay at MIT. I have always felt that no other

group could have given me as much opportunity for

independent growth as was provided by them. I would also

like to acknowledge the invaluable help of Professor Tony

Patera without whom the computational aspects of his work

would not have been possible. Thanks also to Professor Steve

Orszag for helpful discussions in the earlier stages of this

work.

The friendship and assistance of all the members of the

Fluid Mechanics Lab. has provided a wonderful environment to

work in. In particular, I would like to thank Dick Fenner

for all his help in constructing the experimental apparatus.

Finally, the love and support of my family and my good

friend, Martin made this all possible.

-4-

TABLE OF CONTENTS

PAGE

TITLE PAGE ............................................. 1

ABSTRACT ........................................... 2

ACKNOWLEDGMENTS ........................................ 3

TABLE OF CO NTENTS ...................................... 4

I. INTRODUCTION ...................................... 5

II. EXPERIMENTS....................................... 16

2.1 Experimental Methods ......................... 16

2.2 Experimental Results ......................... 24

2.2.1 Calibration Run ........................ 242.2.2 Turbulent Flow Regime .................. 252.2.3 Effect of Re and A ..................... 48

2.3 Discussion ................................... 78

III. NUMERICAL SIMULATIONS . . ........................... 84

3.1 Problem Formulation and Numerical Methods.... 84

3.2 Small Amplitude Disturbances (Linear Theory) 92

3.2.1 Two- and Three-Dimensional Disturbances 943.2.2 The Linearized Problem ................. 963.2.3 Effect of Initial Conditions ........... 983.2.4 Results ................................ 104

3.3 Finite Amplitude Disturbances ................ 121

3.3.1 Two-Dimensional Disturbances ........... 1223.3.2 Three-Dimensional Instability......... 1343.3.3 Secondary Instability and Transition... 141

IV. SUMMARY AND CONCLUSIONS ........................... 147

REFERENCES............................................. 152

-5-

I. INTRODUCTION

The flow induced in a semi-infinite body of fluid by a

plate undergoing sinusoidal oscillations in its own plane

was originally investigated by Stokes [1851). If the plate

moves with velocity Upcoswt, then the unidirectional flow,

parallel to the plate, in the surrounding fluid is given by

u(z,t) - Upexp(-z/6)cos(wt-z/6), where 6v2/w; is a measure

of boundary layer thickness. The Stokes layer is the

representative boundary layer in a large class of problems

involving periodic flows within or around rigid boundaries.

Our concern is with the stability and turbulent motion of

the bounded oscillatory Stokes layer.

Previous work on transition in purely oscillatory flows

is rather scarce. Here we make the distinction between

"purely" oscillatory flows (about a zero mean) and flows

modulated about some non-zero mean. The two problems are

quite different, as the instability of flows modulated about

a non-zero mean is usually associated with the (steady) mean

flow. In this case, the modulations can usually be accounted

for by perturbation methods about the mean flow. This

contrasts with the situation for purely oscillatory flows

where the basic flow is unsteady.

-6-

A number of experiments have been reported in

literature on the problem of transition in purely

oscillatory flows. As these experiments are generally

performed in bounded geometries (such as in pipes or in

channels), a new parameter describing the ratio of the

physical dimensions of the flow (for example, pipe radius or

channel half-width) to the Stokes boundary layer thickness

is introduced - the problem; A-d/2vw/2v . Alternatively, A

can be thought of as the ratio of the diffusive time scale

of the flow to the period of oscillation. Note also that A

is related to the parameter known as Womersley number,

r=d/2v , by the relation A-/ . All investigators agree

that for sufficiently large A (2£A) transition is governed

by a Reynolds number that is based on the Stokes boundary

layer thickness. Furthermore, a value of Re6 on the order of

500-550 is more often cited as the transitional Reynolds

number. Here, Re =-U06/, where Uo is the amplitude of cross-

sectional mean velocity and 6-vFi/w is the Stokes boundary

layer thickness. For example, for flow in a straight pipe

Sergeev [1966] found the transitional Reynoldz number to be

Re -500 for the range of 2.8<A<28; while Hino, et al. [1976]

and Ohmi, et al. [1982] each found the transitional Reynolds

number to be Re 550 for the range of 2A<6 (Hino, et al.)

and 2<A<29 (Ohmi, et al.), respectively. In a different

geometry, Li [1954] observed turbulence for Re6>565 for flow

in a channel with an oscillating bottom. The value of 500

for the transitional Reynolds number, however, is not

-7-

universal; for example, Merkli and Thomann [1975] observed

that the flow in a resonant pipe became turbulent at a Re6

cf 280 (30<A<50); while Vincent [1957] and Collins [1963]

found the flow due to periodic gravity waves on a smooth

plate to become turbulent at Reynolds numbers of 110 and

160, respectively.

Of the various experimental studies, the works of Hino

and coworkers are the most detailed. They distinguish four

classes of flows in their experiments: (I) laminar flows,

(II) distorted laminar or weakly turbulent flows, which are

characterized by the appearance of small amplitude

perturbations in the early stages of the acceleration phase,

(III) turbulent flows, which are characterized by the

explosive appearance of turbulence with the start of the

deceleration phase and which recover to "laminar" flow

during the acceleration phase, (IV) turbulent flows that

remain turbulent throughout the cycle. Flow regime (IV) has

not been observed by any of the investigators, although

Ohmi, et al. observed that as Re was increased, turbulent

bursts occurred in the later stages of the acceleration

phase as well as in the deceleration phase.

In the experiments of Hino, et al. and Ohmi, et al.,

the transitional Reynolds number of 550 marked the onset of

flow regime (III) independent of A. Flow regime (II) was

observed by Hino and coworkers over a range of Reynolds

-8-

6 6 6numbers Re mitRe 550, where the value of Re min was

dependent on A; for example, for A6.2 flow regime (II) was

observed for 70Re (550 while for A-1.9 the same flow regime

was observed for 460<Re 550. Ohmi, et al. on the other

hand, found that onset of flow regime (II) in their

experiments agreed well with the critical Reynolds number of

Re -280 which was proposed by Merkli and Thomann. In other

words, flow regime (II) was observed by Ohmi, et al. for

280<Re <550 independent of the value of A. Merkli and

Thomann [1975] themselves do not make any distinction

between flows of type (II) and (III), and therefore it is

not possible to determine in giving the value of 280 for the

transitional Reynolds number which type of flow they were

referring to.

In general, when reference is made to turbulent

oscillatory flows, the regime (III) described above is

intended. The details of the flow structure in this regime

was studied by Hino and coworkers [1983] for flow in a

rectangular duct at Re =876 and A12.8 . The profiles of

mean velocity, the turbulence intensities, the Reynolds

stresses and the turbulent energy production rate were found

to be completely different between the accelerating and

decelerating phases, and in either case quite different from

turbulence in flows which are steady in the mean.

Nevertheless, they concluded that the basic processes such

as ejection, sweep and interaction towards and away from the

wall are the same as those of 'steady' wall turbulence.

-9-

Theoretical investigation of the stability of Stokes

layers is also relatively new. The mathematics of the

problem, even for the 'linearized' case is quite

complicated. The complexity arises because in the presence

of a time-periodic basic flow, the linearized equations for

the disturbances have coefficients that are periodic

functions of time; consequently, the time dependence of the

perturbation is not separable and the method of normal modes

is no more (formally) applicable. Two classes of solutions

have been suggested in the literature for this problem;

analysis by Floquet theory and solution based on the

assumption of quasi-steadiness. Von Kerczek and Davis [1974]

noted that since the linearized equations have coefficients

that are periodic functions of time, the results of Floquet

theory [Coddington and Levinson (1955), Yih (1968)] can be

employed to predict that in the periodic steady state the

disturbance stream function should be of the form

(x,z,r)=e x+Ar(z,)+complex conjugate. Here, r=wt is the

time normalized to the period of oscillation of the basic

flow, a is the streamwise wavenumber (with x normalized to

the Stokes layer thickness), A is the Floquet exponent and

represents the net growth (and phase change) of the

disturbance field from one cycle to the next, and is a

periodic function of time with the same period as the basic

flow and represents the variation of the disturbance field

within each cycle. The disturbances experience a net growth

from one cycle to the next if Real(A)>O . Von erczek and

-10-

Davis used a Galerkin technique to numerically integrate the

resulting system of equations for the geometry consisting of

an oscillating plate with a second stationary plate placed a

distance of eight boundary layer thicknesses away. For the

range of wavenumbers 0.3<a<1.3 and Reynolds numbers

O<Re <800, they found A to be real and in all cases

negative, indicating a net decay of disturbances from one

cycle to the next. The periodic part of the stream function,

O(z,r), was also found to experience no considerable growth

within a cycle; thereby ruling out the possibility that

within a cycle the disturbances may grow to such an extent

that the flow might become nonlinearly unstable. Therefore

von Kercz.k and Davis concluded that the Stokes layer is

absolutely stable to infinitesimal disturbances for the

range of parameters investigated and possibly for all

Reynolds numbers.

As a corollary to their work, von Kerczek and Davis

point out that since the principal Floquet exponent A found

in their numerical results was in all cases 0(1), one must

conclude that the time scale for the growth or decay of

linear disturbances is of the same order of magnitude as the

time scale of oscillation of the basic flow. This would

serve as a warning against the use of quasi-steady theories;

as the quasi-steady assumption would be valid only if the

disturbances varied much faster than 1/w.

-l-

Hall [1978] investigated the linear stability of the

infinite Stokes layer using Floquet theory, and found this

flow to be also stable for all Reynolds numbers investigated

(O0Re <300). In addition to a discrete set of eigenvalues,

the Orr-Sommerfeld equation for the infinite Stokes layer

was found to also have a continuous spectrum of eigenvalues

for all Reynolds numbers. The most dangerous modes were

associated with this continuous spectrum. Hall also

considered the effect of introducing a second boundary a

long way from the Stokes layer, as was done in the problem

addressed by von Kerczek and Davis. For the latter problem

only a discrete set of eigenvalues exists. Hall showed that

the least stable disturbance of the 'finite' Stokes layer

was not the same as the least stable discrete eigenvalue of

the infinite Stokes layer; rather the eigenmodes of the

'finite' Stokes layer were found to 'merge' into the

continuous spectrum of the infinite Stokes layer as the

separation between the two plates tended to infinity. Thus

he concluded that the disturbances given by von Kerczek and

Davis have no connection with those of the infinite Stokes

layer.

Despite the objections raised by von Kerozek and Davis

against the use of quasi-steady theories in studying the

stability of Stokes layers, one can show that quasi-

steadiness is a fair assumption if one is considering the

inflectional instabilities of the Stokes layer. First, note

-12-

that at all time during the cycle the Stokes layer has an

infinite number of inflection points, each satisfying

Fjortoft's criterion for instability. For example, for a

flat plate oscillating with velocity Upcoswt there is an

inflection point on the plate (y/-0=O) at wt=O (as well as

other ones at y/6=nr) which move away from the wall as time

progresses within a cycle. At asymptotically large Reynolds

numbers, these inflection points may become inviscidly

unstable. Since the time scale for the growth or decay of

inflectional instabilities (6/Up) is much shorter than the

period of oscillation (11/w); (6/Up)-/Re (l11w), one can

neglect the time dependence of the base flow and study the

stability of the flow by solving the (inviscid) Rayleigh

equation for a succession of quasi-steady velocity profiles

for different times during the cycle. This approach has been

adopted by Cowley 1985] and by Monkewitz and Bunster

[1985]. One problem with this type of analysis is that the

normal modes, which are associated with the inflection

points, completely change character from one cycle to the

next as they move away from the wall. This makes it

difficult to compare these results to those of the Floquet

theory (which assumes the flow has reached a periodic steady

state). Monkewitz and Bunster suggest a scenario where a

mode associated with an inflection point moving away from

the wall would pass its energy to a mode associated with an

inflection point closer to the wall, before it gets damped

out in the free stream where inflection points are

exponentially weak.

-13-

Cowley [1985] carried out an inviscid quasi--steady

analysis of the stability of the Stokes layer for

asymptotically large Reynolds numbers where the effects of

interaction between different modes was included in the

analysis. His results showed that disturbances with

wavelengths comparable to 6 can grow. The most significant

growth rates were observed between r/4 and r/2; i.e, ust

before the start of the acceleration phase of the cycle.

None of these modes were found to evolve in such a way as to

return to their original state (multiplied by a constant

factor) after one cycle, as would be required by Floquet

theory. Therefore, although these results are not

necessarily incompatible with the results of the Floquet

theory, no direct comparison between the two theories is

possible either.

In short, the present status of knowledge with regards

to the stability of flat Stokes layers can be summarized as

follows. On the one hand, experimental results show that the

flow undergoes violent transition to turbulence with the

start of the deceleration phase of the cycle at Reynolds

numbers on the order of 500. Theoretical studies, on the

other hand, have so far been restricted to infinitesimal

disturbances and either predict the flow to be absolutely

stable in the periodic steady state (Floquet theory), or

consist of analyses which are valid for asymptotically large

Reynolds numbers and predict transient instability but at

-14-

phases during the cycle opposite to where turbulence is

observed in experiments (inviscid quasi-steady theories). In

either case no critical Reynolds number can be identified

for comparison to the experimental value for the

transitional Reynolds number. Clearly a large gap exists

between theory and experiment in the problem of stability of

flat Stokes layers.

In the present work we have tried to bridge some of

this gap by performing simultaneous experimental and

numerical studies of bounded oscillatory Stokes flows at

transitional and turbulent Reynolds numbers. The experiments

were performed in circular pipes. We felt these experiments

were necessary; because even though a number of experiments

have been reported in literature on the stability of

oscillatory Stokes layers, the only detailed work is that of

Hino and coworkers 1983] which is restricted to a single

flow condition, Re =876, A=12.8 . As we shall see, our

experimental results are for the most part in agreement

with, and serve as a confirmation of the earlier work of

Hino and coworkers. We believe this confirmation by itself

is quite important. Furthermore, comparisons between the two

sets of experiments has allowed us to draw conclusions upon

key features common to both sets that were neglected (or may

have been rejected as noise) by Hino and coworkers.

-15-

To identify the nature of disturbances that could

explain the experimentally observed values of transitional

Reynolds numbers, and to gain an understanding of the

physics of the transition back and forth between the laminar

and turbulent states that is observed in experiments at

values of Reynolds number above Retrans ; we have performed

direct numerical simulations of the full, time-dependent

three-dimensional Navier-Stokes equation for oscillatory

flow in a channel. The channel geometry was chosen for the

ease of computation and in light of the close agreement

between our experimental results for flow in a pipe, and the

experiments of Hino, et al. which were performed in a

channel. The channel flow is subjected to initial conditions

that consist of various combinations of two- and three-

dimensional infinitesimal and finite-amplitude disturbances

and the time evolution of the disturbances are followed. The

disturbances that most closely reproduce the experimentally

observed features are identified, and the dynamics and

mechanisms of transition is discussed.

-16-

II. EXPERIMENTS

2.1 Experimental Methods

The experiments were performed in a quartz circular

pipe, 2 cm in inner diameter and 8 meters long using the

experimental setup shown schematically in Figure 1. The

length of the quartz pipe was chosen such that under all

flow conditions the tube length was at least three times the

stroke length of the flow in the quartz pipe; thereby

ensuring that the flow in the middle section of the pipe was

not suffering from any end effects. A tube geometry was

chosen for the experiments over a two-dimensional channel or

an oscillating plate, because of the great care required to

ensure two-dimensionality in a channel and the complex flow

systems that would be required for an oscillating plate. The

choice of a particular geometry does not seem to be too

critical, as earlier experimental studies show a close

correspondence between the structure of the turbulent

oscillatory flows in all three geometries.

The sinusoidal flow rate into the quartz pipe was

generated by a piston whose motion was controlled by an

electrohydraulic servovalve ( MOOG model 76-102 ) with

feedback control on the velocity. The sinusoidal output of a

function generator served as the input command for the

control circuitry. The quality of the sinusoidal flow rate

j

es

tb

.

·,t2o QQ3

ifI-

I

I I

I

3

o v0

v

-17-

RA

E000TOTOaO

Cd

Cdp,

0))

0.,10

-i

0

Ca

I

-1

I

I -- -

-18-

generated by this flow system was checked by monitoring the

motion of the piston with a velocity transducer (LVDT,

Schaevitz model 6L3-VTZ) and also by computing the flow

rates based on LDA measurements of the velocity profiles. In

each case an FFT fit was performed on the output. The

results of these two measurements for the flow rate of

poorest sinusoidal quality is shown in Figures 2a and 2b. As

seen in these figures, the maximum deviation from a pure

sinusoid occurs near the point of zero flow. At this point

because of some backlash in the O-ring seals on the piston,

the motion of the piston could not follow the input command.

Despite considerable effort on our part to overcome this

backlash, the problem was not eliminated. There was no mean

flow.

Velocity measurements were obtained using a DISA two

color laser doppler anemometer in which the blue (488 nm)

and green (514 nm) lines of an Argon laser were used to

measure simultaneously the axial and radial velocity

components. The LDA was equipped with a Bragg cell and was

operated in the backscatter mode. Counters were used for

signal processing. The probe volume had dimensions of

20x20x100 microns and fringes were 2 microns apart. The

front lense in the LDA system was mounted on a two-axis

linear translation stage with micrometer control of position

to within .0001". This traversing mechanism was used to

position the probe volume at selected points along a

diameter within the quartz pipe.

-19-

.6

.4

.2

-. e-E 0-. 2

-. 4

-.

7r/2 37r/2

wt

Figure 2. Cross-sectional mean velocity calculated from the

output of the LVDT that was used to measure the

velocity of the piston.

-20-

- 2 7 /2 3r/2wt

Figure 2ab Cross-sectional mean velocity calculated by

integrating the velocity profiles that were

obtained by the LDA system.

.U

.8

.6

.4

0-

-. 2

-. 4

-. 6

-.8

-1 n

-2 1-

Because the basic flow was unsteady, the time of

arrival of each doppler burst in relation to the periodic

motion of the basic flow had to be determined. This was

accomplished by a MINC PDP11/23 data acquisition computer.

The computer was equipped with an internal clock, a DMA

parallel interface board (DRV11-J), and A/D converters. In

every cycle, at a point corresponding to (positive-going)

zero flow rate into the system, the computer clock was

triggered to start counting. The trigger point was

determined from the output of the velocity transducer that

monitored the motion of the driving piston. For the first

few cycles, the time between successive triggers was

measured to evaluate the precise period of the basic flow.

This period was then used to set the clock so that every

1/240th of a cycle it would overflow and increment a

register within the computer. This register would then

indicate time in fractions of 1/240th of the cycle period.

The register was reset at the beginning of each cycle. The

data acquisition protocol can then be summarized as follows.

Once both LDA counters had verified a doppler burst (within

15 se of each other), the computer would be interrupted

via the parallel interface board. The time in increments of

1/240tb f the period of the cycle would be read from the

appropriate register, and the doppler information consisting

of the doppler frequencies, the number of fringes associated

with each doppler burst and the time would be stored in the

computer for later processing. In addition to the above

_9 --

information, the instantaneous pressure drop through the

quartz pipe (measured with a Microswitch pressure

transducer) and the flow rate through the system (as

measured by the velocity transducer attached to the piston)

were sampled through the A/D converters 240 times per cycle

and stored.

Typically on the order of 2000 doppler bursts per sec

were processed in this manner. The rate limiting factor was

the amount of seeding that could be used before the flow

became opaque.

The optical distortion of the quartz pipe was

eliminated by index-matching the working fluid to that of

quartz (nd - 1.458) using an aqueous solution of ammonium

thiocyanate. The solution was mixed until the correct index

of refraction was achieved (approximately equal weights of

ammonium thiocyanate salt and distilled water).A hand held

refractometer (Extech, model K502192) with 0-90% sucrose

range (nd-1.333-1.512) and +0.2% accuracy was used to

measure the index of refraction of the solution. Otherwise,

the solution had properties very close to water (pl.1 g/cm3

, v-1.32 centistokes) . At the measurement site the quartz

pipe was immersed in a rectangular chamber with quartz

windows which was filled with the same index-matched fluid.

-23-

A variety of seeding particles including alumina,

latex, silicon carbide and titanium dioxde were examined.

Titanium dioxide (with particles of diameter from .5 to 1

microns) was chosen over the others because of its good

signal to noise ratio and ability to follow the smaller

scales of turbulence.

-24-

2.2 Experimental Results

2.2.1 Calibration Run

To check the reliability of ouz, experimental methods,

the velocity profiles for an example of laminar flow

(Re -233, A-3.6) were measured. In the laminar regime, the

equations of motion have a closed form solution as given by

Womersley 1955] and Uchida [1956], and provide a direct

check on the experimental results. For a sinusoidal flow

rate, Q=Qocoswt, the instantaneous velocity profiles are

given by,

u(r,wt) B cos(wt+6) + (l-A) sin(wt+6)

U0 I (1-C/) + CD/)(1)where, v - A/V2 (note that x c Womersley number)

ber(c) ber(rlv) + bei(c) bei(rv W)A 2

ber () + bei 2 ()

bei(x) ber(rvV/) - ber(c) bei(rvlw)B

ber2 (n) + bei2 (c)

ber(x) bei'(x) - bei(c) ber'(c)C 2

ber2(i) + bei2(I)

ber(c) ber'(c) + bei(x) bei'(n)D

ber2 (c) + bei 2(c)

1-C/v6 tan-1{

D/v

-25-

The comparison between experiments and the theoretical

solution of Uchida is shown in Figures 3 and 4. Figure 3

shows the time evolution of axial velocity at various radial

locations. The theoretical solution predicts sinusoidal

variation at each location, however the points in the

boundary layer have a phase lead with respect to the core.

In Figure 4, the experimental velocity profiles for various

phases during the cycle are compared to the theoretical

curve. The agreement between theory and experiment is in

general very good. The maximum deviation from theory occurs

at wt-r/2, which corresponds to 'turnaround' in the motion

of the driving piston. At this phase because of some

backlash in the O-ring seals, the motion of the piston

deviated from a pure sinusoid as already discussed in

section 2.1 .

2.2.2 Turbulent Flow Regime

Hino et al. [1976] have fairly extensively investigated

the transitional regime in an oscillatory pipe flow. As

already discussed in detail in the introduction, they

observed two classes of non-laminar flows in their

experiments. One class can be identified as "distorted

laminar" or "weakly turbulent" and is characterized by the

appearance of small amplitude disturbances that are

superimposed on the laminar profile during the acceleration

-26-

-/ 2 ir/2 37r/2

wt

Figure 3. Phase variation of axial velocity at fixed radial

locations compared to theory in the laminar flow6regime. (Re -233, A-3.6)

* Z-

.1

0

C-u

_E

1::

1%

-27-

.i

.05

0

-. 05

-. 10

-. 15

-l

-1.0 -. 8 -. 6 -. 4 -. 2 0 .2 .4 .r/R r/R

Figure 4. Experimental velocity profiles at various phasescompared to theory for laminar flow. (Re -233,

A-3.6)

.1

.I

'C

-. 1

-. 1

-. 2

6 .8 1.0

An. .

I

-28-

phase of the cycle. The other type of flow is what is

generally considered "turbulent" and is characterized by the

sudden explosive appearance of turbulent bursts with the

start of the deceleration phase and recovers to laminar flow

during the acceleration phase. Our main concern in this work

is with the characterization of the latter "turbulent" type

of flow.

The stability boundary for "turbulent" flows (of the

second kind described abcve) is very well defined in our, as

well as other investigators experiments. In particular,

these flows were observed in our experiments for Re >500

(3<A<10) consistent with observations of earlier

investigators.

In contrast, the region of dominance of "distorted

laminar" flows seems to be strongly dependent on the

specific experimental apparatus. For example, Hino, et al.

observed these flows for 70<Re <550 at a A of 6.2, and for

460<Re <550 at a A of 1.9; while the same flow regime was

observed by Ohmi, et al. for 280<Re '550 independent of the

value of A. We have not made any attempt to identify these

flows in our experiments.

In the turbulent regime we have made detailed

measurements of the flow field for four flow conditions;

A-10.6 Re -530, 1080, 1720; and A-5.? Re -957. The

-29-

characteristics of the resulting turbulent flow will first

be discussed in detail for the flow condition corresponding

to Re6-1080, A-10.6 . Most of the features demonstrated by

this flow are common to all the flow conditions studied, and

as such this data point may be considered as a prototype

flow. The effect of different values for Re and A will then

be discussed.

Figure 5 shows the phase variation of the ensemble

averaged axial velocity at different radial positions. The

ensemble-averaged velocity is defined as,

1 N

u(r,wt) - - u (rwt±+t) (2)N J-1

where N is typically on the order of 100, and phase of the

cycle is resolved to within 2t=1/240th of the period of

oscillation. The averages are corrected for the bias errors

associated with the finite transit time of the seeding

particles through the probe volume Buchave, 1979]. The

solid lines are the theoretical laminar curve, and are given

for reference.

The most striking feature of these velocity traces is

the explosive appearance of turbulence with the start of the

deceleration phase of the cycle, which transfers the

momentum of the high speed fluid towards the wall and the

momentum of the low speed fluid towards the center. This

momentum exchange appears in the velocity traces of Figure

-30-

r/2wt

Figure-fa. For caption see Figure 5c.

2

1

0

Q

"

U

3r/ 2

-31-

7r/2

wt

Figure 5b. For caption see Figure 5c.

2

1

0

U0Ua

1E1

0

37r/2

-32-

-X/2 r/2 37r/2wt

Figure 5c. Phase variation of the ensemble averaged

turbulent axial velocity compared to the

laminar theoretical curve for various radial

positions. (Re6-1080, A-10.6)

I

O

-

4)D

A

-33-

5, as a dip at wt-O for 0.7<r0.9 and a simultaneous hump in

the velocity trace for 0.9<r<1.0 .

A direct consequence of this enhanced momentum exchange

is that the phase difference in the motion between the

boundary layer and the core is significantly reduced by the

end of the deceleration phase. This point is best seen in

plots of instantaneous average velocity profiles (Figure 6).

Once again the solid line represents the laminar solution.

Note, for example, that at the time wt-r/2 (corresponding to

zero flow rate) there is practically no phase difference

between the boundary layer and the core.

In general, the ensemble-averaged velocity profiles

during the acceleration phase of the cycle ( -/2<wt,0 ,

r/2<wtr ) are similar to the laminar profiles that would

result in an oscillating pipe flow, starting from initial

conditions u(r,O)-O . These profiles, however, are different

from Uchida's solution because the flow has not yet reached

a periodic steady state. As confirmation of this point, note

that the thickness of the boundary layer in the acceleration

phase grows approximately in proportion to v, as in

Stokes' first problem.

During the deceleration phase of the cycle, ( Owt(r/2,

r<wt<3r/2 ) the average velocity profiles are inflexional

Just outside the boundary layers. The shape of these

-34-

0 r/6 r/3 r/2

U (m/sec)27/3 5/6 r 7r/6 4r/3

U (m/sec)

Instantaneous ensemble averaged velocity profilesin the turbulent flow regime compared to thelaminar theoretical curve. (Re -1080, A-10.6)

Wt -

1.0

.a

.a

.37

.4

.3

. 2

.$

0

1.0

.0

.

I .

- . 4

.

.2

.1

0

-r/3

-

-35-

oo

aga0

a

a

a

a

o

-7r/6

aa

aa

oo

a

oo0

a

0o

0

aaaaaaaaa

aa

a 'OC.

-0

0

0

2r/ 3 5r/6

0

o°

o

o0a

00

o

o

o

o0

o* aC.

- a

aO

-2 -

o

o

o

o0

0

o

oooo

o

a*OOCaOaOC

a

7r/6 r/3 7/2

ooo

oa

o

ooa

aoa

0a

a

a

a

aC.

7r/6

oooo

oo

o

o%o)

ooa

oo

aoo

oo

o

a

aaaa0

a

47r/3

o

o

U (m/sec)

Figure 7. Semi-logarithmic plots of ensemble averaged

axial velocity in the turbulent

(Re -1080, A-10.6)

flow regime.

wt - - /3

o00

10-II.

o0

o0'-

C10

o10

U (m/sec)

I-

0

0-t

I 0-1

I 0'

L ^_ · _ ._ - - - 1 ·

I

II

4

I1I

4

44

1

I_

-36-

profiles is very similar to instantaneous profiles that

obtain in steady wall flows during the ejection phase of a

turbulent burst, and suggests that the fundamental process

of generation of turbulence in oscillatory pipe flow may be

the same as that of steady wall flows. This point will be

discussed in more detail later.

The contrast between the velocity profiles during the

acceleration phase and those for the deceleration phase is

best shown in Figure 7, where these profiles are plotted on

semi-logarithmic scale. As seen in this figure, during the

acceleration phase the regions dominated by the semi-

logarithmic law are very narrow, while for the deceleration

phase the semi-logarithmic region extends across a much

larger portion of the tube.

Instantaneous turbulent velocities are defined as the

difference between the instantaneous and the ensemble-

averaged velocities,

u'(r,wt) - u(r,wt) - u(r,wt) (3)

v'(r,wt) - v(r,wt) - v(r,wt) (4)

The turbulent intensities (u '2) and <v'2 >) and the Reynolds

stress (u'v') can then be defined as,

2 1 N 2u'2 (r,wt) - - u1 2(r,wt+±t) (5)

N j-1

-37-

1 N,v'2 (r,wt)> - v- 2 (r,wt+6t) (6)

N J-1

1 N

(u'v'(r,wt), - uj(rwt+±t) v(r,wt+t) (7)N J-1

The phase variation of axial and radial turbulence

d)2intensities (<u'2 , <v >) at various radial locations is

shown in Figures 8 and 10. Throughout the pipe, turbulence

intensities remain at a very low level during the

acceleration portion of the cycle. However, with the start

of deceleration, turbulence appears abruptly in the boundary

layer ( 0.93 r ( 0.99 ) and propagates towards the center.

The abrupt generation of turbulence during the deceleration

phase seems to first occur at r - 0.94 wt--r/8. Thereafter

turbulence levels rapidly intensify throughout the boundary

layer; with the axial turbulence intensity reaching its peak

value around r-0.98, wt-ir/10. During the later stages of the

deceleration phase turbulence intensities gradually drop to

low levels.

Spatially, axial turbulence intensities are

substantially reduced as the center of the pipe is

approached. The distribution of axial turbulence intensity

across the pipe at different phases during the cycle is

shown in Figure 9, and confirms the above arguments.

I ~~~~~~~~I I I

r-O 1- _ ~ - s

r -. 1

0-^"- r- -. 2 r- . 2

r - . 3

r-.4,,,^_ _ A . , ;·> .

r -.5__ W -0 -'I-~CI~·._~

r.6_r 6. .d_.:= CB's~,'·*PC~e'O wo , " ns~~a~~~~~ ,37r/2

Figure 8a. For caption see Figure 8c.

-38-

.1I

c4

U40I.

SI--

N

7r/2

wt

~--~----- -- C1i - __ _ _ __

of,-%__l_ _ ,~L __C ---- I· U-IDr

.1 Al__

--- ·I

I·u~~hC~

-39-

I I I

K r-- .7 --. -. 7

r-.8

r.86

*.,. r-. 895-p~ ~ *·~2·

- r~·~LI~ks,,

L* _ _X* '= as * r.92,,*q. . 's . · . ·

db

'

r~~~~~~~ as~ gsA

-7r2r/2 /2 3/2wt

Figuze 8b- For caption see Figure 8c.

.1

o0

a,C)

N

,,

I

I I I I I

-40-

s.

,r· .sr-.93·· · i

.. -. 96. . r - . 9 60*~~~~~~~~~~~~' "4S~,0* ~ 0*~~k·C~~ChlP ~ ~ *%dsdPu1.,.*.*pAJ c ~AVd~~' R

* w 00 --

0 P .f.

P·Bq.~1 p~muAin n _ _'S *6-%' --,P --%,·~~,qL · · . ,%t16· ?·

X 04111

,., , o .0 ;

0. 0

r a· C r-. 97, A. . :W I -'? +r , - - e

r 0P 0

0 % * 00 .s '0,%e * . ;..- 0 -0 . o * · 0.

·E ·.

1.

r - . 98

' ;dla00.. 0 * ry -0 s..-. ....... '_

_

0 0~ ~~~~~~.0 r-.992 -. _c~rc~pe.. . -.....W" - .- .

7r/2 37r/2

wt

Phase variation of axial turbulence intensities

at various radial positions. (Re -1080, -10.6)

.1

0

0

,::J

C4

0

__ ·I

.

e-ft.

-% .1 %ft-. Ia * . . -140' N S4.I

A- ._ no

, . .

~b~b~sC~'NOWE v�c�L�

I I I

-41-

wt -r/3 -7r/6 0 7r/6 7r/3 7r/2

u ' (m/scc )27r/3 57r/6 7r 77r/6 47r/3

u' (m/sec)4

Figure 9. Distribution of axial turbulence intensity across

the pipe radius for various phases. (Re -1080,

A-10.6)

.0O

. a

.8

.7-

.

. 2

.o

0

L.0

.

.

.3

I .

.4

.3

.Z

.1

O

,*t

-42-

Radial turbulence intensities (<v'2>), in general have

magnitudes on the order of 1/5th of u'2) (see Figures 10

and 11). In addition, the distribution of <v'2> across the

pipe is more uniform at a given phase (Figure 11).

Notwithstanding these differences, the phase variation of

<v' 2, is very similar to that of u'2 > .

The radial distribution of Reynolds stresses (<u'v'>)

is shown in Figure 12 for selected phases during the cycle.

The Reynolds stress is almost negligible over the whole

cross-section throughout the acceleration phase of the

cycle, but with the start of the deceleration phase it

gradually begin to grow within the boundary layer and

spreads upwards towards the center as time progresses within

the cycle. By the end of the deceleration phase the Reynolds

stresses once again reduce to very low levels.

-43-

- r2 ir/2 37r/2wt

For caption see figure 10c.

.u-

0

14)

o

ce-

I . I I

r=O.

r=.l 1

r=.2

. r=.3

r= .4

.0 .-*. r=. 5 -

.0 1 * * *. r- .60~e~cs' · . ·.. ·· * . . .. -

. · w0C% 0 ~Waq4_-Idbl emL cr·oI I I I

Figure la.

I I

I

I

_...· -%- .- .' r=.7. ~e · --qt~ e44--- · y · · ·%A%4 a_ ·

0 0=~~. % ~ 'O0Ldrrrsrc~ 0-~ 0

_ */~~~~S.

0.0_ rs·~~u~hw~

* 0~0 I·I.

I · B ·* 0a p * 0*e:d·PI:¶·..o; . ...

~;·r _ '.l/

%'.- r=·~~t~· T·~~ p· · ·~~~.. s. , ~vr ~~·r~A~~ · $ .V~6 a~~aP~I 0 . ..., % ,-..= :I-,... -. · 0. .84·· dpm *.

· , .d · o+ r lad· ~ ~ ~ I " .. I

0 0:

00 _....~pmmAc I miP .

I.

' . .0 ft.. .* . r=.86 -b* I* - *-hw,'

000V

ii 0 'S-~., %-,, ..".. -00000 * .~r. 89 -e. 0

bA·· · ° qE'*oedJr°[ ..,

0e% W f- e.am I r=. 929~~_ it @ *< *wX~~~~bs° * e~

7r/2 37r/2wt

For caption see figure 10c.

-44-

.02

0

I4

- r/2

_ I · ·

III --

I I

,-- 79

s bhr1Z0~

I0 00 "I.0

-45-

0 0 r=.93:,'N,,~~~~~~~* 0. . *

e . qr · _· · r=94 .. ,.p? · L % ._ .,,~..S o .j,~.,P- 0 ' ~,~.~.,

· , ~~d~~b~pp:

' *-.; .: : * .· 0, * * * 0 I·orp ··~Bb~d. . .'. . - . . .. . %-'· "

!l.

F

..* r=.96

. .* % %'6. .: -,--a~

r=.97-amp'...

- . -- %. "OF

. .f...00~%r ** ·0 0. o /. r=.98 -ee I: M b L k=C·· ~P;C er

r=.99.· i·~~·

r= .992

I- I- -

r/2 37r/2

wt

Figure 10c, Phase variation of radial turbulence intensitiesat various radial positions. (Re -1080, A-10.6)

.02

0

N4)0

vaa

0 0. 0·_~_ ~__~L_~,s Ul1 iyM%

_ 1 1

w a_ ,_ - ....

------- -r�IIL�L�- s r~- L~e -F -- 1W

I I

i!

.

r·

-46-

wt - -X71/3

.0o

.8

.. .S

.4

.,

.2

.1

0

0 7r/6 7r/3 7r/2

v' 2 (m/sec) 2

27r/3 57r/6.0

.a

.

.s

.-

.

.2

.1

0

7- 77r/6 4r/3

v' 2 (m/sec)

Figure 11. Distribution of radial turbulence intensity

across the pipe radius for various phases.

(Re -1080, A-10.6)

Raec.

-47-

wt --/3 -r/6

1.0

.

.-

.

.-

.- 4

.

.2

.

ao

0 7r/6

2r/3 5r//6 7/6 4X/3

u'v' (m/sec)'

;Figure La Distribution of Reynolds stress across the piperadius for various phases. (Re -1080, A-10.6)

?r/3 7r/2

.a

.6

..

.a

.s1-

-

.4

.a

.2

o 1

a

r

u'v' (m/sec)"

11

-48-

62.2.3 Effect of Re and A

The effect of Re and A on the characteristics of the

turbulent oscillatory pipe flow has been briefly considered

by studying two other flows; one with the same A (-10.6) as

the case discussed above, but at a higher Res (-1720), and

the other at roughly the same Re (~1000) but at a lower A

(A-5.7) .

The experimental results for the case Re -957, A-5.7

are shown in figures 13 through 20. This flow differs from

the case discussed in the previous section in that, due to

the lower value of A, the adverse pressure gradients that

are set up during the deceleration phase of the cycle are

not as severe. Accordingly, turbulence is observed only

during the later stages of the deceleration phase (starting

at wt--r/8, r=.93) when adverse pressure gradients are

sufficiently high. Furthermore, the observed turbulence is

not as violent in the boundary layer but tends to be more

spread out through the pipe cross section. The "glitches"

seen in the velocity traces of Figure 13 at wt-r/2 are due

to the backlash in the motion of the driving piston as was

discussed in section 2.1 . The curves of instantaneous flow

rate for this flow condition were shown in Figures 2a and

2b, and manifest the same "glitch".

-49-

ir/2wt

Figure 13a. For caption see Figure 13c.

1.0

.5

UVon

la

IP

-7r/2 37r/2

-50-

7r/2

wt

Figure 13b. For caption see Figure 13c.

1.0

.5

TO

6)IS

37r/2

-51-

-7r/ 2r2 37/2

t

Figure 13cL Phase variation of the ensemble averagedturbulent axial velocity compared to thelaminar theoretical curve for various radial

positions. (Re6-957, A-s.7)

. U

.5

0

O

ID

-52-

0 7r/6 ir/3 7r/2

U (m/sec)2X/3 5r/6 X 7r/6 4r/3

.0

. a

U (m/sec)

Instantaneous ensemble averaged velocityin the turbulent flow regime compared tolaminar theoretical curve. (Re -957, A-5.

profiles

the

7)

wt - -X/3

1.o0

.7

- .

. 4

.3a

.2

.

0

-

1.4

-

1-

.I

.

.3

.

. 2

.

-1.0 -. S 0

0 7/6 r/3 r/2

_- o

o- ao

oo

- o

a- a

a- 2- C

oa

aaa

aa o0

o

o

ao

.6 1.0

27r/3 5Sr/6

aa

aa

aaoa

oaa

oaaa

0

7r 7r/6

U (m/sec)

Figure 15. Semi-logarithmic plots of ensemble averaged

axial velocity in the turbulent flow regime.

(Re -957, A-5.7)

-53-

wt - -7/3o10

to0-

.4I_0

-oo

1 0 --10'

aaaaaa

ao 0

o

CI

c)oaa

aa

o

0

:)o)

4

C

CaC

'C

,C

CC

c

41r/3U (m/sec)

4

0

I

I

I

.0313.

II

-54-

7r/2

wt

For caption see Figure 16c.

.05

0

Nlu000

-1S

(4

Pd'. -I Wp

r=.lr-. -=b .2 -

-r=. 2

0.. 0~~~~~-- 0% V

b~Y~· ~Q~ l~bWr/nL~L~L~ l~ia .,^ f W%,%-·C

r=.4_'- i % 0 -0 ' r=.5

.r= .6~4~~.ch~hu~~~·r~kJ:I._r· -__A·l~s-r *Sw;,s b`·.~

37r/2

_ __

w --- ~ ~ -

~~~~~~~~~~~-A

=-I-Lc-~-~-~ �Ld �P-6- -� IPBbs� 1U

I I

%.r=. 3 .- . :a

I I

-55-

r=.7

r=.76

r= .8

r=.844a·-, ~ ·--- ·~~~=:. i~~~ ~~gbg% ;

r=.86

-' r-.89· 0 0 · 1 4

0' ' TV,. .v ...... - *~o

r=.92

h0, E P.A U~ E % % 0 % > ,I. OI %011 1-P

7r/2 37r/2wt

Figure 16b. For caption see Figure 16c.

. 05

TO

M

011

11

9p-

_ __

b�c"-l�b -�--r -- � 1�Re -

-%WdV%#M-a.

.

V.

16./ *

. n r-- - I ~a*%I

-56-

_ .w·.9_e,_g , _ _ w 0./ -A10*ms~ ~~~~~Ib . %

~C. % ' r- 93.Cc'~.B'-~ C

' . .. k~~ ·- r=.96-$·· - ·- "--I'---a ct ., -

I%* 4__ AI~ h4_q___S,^.b*S *_" . I

:~~~~~~~~~~~~~~~~~~~~= 98-r= .99-

r=.99;

7r/2 37r/2wt

Figure 10c. Phase variation of axial turbulence intensitiesat various radial positions. (Re -957, 5A5.7)

.05

0

.4

00

C4

I1

_ ,_c--� -r--as�- -- -- hCJ

I II - - - - ---

r

0

d~~ci~c r=.97-

r=.98-

-57-

wt - -7r/3 0 r/ 6 r/3

2/3 (m/s e c )27r/3 57r/6 Xir ?~/6 47r/3

u' (m/sec)2

Distribution of axial turbulence intensity across

the pipe radius for various phases. (Re -957,A-5.?)

L.0

.o

.a

7r/2

I

11

.6

.

.4

.5

.2

.1

0

L.O

oO

.8

.7

.6a

.5

.3

.2

· I

0C: A

-58-

-X7r/2 7T/2 37/2

wt

Figure 18a. For caption see figure 18c.

.UU5

goA_

I I I

r=O.

: A

r=.l

* .. r=.2 7

r=.

~~~~~r- A ,~~~i '=.h

I i I

--- -

-59-

7r/2

wt

For caption see figure 18c.

.005

0

Nt)

0

;l

C4

r=. 7

' r=.76

*:, -... · , . _, .

-l'e~~~= . ~ .- ·~~~ c 0'~ * qmV'

f* . .. r=.84 '

_________·--~c~ e~-~ ck-~b4I~;5~4IELH'. m ft 1% · .. ;I, -/. . r=.86-_'f" - b h idP

: : . r=.92.2

-

3r/2

_ ·

A- - ^ur

- ---~---- -e --4U~L~ I-t~l·F· -- A O--CM · ·---0

--

.. ,:/ , -% ~ -r.89-~~CL·/· · CC/P dll

m~w

I !

-60-

7r/2 37r/2

Figure 18c. Phase variation of radial turbulence intensities6

at various radial positions. (Re -957, A-.?7)

.005

0

m

v

C;

r=.93_*

,.. ..: r=.96e' ~· m*.* ardh

r=.98

-r=.99

r=.992

I· _ _ _ _ _

I~C~ -- 13 -- Y

ww-" v~-- -r -- I-- -

- -�c-r -- -�--- -CT---C --

I -- 1 --- -RAN -- - - ---- -·- - --- - -

�r�Lvrr�hr

r=. 94A,1 e .V-WOc~te",

7r/2

v' (m/sec)42r/3 Sr /6 7r/6 4r/3

V,2 (m/sec)2

Distribution of radial turbulence intensity

across the pipe radius for various phases.

(Re -957 ,-5.7)

-61-

wt - -/3 0 7r/6 7r/3

L.O

.O

.

.

1.4- . 4* 4

.2

.1

0

1.0

.

.a

.7

-I

-

.

.5

.4

.5

.2

.I

0 O .00

,'

-62-

0 r/6 7r/3 7r/2

u'v' (m/sec)'21r/3 57r/6 X 77r/6 4ir/3

u'v' (m/sec)2

Figure 20. Distribution of Reynolds stress across the pipe6

radius for various phases. (Re -957, A-5.7)

- -"r/3wt

1.0

.

.

I

q,-4

.9

.

.

.2

.1

0

1.0

.

.

.7

.

..

.

.2

.1

0- 01 0

rr

-63-

In contrast, the characteristics of the flow at the

same A but at a higher Re6 is shown in figures 21 through

28. In this case, the region of turbulent flow extends to

the later stages of the acceleration phase ( when the flow

is still experiencing a mild favorable pressure gradient),

suggesting that if the Reynolds number is sufficiently high

the flow may remain turbulent for a substantial portion of

the cycle. For an oscillatory flow, however, there will

always be a region near the phase of zero flow where the

instantaneous Reynolds number is low and the flow is

experiencing a favorable pressure gradient, and as a result

the bursting process has to stop. If the time scale of

oscillation is large compared to time scale of dissipation

of turbulence so that no residual turbulence is left, the

flow has to relaminarize during this period. This point will

be discussed in more detail later.

-64-

7r/2wt

For caption see Figure 21c.

q

2

0

am

37r/2

Figure 21a.

-65-

r/2wt

For caption see Figure 21c.

2

0

ar1-10

37r/2

-66-

-X/2 r/2 37/2

wt

Figure 21c. Phase variation of the ensemble averaged

turbulent axial velocity compared to the

laminar theoretical curve for various radial

positions. (Re 6-1720, A-10.6)

q

2

0

O

I.:

-67-

wt - -/3

1.0

.

. a

54 .

.4

.2

.2

.I

0

0

U (m/sec)27r/3 57r/6

L.

Ide-4-

- !-

7r1 7 r/6

U (m/sec)

Instantaneous ensemble averaged velocity profilesin the turbulent flow regime compared to the

6laminar theoretical curve. (Re -1720, A-10.6)

7r/6 7r/3 7r/2

4ir/3

o

e

-68-

wt - -7r/3 -7r/6 0 ,r/6 r/3 7r/2

U (m/sec)7r

8o4ooo

oaooooooO0CCo

9

77r/6

aao

ooooaaaa0 -0'C,'

I*C

47r/3

ooooaoa o

o

oo~~~o~~~o~~~

o° ~ ~

oa0'

10

-0

aO

aO

Semi-logarithmic plots of ensemble averagedaxial velocity in the turbulent flow regime.

(Re 6-1720, -10.6)

o

- 0 o

0oooo: - a

- 9

0141

-4

0-4Oa

·-(

LO'-

10-

o

0

0

o0

O

oao

o

oooooo

ooo a

ooo

aoo

oooooo

o

oo

ooooo

oao

o

I

C

C

,C,CI

C

a 2

100

27r/3 57r/ 6

I8I%

0o

O-I

0-1

o

o

o

o

oo

oo

o

oo

o

o

aaaa0laM

a0

-2

1

o

oo0

oI10--

U (m/sec). . * - . . , . I . . . ._

10

· ~mm. .-

i

4

-69-

7r/2

wt

For caption see Figure 24o.

2

0

ToNpgo

El

1-

-7r/2 37r/2

-70-

r/2wt

Fiwure 24b. For caption see Figure 24c.

.2

0

oo

,=000E!

(4

-7r/2 37;/2

-71-

- _ ,~?~, _ ~ . ~. r=.93

r=.94

r=.96.sPr .97_

oil,.0 ~~~~~~~~d

I I I

lr/2

. %'"-,"'.._ r=. 99.0 ~~, He_~~'~~~,

37r/2wt

Figure 24c. Phase variation of axial turbulence intensities

at various radial positions. (Re6-1720, A-10.6)

.2

0

c)

'-S

-7r/2

UI ·

-72-

0 ir/6 7r/3 7r/2

u' (m/sec)'2r/3 5ir/6 77r/6 47r/3

u,2 (m/sec)

Distribution of axial turbulence intensity acrossthe pipe radius for various phases. (Re -1720,-10. 6)

wt - -7/3L.0

.o

.a

.7r

.

.

.

.O

.O

.a

.I

I_ . 4

. 3

.2

.o

0

-73-

7r/2

wt

For caption see figure 26c.

. 04

.02

0

0

N.1-1

I- I

r=O. 'S. v e'/be.S4h u. .AwW 4. - . fto

:~.*... - r=.1 -, . ~ ' ~-. /... ~ .... .

** *%:. 0%

r=.4·0-A -' . r W M

**~A'I*.~~·i%

* *O b- -r=

%W *J ., .9 ._14 0·_·CI · P~~~~~__ _ _~~~_ ~~·_·(~~~. _· itP

*,.0 r=. 6

e* -J37r/2

.-. ... · ri- .-

ftA

-- fwok

lE -- i--

- - -- , - Jrr

.,O.: "" ~ ~ ~ ~ ~ rI %- . .01

C -t,--Jb~A

_ _ a - -- - S

%O O.rcr r · 4-ks,V..O~ r~e %:r =rb: B

r= q

. "- -: 'A.-O. m.

I

-74-

.... - .%', * *,, ~,.

I0 -0

;S'~~~~~~0

I-~ ~~~ 0f-· -, S·,,.,..* ~, · -,~"'.-*.,,,~,,~

-e. r= . 7 W u'*

. -0: *, r=.8a A 0 0 O

· 0 * S f~b01., -.~. o.

-P··k. ~ · 500 V'r=. 84-'Yl~ru, r;Y; , r. a -eel

... : ..r=86 -hmm..m.mmh'" __-_·.-____ _____ , L:..- .·d=' % . .C-· -r0r~h~· :

W s.s-.1I.r=. 89 -

lvvlv * 0 0m-p

- _--

q.V".hc *~*~e p * ¶ * I, r=.92 -

~~hIEI..%lPU.~l~VdIP'r 00 V

*4 6w" "ee

I I I

7r/2

wt

Figure 26b. For caption see figure 26c.

.02

0

N(400ao

37r/2

_ __ _ _

-r/2

-75-

ir/2 37r/2

wt

Fiure 26o, Phase variation of radial turbulence intensities

at various radial positions. (Re -1720, A-10.6)

.04

.02

0

U

C

I I I

'_ ., · .% %'-%,.%. ~~~r= .94

,- .7:WI~;I(L~~b~L) a~ '~_96_ ~ P)Erg~anoCI*4. %,O% 4q r=.97% ^6m *040d'O* * .-el . !-%AO

I r ~~~~%ft .%16-,V, .% 1%. 0-_VIP ~ ~ +.+~

_ __ ~ ~ ~ ~ ~ ~ ~ ~ ' . =

__ _

1 C ~ ~ ~~~~_ -

__ 1~~~~~- -- - - - - - - - -

I

rr/6 7r/3 7r/2

27r/3 Sr /6 7r/6 47r/3v' 1 (m/sec)'

2 2v' (m/sec)2

Distribution of radial turbulence intensity

across the pipe radius for various phases.

(Re -1720,A-10.6)

-76-

0wt - -r/31.0

.a

.O.

.7

. .

.-.I

..

.2

.1

0

1.0

.a

.6

.O

.6-

.5

.2

.1

0 o

Figure 27.

1%

-77-

- -/3 -7r/6 0 r/6

27r/3 51r/61.0

.8

.8

.7

..8

I .5

-

.

.2

.1

0

Wr 77r/6 4r/3

u'v' (/sec) 2

Figure 28. Distribution of Reynolds stress across the piperadius for various phases. (Re -1720, A-10.6)

wtr/3 Ir/2

1.0

.

.7-

i1>% .-

.

. 4

.2

.1

0

u'v' (m/sec)4A

-78-

2.3 Discussion

One of the major objectives of the present work is to

gain some understanding of the fundamental physical

processes that underlie the transition back and forth

between the "laminar" and turbulent states of oscillatory

pipe flow.

We have already mentioned that the ensemble averaged

velocity profiles during the deceleration phase of the cycle

(when the flow is turbulent) look very similar to

instantaneous profiles that obtain during the ejection phase

of turbulent bursts in steady wall-bounded flows

[Kline,et.al (1967), Kovasznay, et.al (1962)], in that the

profiles are inflectional ust outside the boundary layer as

a result of momentum exchange between the low speed fluid

near the wall and the higher speed fluid further away. In

contrast, the profiles for the acceleration phase of the

cycle are full, and suggest that the relaminarization of

flow during this period may be due to cessation of turbulent

bursts.

Let us start by reviewing the bursting phenomenon in

steady wall flows. We can then consider the effect of

-79-

acceleration or deceleration

process and decide if the

relaminarization of the flo

bursting is ustifiable.

of the flow on the bursting

assumption with regard to

w because of cessation of

Consider the (inviscid) vorticity field associated with

a uniform shear flow. The vorticity field for this

undisturbed flow consists of spanwise vortex lines forming

sheets of transverse vorticity (Figure 29). If this flow is

perturbed by a disturbance which has a streamwise vorticity

component the vortex lines which were originally in the

spanwise direction are deformed into U-shaped loops in the

xl-x2 plane (see Benney and Lin [1960] for the origin of

these disturbances). Due to self-induction of the U-shaped

vortex a positive u3 velocity component is induced at the

tip of the loop (points T) while a negative u3 velocity

component is induced at points V (valleys). This induced

velocity lifts up the tip of the vortex bringing it into

regions of higher axial velocity, and pushes down the vortex

filament near the valleys into regions of low velocity. As a

result, the tip of the vortex loop ends up in a region of

higher axial velocity relative to the valleys and therefore

would have to move faster compared to the valleys, giving

rise to further stretching

formation of an -shaped filament. At

original vortex filament is no more

direction but acquires a strong

of the vortex filament and the

the same time, the the

purely in the spanwise

streamwise component

-80-

Figure 29. Conceptual model of the bursting process in

steady wall-bounded shear flows (from Hinze,

Turbulence).

-81-

especially around the legs of the vortex loop. This

streamwise vorticity pumps the low speed fluid away from the

wall and results in a defect in instantaneous axial velocity

profiles which is accompanied by very intense shear layers

in the velocity profiles. A turbulent burst is the result of

local inflectional instability of these intense shear

layers.

when the flow is accelerating, because of the favorable

pressure gradients that are superimposed on the flow the

profiles at the "peaks" cannot become as inflectional. As a

result, the streamwise vortex filaments cannot lift and

stretch as effectively and the bursting process does not

occur.

Deceleration of the fluid, on the other hand, would

tend to enhance the "defect" in the velocity profiles at the

"peaks", and would result in more violent bursting.

For the oscillatory pipe flow, the (adverse or

favorable) pressure gradients are not constant, but vary

with time. This variation, however, is immaterial during an

individual bursting process because the time scale of the

burst is much shorter that the time variation of the flow.

To see this point, recall that the turbulent burst has

convective time scales on the order of (d/U), whereas the

oscillatory pipe flow varies on a time scale (1/w). The

-82-

ratio of these two time scales is the Strouhal number - U/wd

which is of the same order as Re /A ,, 1.0 .

The relaminarization of the flow during the

acceleration phase requires, in addition to cessation of

bursting during this period, that no residual turbulence is

left in the flow from the deceleration phase. Thus we need

to estimate the time scale of decay of turbulence. In this

regards, one must recall that dissipation is a very passive

process in the sense that it proceeds at a rate that is

dictated by the inviscid inertial behavior of the large

eddies (Tennekes and Lumley). In other words the time

limiting step in the dissipation process is the rate at

which large eddies can transfer their energy to the smaller

scales. This time scale is given by u'/l, where 1 represents

the size of the largest eddies. These eddies, on the other

hand, maintain their vorticity as a result of interaction

with the mean shear flow and thus have characteristic

vorticities of the same order as the mean shear; u'/lU/d.

This reasoning is the basis of mixing length theories in

steady flows, and can be extended to our nonsteady problem

because d/U,,l/w. From the above arguments, the time scale

for decay of turbulence can be estimated as d/U, which again

is much shorter than 1/w if Re /A>>l. Therefore, as long as

the Strouhal number=U/wd-Re /A ,>1, the turbulence generated

during the deceleration phase does not get carried over to

the acceleration phase, and in the absence of bursting

during this period the flow has to relaminarize.

-83-

In summary, it appears that the transition back and

forth between the laminar and turbulent states is mainly due

to cessation of "bursting" during portions of the

acceleration phase of the cycle, as a result of favorable

pressure gradients superimposed on the boundary layer. Our

experiments present indirect evidence that the fundamental

bursting process is the same as that for steady wall-bounded

flows. Further experimentation, (in particular in the form

of Lagrangian flow visualization) can provide better

evidence for this claim.

We will instead proceed to investigate the transition

process by direct numerical simulations of the Navier-Stokes

equations. These results will be discussed in the next

section.

-84-

III. NUMERICAL SIMULATIONS

3.1 Problem Formulation and Numerical Methods

In this section we present results of a numerical

investigation of the stability of oscillatory flow in a

channel which is subjected to various classes of

infinitesimal and finite-amplitude two- and three-

dimensional disturbances. The full three-dimensional time-

dependent Navier-Stokes equations are solved in each case,

using the pseudospectral formulation suggested by Orszag

(1971].

The flow geometry is a two-dimensional channel which

has rigid walls at z - +h and extends to infinity in the x

and y directions. In all that follows the channel half

width, h, is taken to be ten times the Stokes boundary layer

thickness ( h 10v2v/w ). The flow rate per unit width into

this channel is imposed as Q -Qocos(wt )

The choice of a channel geometry for the theoretical

study versus a pipe for experiments is mainly a matter of

convenience. The characteristics of the turbulent flow in a

pipe and a channel are very similar, because the turbulent

flow structures are in general much smaller than the

curvature of the pipe. For steady flows the large pool of

-85-

experimental data is in support of this fact. For the Stokes

problem, we base this argument on the close similarity

between our experimental results for flow in a pipe and the

results of Hino and coworkers which were obtained in a

channel.

We solve the Navier-Stokes equations expressed in

rotation form,

a* * * 2

-V x W -V( -- + | ) + f xat p 2

I 2* * p* 1v1 2

where - Vxv is the vorticity and ( - + ) is theP 2

pressure head. The mean pressure gradient is not included in* * *

p but is represented by the external forcing f , (f -

VP /p). Here, * denotes dimensional quantities.

If one further introduces the non-dimensional

quantities,

t - wt

v --- , where U -Uo 2h

*

x - , where 6 V2iv

* * 2p Iv I

P 2

11 2

0

f -

-86-

the Navier-Stokes equations can be represented in the

nondimensional form,

av Re Re 1 Re-- (--)(xw) - ( ) + - Vv + (--)f At 2 2 2 2

This equation is solved numerically subject to no slip

boundary conditions at z-+10, and with periodic boundary

conditions in the horizontal directions;

27rn 27rm

v( x + - , y + - , z , t ) - v ( x,y,z,t )

for all integers n and m, and specified streamwise and

spanwise wavenumbers a and P.

The numerical scheme is that used in earlier studies of

transition in steady wall-bounded flows by Orszag and

coworkers. Various details of the the numerical code may be

found in the literature [Orszag and ells (1980), Orszag and

Patera(1983)]. Here we summarize the basic discretization

and time-stepping procedures.

The velocity field is represented using Fourier series

in x and y and Chebyshev polynomials in z,

P N Mianxv l E vnmp (t) eia eip my T(z)

p-0 n--N m--M

N 1m

E E Vnm(Z,t) ei nx eiImyn--N m--M

-87-

The time-stepping procedure uses a fractional step (or

splitting technique), where incompressibility and viscous

boundary conditions are imposed in different fractional time

steps. The time-stepping procedure is given by,

'n+1 vn

i) 6Re(- At)

2

-n+l V -(ii) -

Re

2

V.vn + l

I I w+l1o

3 1-1--- xw-(v -(vxw) - n - + f

2 2

rn+l

--VH

-At)

-

at z-+10

(iii)VA E

At

I Iv n +

-, V _ _ 2_n+ 1pyrL 1~r

2

= 0 at z=+10

The first step incorporates nonlinear effects. The time

stepping procedure is an Adams-Bashforth explicit scheme,

which incurs errors of O(At2 ).

The second step imposes incompressibility. The vector

equations are reduced to a Poisson equation for w which is

(

L 1 -M.& 1

_ - v

-88-

solved subject to the inviscid boundary conditions (i.e, w-O

at z-+10). At the end of this step the flow is

incompressible but the viscous boundary conditions are not

satisfied.

The last step incorporates the viscous effects and

imposes the viscous boundary conditions. The time-stepping

scheme is only first order but implicit.

In the present problem the flow rate (Q=cost) rather

than the external forcing (pressure gradient) is specified.

This can be implemented by noting that since f is

divergence-free, it merely has an additive effect on vn+l

and vn+l in steps (i) and (ii). In other word the effect of

the external forcing is to introduce a contribution to vn+

in step (iii) given by v n+ 1 where,

v n+l 1 Re6v V2vn+l + ( ) f

At 2 2

lv ' n + 0 at z +10

One could thus solve for v corresponding to a unit

forcing in a pre-processing stage,

v 1 2 n+l- Vv +

At 2

I v - 0 at z +10

-89-

At each time step, equations (i) thru (iii) are then solved

without the effect of external forcing to give v" n+l The

correct solution to vn +l is then,

n+l - v n + l + (Re /2) fn+ v

where fn+ is determined by the imposed flow rate at time

(n+1), i.e;

f v n+ l dA + (Re6/2) fn+l f v dA n+

or,

Re6 Qn+l - f v n+ l dA

2 fv dA

This completes the description of the main part of the

program.

Initial conditions for the runs reported in this study

generally consist of an unperturbed laminar flow on which

are superposed various combinations of infinitesimal or

finite-amplitude two- and three-dimensional eigenmodes of

the (time-independent) Orr-Sommerfeld equation for the

"frozen" initial profile. Note that even though the initial

disturbance is strictly valid only if the flow was time-

independent, the results obtained in the steady state using

the full time-dependent simulation and the initial

-90-

disturbances described above should accurately represent the

full time-dependent behavior of the flow.

A number of tests have been performed to verify the

accuracy of the direct numerical simulation. As a start, the

evolution of the least stable eigenmode of the Orr-

Sommerfeld equation is followed for a frozen profile using

the direct simulation and the results are compared to those

predicted by the Orr-Sommerfeld equation. A few of these

results are summarized in table 1. In other tests results of

the full simulation for small disturbances were compared

with those of the linearized equations; and finally direct

simulations of two- and three-dimensional infinitesimal

disturbances were compared to verify that they correspond to

each other according to Squires' theorem. These results

will be discussed in later sections.

Typical parameter values for direct simulations are

P-64(or 128), 2N-32, 2M-2, Courant number - 0.1 .

Convergence in all these parameters has been verified by

either halving or doubling the resolution.

-91-

Re6

3

Real(w)

Im(w)

Resolution2N x (P+1)

Courant

FinalTime

predictedamp. chni

frozen profile of wt=0

2Ddisturbance

1000

0.5

.54467293

.00175319

8x65

.1

r/10

1.309

e(Re/2)Im(w)t

computedamp. chng 1.346

predictedphase chng -85.557e-i(Re/2)Real(w)t

computedphase chng -85.813

3Ddisturbance

1000V2

0. 5/X/

0.5 /A/i

.38514192

.00121291

8x65

.1

7r/10

1.309

1.341

-85. 557

-85.813

frozen profile of wt-7/2

2Ddisturbance

1000

0.5

-. 06858357

.02015786

8x65

.1

7r/10

23.722

23.406

10.773

10.772

3Ddisturbance

1 000\/

0.5 //

0o.5/v

-. 04849591

.01425378

8x65

.1

r/10

23.722

23.446

10.773

10.772

Table 1. Comparison of evolution of small-amplitude

disturbances for frozen profiles to prediction

of Orr-Sommerfeld equation.

-92-

3.2 Small Amplitude Dsturbances (Linear Theory)

The linear stability of oscillatory Stokes layers has

been the subJect of a number of investigations. The simplest

and most popular approach is the quasi-static assumption,

where the time dependence of the basic profile is ignored

and the stability of instantaneous "frozen" profiles is

studied using .the the classical (time-independent) Orr-

Sommerfeld eigenvalue equation. One question is the choice

of which frozen profile to use. For the particular example

of oscillatory flow in a channel with an imposed flow rate

of Q - cost, two such quasi-steady stability maps, one for a

time of t-0 and the other for t/2, are shown in Figures

30a and 30b. As one would expect, the two stability maps are

quite different. The minimum critical Reynolds number, for

example, has a value Re crit. 80 for the profile at t=r/2;

but a significantly larger value (Re rit. - 500 ) for tO.

The most dangerous streamwise wavenumbers, however, are in

both cases close to a of 0.5. If the quasi-steady problem is

at all relevant to the problem of stability of oscillatory

Stokes layers, the stability maps of Figure 30 suggest a

search for unstable modes (in the full time-dependent

problem) in the range of wavenumbers a - 0.2 to 0.7.

-93-

wt-7r/2"o,

;U

10-'

10-2

Rev

wt-Olno

10-'

1 10 . I jI ,

10-' lo 03 10 '0Re

Figure 50. Iso-growth curves for the frozen profiles at

wt-r/2 and wt-O.

Aui

i%J

i

i

-94-

3.2.1 Two and Three-Dimensional 1Disturbar&es

Squire's theorem provides a very useful transformation

for the linear stability problem of "steady" unidirectional

shear flows, which reduces the linear stability problem of

any 3-D disturbance to that of 2-D disturbance at a lower

Reynol.ds number Drazin and Reid,1981]. Exactly the same

transformation can be used in the "unsteady" linearized

equations von erczek & Davis,1974] with the conclusion

that the evolution of any infinitesimal 3-D disturbance is

related to a 2-D one at a lower Reynolds number. This is

confirmed in the results of the numerical simulation, one

example of which is shown in Figure 31. In this figure the

evolution of a 3-D disturbance and its equivalent 2-D

disturbance are compared. Here En2 is the total two-

dimensional disturbance energy (in all the streamwise

fourier modes l<n<N ) and is given by,

N 10 (2) *(En2 - t J(Vn vn )ea f (Vn(2) ( ))dz

n-l -10

where * denotes complex conjugate.

Similarly En3 is the total three-dimensional energy,

M N 10 *

En3 - £ (3) (vi d( zm--M n--N -10

U

-2

-4

-6

-8

M -100;.10

-12-4

-14

-16

-18

-20

-22

-_"m

U

-2

--t

-6

-8

e -10

-12U-4

-14

-16

-1is

-LJ

-22

-95-

o Ir/2 X 3r/2 2x

-V

a 7/2 7 33r/2 2r

F gure B1. Equivalence of two- and three-dimensional infinitesimaldisturbances by Squires Theorem. For 2D disturbances

Re -1000., a-0.5 . For 3D disturbances Re -1000\ ,a-I-0.5/Vr. Initial conditions are the least stableeigenmodes of O-S equation for the frozen initial profile.

-- - A , I -1 I

I I , -i- T- r=- - I - I - I . .. . ... . . . . . . . .

I . . I I I I . . . . . . I I I 1 I I I . I . I I I I I - I I I I . I I

..,,..; .,,,.,..,,,,,.,..., ....,...,,..,,..,

-96-

The comparison also serves as an extra check on the

integrity of the numerical code.

Consequently, only two-dimensional infinitesimal

disturbances will be considered in this section.

3.2.2 The Linearized Problem

For infinitesimal disturbances, a substantial savings

in computation time is achieved by solving, instead of the

w-Oill nonlinear problem, the linearized disturbance equations

given by:

a Re-(u',v',w')= -- (Uu' +w'U , Uv'x ,Uw' )~t ~2 z x

Re 6 1--( 1, H 7 ) + -V 2 (u',v',w')2 X 3 Z 2

Here v' is the disturbance velocity, and only one fourier

mode is kept in the streamwise direction;

v' = v'(t) einx T (z)n-l,+l P

- v'(z,t) eianx

n--l,+l

Except for the implementation of the nonlinear terms, the

method of solution is the same as that described earlier.

-97-

0

-2

-4

-6

-8

CN -10

-12

-14-I i

-16

-18

-20

-22

-;)40 W/2 Ir 3r/2 2r 5r/2 3T ? x/2 4 9x/2 5r

Figure 32. Comparison of the results of the full three-

dimensional simulation to the linearized problem

for the evolution of two-dimensional disturbances

at Re -1000.0 a0.5 .

-98-

For infinitesimal disturbances, the results of the

linearized equations should, of course, be in agreement with

the full nonlinear problem. A comparison of the two is shown

in Figure 32 for Re =1000 and a-0.5 . The slight wiggles at

large times in the full simulation are due to the excitation

of higher order modes by roundoff errors.

The results to be discussed in the rest of this section

were obtained using the linearized equations.

3.2.3 Effect of nitiat Conditions

It was mentioned earlier that the Navier-Stokes

equations are solved subject to initial conditions that are

the least stable eigenmodes of the Orr-Sommerfeld equation

for the initial "frozen" profile. It was also mentioned that

for large times the solution becomes independent of the

initial condition. This point is verified by the results

shown in Figures 33(a-d), where the flow is started at two

different times ( t=O, and t=w/2 ) and in each case

subjected to the least stable disturbance of the initial

"frozen" profile. In addition to the overall energy of the

disturbance, the distribution of the disturbance energy

across the channel width is also shown for selected times.

Comparison of the growth rates in the two cases (as shown in

Figures 33a,b), or the distributions of the disturbance

energy for large times (as shown in Figures 33c and 33d)

-99-

U

-2

-6

-8

C -10

-16

-18

-20

-22

-240 1/2 r 3/2 21 51/2 3I 7r,'2 41 9 /2 5

Figure 33(a.b). Evolution of two-dimensional disturbances at

Re - 1000.0, a - 0.5 for two distinct initial

conditions. Note that at large times the growth

rates ( Figures 33a , b ) and distributions of the

disturbance energy across the channel width

( Figures 33c , d ) become independent of the

initial condition.

-t0-

Distribution of E2 at time 0 Distribution of E2 at time r/2

. , . . . I I . I . ..0O O o a In

0 0 a t 0 0 0 0 0 1 X Jr X X X Y Y K X

N c' v U ED a, 0 K? Oa -C~ x x x x x 7 c ~ ~ (n VI I w r- co 0)

Distribution of E2 at time Distribution of E2 at time 3r/2

w w 0 0 0 0 w M 0 0 0 0 x K X ;A K x x S x K AN r W O 0 N r u N cr 9

eVN

LJ

18

16

14

12

>- 10

8

6

4

2

0¢

.Elg==,re 33 For caption see Figure 33a.

18

16

14

12

10

8

6

4

2

n I

18

16

14

12

- 10

8

4

2

0C

18

14

12

10

0

6

4

2

0

I t r- r r - r --- -rr --. ._�--7---------- -I .I I I r III-- I -

I

.1nIC_t

--- I,

-101-

Distribution of E2 at time 2r

o 0

_ _-r (0

. . I .. . .M a en a) a

Y x x0 0 0 r tD_ _ _ _

0 0(14

Distribution of E2 at time 57r/220

16

14

12

10

6

4

2

O o C3o C C o C3 o C 0

e T q _ _ _ 0 ( N

Distribution of E2 at time 3rT II ' T r7 ..... I . ... r r.. . . .. r. I . r I * I T ....... .

. ....,.....a..... . ..... i......... , ...........-"" ''' · · · ·'··'· · · ·'10 00 0 0 0 0 0 0 0 0 _ 0 0D

r o "' R 4 g o p o w S For caption see Figure 33a.

Fig.re 330. For caption ee Figure 33a.

CU

18

16

14

12

>- 10

8

6

4

2

20

18

16

14

12

8

6

A

2

n

I

I

.1

>-

o I

"N

-102-

OISTRIBUTION OF E2 T TIME= ?r/220

18

16

14

12

> 10

8

4

2

0 - _ - - - - - -' ' -

XDISTRIBUTION OF E2 T TIME= r DISTRIBUTION OF E2 T TIME= 3 r/2

a 6 a, b a, 6 6, a, b a, 6 a X g X 3 X x W x x x xU*) C fl r tfl

Lu

1E

16

14

12

>- 10

8

6

4

2

0

*u ·. F o- r an atn ; = l� i - W

Eigure 33d. For caption see Figure 33a.

"u

18

16

14

12

>--10

8

6

4

2

0

^^

-103-

OISTRIBUTION OF E2 AT TIME= 27

I I i i i i i i i I ! i i i i 0a a 0 0 0 0 0 0 0 a a 0 0

(N q tO s cs t 0 (N W N N 0 R nX

OISTRIBUTION OF E2 AT TIME= 37

20

18

16

14

12

10

8

6

4

2

0

0 0_ _A A

OISTRIBUTION OF E2 AT TIME- 57r/2. . . . . . . . . . . . . .f I s w W W I W T l / r w T E s s U s W |

............. 0T t I

o 0 0x x x

0

J ; I ; ; ; ; If~~~~~~~~~~~ i ~,tt t a n o i , a

x x x x x x x xiN I C I 0 ! to

Figure 33d. For caption see Figure 33a.

20

18

16

14

12

10

8

6

4

2

0

20

18

16

14

12

> 10

8

6

4

2

0

.........

o 0x xO (ncu ru

....... 0.T o

_ ax xa' ('1

0.0x0

,.... .

x xLI 0VI U,

X

c

· 1·1·1·1 11·1·(·)1(1(·(·1· )·)·)·( , I

. v . I I T - r - T- - --

-104-

proves that the final solution is independent of the form of

the initial disturbance.

3.2.4 Results

In Figures 34 and 35 we show how a typical two-

dimensional disturbance which starts off as the least stable

eigenmode of the (time-independent) Orr-Sommerfeld equation

for the initial "frozen" profile, evolves to a steady state

solution which is that predicted by a Floquet analysis of

the time-dependent Orr-Sommerfeld equation.

In addition to the overall energy of the disturbance

(shown in Figure 34), we find it useful to consider the

distribution of the disturbance energy across the channel

width at various times (Figure 35). From Figure 35 it

immediately becomes obvious why quasi-steady theories give

results that are so different from those of the Floquet

analysis of the Orr-Sommerfeld equation.

The initial disturbance field (at time r/2) is

specified as the least stable eigenmode of the Orr-

Scmmerfeld equation for the "frozen" initial profile. The

plot of distribution of energy for this time has a major

peak at the "critical layer", with two less prominent peaks

at successive inflexion points. Between r/2 and r, the

-105-

-2

-4

-6

-8

CCq

a -10

a -12

-14

-18

-20

-22

-P4

-1

I1

1

1

I I I I I I I

O T 2W 3W 4V SW 6r It

Figure 34 Evolution of the energy of two-dimensional

disturbances at Re 1000.0, a=0.5 . The initialconditions correspond to the least stable eigenmodes of the Orr-Sommerfeld equation for thefrozen initial profile.

-106-

r... aT TIME= rc-.. , o T Et= 2

2a

16

t2

X .1 - 0 Ln -Co i "Ur -WLfl 0 T ,c 4..... rr..l, n¢ '? AT TIME- 21

20

18

It6

C* Ir~ (4C6* -~U~N

a;01c:.,

1;

x .r ~ t!= ,- .- -- CN r o 6o

-10 7-

DISTRIBUTION OF E2 T TIME: 57r/2- - --- - - -- - - -- - - - - - - . . . . . . . . . . ,. . . .,,. I . - . 20

18

16

14

12

10

8

6

4

2

n*. En o n a n l ED m a l a) ol ci' i T - 7-_ q-, -_ x x x x x x x x x , x

0 0 0n 0 0 0 0 0 0 0 u_ _ X X Xn u) X

DISTRIBUTION OF E2 T TIME= 7/2

.......... 0 .... ... .... 0........ .o o o a o o o o o o CD o

0 0 0 0 0 0 0! 0 0 0 0

_ _U,, n - ' ) .

c

DISTRIBUTION OF E2 T TIME: 37r , I I ' I * , ' I I

, . S . . . . I l

I I DI I I

0 0 .0 00 0

DISTRIBUTION OF E2 T TIME= 47r20 .I

J l i i I r 'i I ' I J . . . . . . . . . . . l

I8

14

12

0

8

6

',x x x x x D 'x 'x x ' x _=b6~b0 _ N N N" IM

X X XX X XX

20

18

16

14

12

>- 10

8

6

4

20

18

16

14

12

O10

8

£

4

2

Q

,

2

_ _

n .. . . . . . . . . . . I . . . . . I . . . . . I . . . . . I

-

>- I

-108-

20

18

16

14

12

>- 10

8

6

4

2

n

0

DISTRIBUTION OF E2 AT TIME: 9r/2. * I I , .

o oo 7x x

o o 0 0 0I I I I I

0o 0 0 0

CD 0 0 N

20

18

lb

14

12

10

8

6

4

2

n

0I

DISTRIBUTION OF E2 AT TIME= 11r/2

1820 . .. ., ., .. . o . , . . . . . .

16

14

12

10

8

6

2

X X X X X X X X X x x x x X x

" VI w m R "

20

18

16

12

10

8

6

4

nn

DISTRIBUTION OF E2 T TIME: 5r... I . ...- - --* -I .**1 -*'1 I" -' I"' -I' '' '' '' .'' '. .". ..'.''. ..'. .. .... .. .... ..... ............................ .... .......

I I II I I I I I Io o o o o o o o o o o 0 0 0x x x x x x x x x x x

- N N ar (DU DISTRIBUTION OF E2 T TIME: 67r

I I I I I I I I I I I I I I0 0 0 0 0 0 0 0m 0 0 0 0 0

X X X X X X X x x X x x X

(~O O O O O O ~ a O OO _

. . . I

I

4 mm

1

I

... I .. . . . .. . ... I.... I ···I·· I .. . I.....· I .....1. .. .......... I·.. I1

I

L

· I . . . I · · I 1CL· C ~ ~ ~ ~ ,,- _ - '- 'II-

-109-

OISTRIBUTION OF E2 T TIME: 1 3/2

. N .... .N.... .. .. I ... ., . N ... .. .... .... .. ,,E JN NN N J o) cm o mo co C D a a a o o o o o a

x _ _ _ x x _x x x x_In O U1 0 n 0 U1 0 fl a 0 U' 0 L

_ N N '1 r Wv n ) n o D

Fiure 35. Evolution of two-dimensional disturbances atRe -1000.0, a-0.5 .

LU

18

16

14

12

'- 10

8

6

4

2

n

-110-

disturbance evolves in ways not too different from what

would be predicted by quasi-steady theories; with the growth

rates of the same order of magnitude and distributions of

disturbance energy across the channel similar to the "least"

stable eigenmode of the Orr-Sommerfeld equation (see Figure

36 for a comparison). During this period, because of the

intrinsic nature of the the basic velocity profiles, the

disturbance has moved away from the wall and towards the

center. In the next half period (for example at a time of

3r/2), the disturbance is too far away from the wall to be

able to excite the same modes that were excited earlier.

Instead, it excites modes that are localized further away

from the wall. This "mode-hopping" continues until the

disturbance has travelled all the way to the center of the

channel, at which point the flow settles into a steady-

state. It is this steady-state behavior that a Floquet

analysis of the time-dependent Orr-Sommerfeld equation would

predict.

In the steady-state, the disturbances are synchronous

with the basic flow,i.e; the disturbance at time t+27

differs from that at time t only by an exponential factor

(with no phase change). This means the Floquet exponents are

real, in agreement with the results found by von erczek

Davis. This is to be expected, since it can be shown from

the Orr-Sommerfeld equation that if A is an eigenvalue, so

is its complex conjugate, . Thus the solution should either

-111-

.030

.028

.026

.024

.022

.018

.016

0o .014

.012

.010

.008

.006

.004

.002

0

-_ - fn?

r/2 3r/2

wt

Figere 36. Comparison of the growth rates predicted by thetime-dependent simulation (solid line), to the

least stable eigenmode of the Orr-Sommerfeld

equation (dots) for r/2<t,3r/2. (Two-dimensional

disturbances a-0.5, Re -1000)

^^^

0

-112-' t 4 7

. . . . . .

· : : ; : .. . . . .. . . . . . . . . .

Ill'l . . . . . . . . ..

. . . . . . . . . ...

I .,. ,. .

,- . .l . . . . ., , _ _ . . . ., , , , . . . . . .W . . i

....... · s · : . i' . .

o . o . . .... .

.... . . . . . . . .

. . . . . o . . . . . .

. . . . . . . .. ~ ~ r r .

. .. . . . . . .

.

\ .. ' o · ·

. .1 . . .... . . . . . . . . .

. .. .. . . . · . . . . .o

. . · o. . . . . . . . ...

. o . . . . . . . .. . ...

i . . t . . .,. '''''I ] 11 1 ll i: ' ['':Irll

It~ ,..,/11 1

I I Iw I I I.. . . . .. o o . . . . . . . . .. . . . . . . . . . . . . . .

. ·· . . . . . . . . . . .

, … ….o. .............., , , _ _ ^ s s >

\ , , , , _ _ _ _ _

t ZF *. __ {/ /

, ,\ v_ _ , *

, v s _ _ _ _ . , .

, ^ ^ ^ ^ _ _ . . . .

Cm-t. I-L9

wt-9r/2I I ii Il l i li I i' II I I

. . . . .

.... ......

.,.........

.. o.......

J//..........

ljll/'''Jl;.......11111'I I Si '[~' ' . .

.....' ,.,..........

..........

..........

........ .............

, , , _........I I , ,, _ 1 1\\\\

I N N _

. .

oooo

. \

-N''

N'''

,~I!ill?

1'II.I,

... ..

. ..

. ..

. ..

. .._ ,

_., ./_ - -

111111. . . . . . . . . ..tt II1r % -

Il II N. . ... o ... ....

. . . . . . . . .

. . . . . . . . . .

. . . . . o . . . . .

. .o o . . . . . . . .

.o . . . . . . .o.o.

,.....,\,,.,

111I///

1 1 1 1 1 t . . . . . .l l l ! . . . .. , , , ~.., . . . , ,. , _ , , , , ............

,..........._ _._

............ ,

1..........., . -

t..........., . -

............ ,,

t............,

. . : : . . .I ! : * i t I I I .

0. cltINU vWTcr

sirll S

--I'll/'Il/

- - L I 6

' . .

-113-w t - 5 71

wt-1 l1r/2I I i i I i i

III' -'

I * 1/'~

Ij 1 ''iLl/I,-11111'`ill'','L I I I''

r

.* . II

)//!---- ''Io o . . . . .

. . .. . . . .

lI I I I I I // . I . .. . . . ' . I

...... . . . . . .. . . ................../ ~ ~ ~ 'l I

S~ ~ ~~ rl 1

R . . w ffI .

. z* ¢9 @ @8 @@

I

0. 4 4.ft" =T

i i i.. . .i i I ; I .

ri S @@

. I ·

t. 7LMrmump v arm~t

-114-w t -67r

wt-13r/2

·. ............ ...

·. . .. ...... .....·. o...............· ·...............

L I~..... . . . . ., /, -,. tI .

l/ I ' ' ,, 1 I I I d' t\\ t % \\ t ~1

I I t I

* \ \… d~ X ' / / '… I I I . .

. a… . ...~~~~~~~~~. . . . . .; ' / . . .

. . . . . . . . . . . . . ... . . . . . . . . . . . . ...

. . . . . . . . . . . . ..

. . . . . . . . . . . . . ..

. . . . . . . . . . . . . ..

. . . . . . . . . . . . . ..

. . . . . . . . . . . . . ..

. . . . . . . . . . . . . ..

. . . . . . . . . . . . . .

.. .. o . . . . . o .

. . . . . .; : ; Z . . '

I .! / . . . . * .lll/ /, o

/d I

I11 1 ,,I t …t i[II I,,,,,___ .t , t I , ~, , . ' ' ' \ \

I I I I \ ~ . . ' . . . J l

I a -. … I '////Il, s v _. . ...

…

... . . . . . . . . . . . ..... . . . . . . . . . . . . ...

. . . . . . . . . . . . . .

. . . . . . . . . . . . . ..

. . . . . . . . . . . . ...

. a1-o

Filure 37 The disturbance field at Re6-1000.0, a-0.5 asseen in a frame of reference moving with thecenterline velocity.

G. 33Z04MXUMiL

. . - - - -| X __ l : : [ ~ ! l ....0;1:;X:r

. . . . : . . .

i - , I i i . . . . !

: : I : : I I ! I

iii 1111111

-115-

consist of pairs of simultaneous disturbance waves e At+iax

· At+iaxand e et which propagate in opposite directions, or

else it should be of the form of a standing wave

At+iaxV(z,t)e t+ , where A is real. Furthermore, the .umerical

simulations show that in the steady state,the phase speed of

the disturbance is the same as the instantaneous axial

velocity in the core. In other words the disturbances merely

get convected back and forth with the mean flow. This point

is shown in Fgure 37, where the disturbance field in a

frame of reference moving with the centerline velocity is

shown.

The results for various other combinations of Reynolds

number and streamwise wavenumber are shown in Figures 38 and

39. In each case the evolution of the disturbance is very

similar to the case discussed above; the disturbance starts

off in the boundary layer and reaches a steady-state

solution with eigenmodes near the center of the channel. The

resulting Floquet exponents are plotted in Figure 40. The

general trends in the behavior of A as a function of Re and

a, are identical to those obtained by von Kerczek and Davis,

but the absolute magnitudes of A are smaller. This is to be

expected, since as we have seen, the disturbances are not

localized within the boundary layers but migrate away until

they settle into a steady-state that is compatible with the

boundary conditions. For a channel flow, this steady-state

consists of eigenmodes that are clustered about the center

-116-

0

-5

-10

-15

Ct

-20

-25

-30

-35

0 2 2r 31 41 5r 61 71

Figure 38. Evolution of two-dimensional disturbances with

streamwise wavenumber a=0.3, 0.5, 0.7 at

Re 61000.0 .

I , i -i

).3

.5

).7_i I I

~~~~- - - - - - -

J

I I I I I

-117-

U

-5

-10

-15

Carut, -20

-25

-30

-35

-dn-- 'U

o f 2s 4r 5r 64 71

Figure U. Evolution of two-dimensional disturbances with6a0.5 at Re - 200.0, 500.0, 1000.0

'l I I I I I ' II I I l T I ] I ji 1 ' I -

Re-200.

I I _ I I I . I I I I

,,

. . . . . . . . .

L

3t

-118-

a,0.0

OO

500

0o

a

1000

Re

The principal Floquet exponent as function of

Re and a .

0

-.1

-.2

-. 3

.4

5

a-0.3

a-0.5

a-0.7-. 6

-,7

.8

-. 9

-1.00

Figure 40.

I I~~~~~~~~~~~~~~~~~~~~

-119-

of the channel. For an infinite Stokes layer, the

disturbances migrate away from the boundary layer and

"merge" with the continuous spectrum of the infinite Stokes

layer, as was shown by Hall [1978]. Von Kerczek and Davis,

on the other hand, consider a bounded Stokes layer, which

consists of an oscillating plate with another stationary

plate a few boundary layer thicknesses away. The requirement

that the disturbances go to zero on this second stationary

plate, results in a problem that is distinct from either the

infinite Stokes layer or the channel.

Nevertheless, our results agree with those obtained by

other investigators for other examples of oscillatory Stokes

layers, in that the flow is found to be stable (in the

steady-state) to infinitesimal disturbances,. Not only are

the principal Floquet exponents negative in all cases

studied, but for a given streamwise wavenumber, the

oscillatory channel flow is more stable than the motionless

(Re =0) state; with the Floquet exponent a monotonic

decreasing function of Re6.

Finally, one should not overlook another phenomenon

revealed by the results of the direct numerical simulation;

namely, that if a disturbance is present in the boundary

layers near the wall, it can substantially amplify during

the acceleration portion of the cycle. In an experimental

setting, this result is probably more relevant than those of

the Floquet analysis, since a certain level of noise is

-120-

always present throughout the flow field. In this situation,

and provided the disturbances are small, they would amplify

during the acceleration phase of the cycle and be damped

during the deceleration phase. Indeed, Hino et al.[1978] in

reporting that two classes of transitional flows are

observed experimentally, describe one as consisting of

"weak" disturbances that appear during the acceleration

portion of the cycle.

As a result of this process,very small disturbances

might be amplified to such an extent that linear theory

fails. In that case, and for tose disturbances that are not

sufficiently small to be treated by a linearized theory, one

must study the stability of the flow to finite-amplitude

disturbances. This case is considered next.

-121-

3.3 Finite Amplitude Disturbances

When the perturbation is infinitesimal a linear theory

results. Such a theory is very attractive in that the

resulting instability is independent of the form or initial

amplitude of the disturbance. However, as we have seen,

linear theory cannot explain the transition phenomenon that

is observed in experiments. We therefore, need to consider

more general (finite-amplitude) disturbances.

We start by a consideration of two-dimensional

disturbances of finite amplitude. Even though a two-

dimensional theory cannot explain all the details of

transition (in particular its three-dimensionality),

understanding the two-dimensional behavior is important

because it has been shown [ Herbert and Morkovin(1980) ,

Orszag and Patera(1983) that the transition process for a

large class of "steady" shear flows can be explained as the

instability of the intermediate state consisting of the

basic laminar profile plus two-dimensional finite amplitude

equilibria (or quasi-equilibria) to secondary three-

dimensional disturbances.

-122-

3.3.1 Two-Dimensional Disturbances

The nonlinear theory for Stokes flow instability has

not yet been developed. One problem is the lack of a

critical Reynolds number according to linear theory.

Nevertheless, subcritical finite-amplitude equilibria

(travelling waves with constant energy) may be possible, if

the nonlinear terms in the Navier-Stokes equations are

destabilizing for disturbances with energies above some

threshold value.

There are two routes to determining two-dimensional

finite amplitude equilibria. In the first approach, the

equilibrium solutions are directly obtained by eigenvalue

techniques. In the second approach, evolution of specific

disturbances are followed by direct numerical simulation of

the Navier-Stokes equations. The eigenvalue approach is more

efficient and would directly locate the equilibrium

solutions within the (E,R,a) space (although it does not

give the time scales of evolution to the equilibrium state).

However, it requires some assumption about the specific form

of the equilibrium solution. In steady flows, for example,

the equilibrium solutions are of the form of a travelling

wave with constant phase speed, and can be represented by

-123-

00 ian(x-ct)the stream function, - V ) (z) e . These

n --o n

solutions are steady in a Galilean frame moving with the

(real) wavespeed c and are periodic in x. Truncation of the

fourier series representing would result in a set of N+1

coupled nonlinear (but time-independent) ordinary

differential equations for An which can be solved to give

the equilibrium solutions [Herbert(1977)].

For the Stokes problem, the basic flow is time

periodic. One might therefore anticipate equilibrium

solutions to be of the general form,

00 ian(x-cr)= = 0 (z£r en=-oo n

where c is real (and possibly zero) and r is the physical

time normalized to the basic period of oscillation of the

mean flow. In other words, the equilibrium solutions (if

they exist) are expected to be time periodic with the same

period of oscillation as the basic flow.

The above formulation would result in a nonlinear

eigenvalue problem for the time periodic functions n', which

can be solved in the (E,R,a) space to locate any possible

equilibria.

In the present work, we adopt the second approach and

follow the evolution of specific two-dimensional

disturbances by direct numerical simulations of the Navier-

Stokes equations. Because of the relatively large amount of

-124-

computation time involved in each simulation, we limit our

search to three wavenumbers ( a=0.3, 0.5 and 0.7 ) all at a

single Reynolds number of 1000. In each case the evolution

of the least stable eigenmode of the (time-independent) Orr-

Sommerfeld equation for the initial (frozen) profile is

followed by direct numerical simulation. The initial

disturbance is normalized to have a total energy of 0.04.

The simulations were obtained with a resolution P128,

2N-32.

The results of the simulation for the case a=0.5 are

shown in Figures 41 through 44. The most important question

is whether the disturbances evolve to an equilibrium state,

or do they decay; and if so what are the time scales of

decay.

Figure 41 is a plot of the overall energy of the

disturbance field as a function of time. After an initial

transient period, the disturbances are seen to settle into a

state of slow decay, suggesting that the initial finite-

amplitude disturbance would eventually die away. The decay

rates, however, are 0(1) (actually, A--.38) on a time scale

which represents the oscillation period of the basic flow.

Recalling that Stokes flows result from an inherent balance

between the viscous terms and temporal acceleration ( and as

such the period of oscillation is the same as the viscous

time scale; 62/v 1/w ), the decay of the two-dimensional

-125-

a-. 5

) 7r/2 37r/2 2r

Figure 41. Evolution of two-dimensional finite-amplitude

disturbances for a-O.5, Re -1000.0 .

U

-10

-20

57r/2· _ ·

r

ll l

-126-

finite amplitude state is seen to be viscous. Transition

phenomena, on the other hand, happen on convective time

scales; therefore, if the mechanism that triggers transition

were to involve the above two-dimensional finite amplitude

state, the viscous decay of the two-dimensional disturbance

field would be of little importance during the transition

process. This point will be discussed further in later

sections.

The development of the disturbance field from its

initial "prescribed" form as the least stable eigenmode of

the Orr-Sommerfeld equation, to the final quasi-equilibrium

state can be seen in more detail in Figure 42 where the

distribution of the energy of the disturbance across the

channel is plotted at various times. The initial disturbance

field is localized near the walls with a major energy peak

at the (linear) critical layer and two less prominent peaks

at neighboring inflection points. Within a very short period

(with convective time scales) though, the disturbances

migrate to the center of the channel and settle into a

quasi-equilibrium state where they basically get convected

back and forth with the mean flow, maintaining their energy

to O(1/R).

Both the disturbance field and the magnitude of the

attenuation rate A, are not too different from the Floquet

theory results of the linear theory. Nonlinear effects tend

-127-

Distribution of E2 at time r/2.,-- --- ---- --- --- -I ., , .. _ _

Distribution of E2 at time r/2+lr/1020

18

16

14

12

>- 10

8

6

4

2

0o '0 a Ln 0 a888888888 88

Distribution of 2 at time 7r20

18

16

14

12

>- 10

8

6

4

2

0

a In C) Lu 0 o In C0 Wi 0 ln C

8 8 8 8 8 8 8 8 8 8 8 8

Distribution of E2 at time 3zr/2

0 .0001 .0002 .0003 .0004 .0005 .0006 .0007 .0008

20

18

16

14

12

10

8

6

4

2

n

20

18

16

14

12

10

8

6

4

2

0

.... I..... .. . :FPF·-· --

. . . . . . . . . . . . . . .

f igure 42.

? ..

U

-,- I~ .T,,. ... ..I.. ..~

3 .0005 .0010 .0015 .0020 .0025 .0030 .0035 .0040

-12 8-

Distribution of E2 at time 2r Distribution of E2 at time 57r/2

18

16

14

12

10

8

6

4

2

0Co " w ITc~ J ( co 0 INr CD

c IN O0 N I t

. . . . . . . . . . . .

-_ 0 0 O ( r- cO CD O O O 0 O Oo o o O o o O oo o C C C C

Figure 42. Evolution of two-dimensioanl finite-amplitude

disturbances for a-05, Re -1000.0 . The initial

disturbance is the least stable eigenmode of the

Orr-Sommerfeld equation for the frozen initial

profile.

20

18

16

14

12

10

8

6

4

2

o L . I . I I I I I I . I

,q, , , , , , - , , , .... . - . v , , . r2

i

-129-

to spread out the perturbation over a wider section of the

cross-section, and to result in slightly larger values for A

( A--.38 for the finite amplitude disturbance as compared to

A--.25 for the linear case ).

The effect of the nonlinear terms on the mean velocity

profiles is shown in Figure 43. The biggest correction

appears near the center of the channel and away from the

boundary layers' as this is where the disturbances are

localized.

Finally, the contribution of various streamwise Fourier

modes to the overall energy of the disturbance field is

shown in Figure 44. Even though 16 fourier modes (2N=32)

were used in the simulation, only the first five are shown

in Figure 44, as higher modes have very small amplitudes. At

the time wt=S5/2, for example, the various modes have the

following energy contents:

n 1 2 3 4En 0.112E-02 0.102E-05 0.630E-7 0.858E-08

n 5 10 15En 0.171E-08 0.621E-12 0.707E-16

It should be noted however, that even though the higher

fourier modes become unimportant at large times, an

insufficient number of streamwise fourier components would

affect the initial transient behavior of the disturbances;

mainly because small round off errors introduced in the

boundary layers could result in new instabilities because of

-130-

wt-7r/ 21.

1.0

.8

.6

.4

-.2

- 0

-. 8

! IC)C1 2 4 6 8 10 12 14 16 18 20

z

Wt lr1.2

1.0

.8

.6

.4

3 .2

-.- .2

-. 6

wt-37r/2I .i

1.C

-.E

-. E

-I ._.

-.-

-.

-I .

0 2 4 6 8 10 12 14 16 18 20

Figure 43

0 2 4 6 8 10 12 14 16 18 20z

wt -27r

0 2 4 6 8 10 12 14 16 18 20z

1.2

1.c

.8

.6

.4

-.8

-1.0

-1.2

. n

& .6

I I I I I I I I I I I

I . I

I

I

i

I

I

I

I

-131-

wt-Sir/2

0 2 4 6 8 10 12 14 16 18 20z

Figure 43. Average velocity profiles in the presence of two-

dimensional finitite-amplitude disturbances

( a-0.5, Re 1000.0 ) compared to the laminar

profiles.

1.2

1.0

.8

.6

.4

-.2

- 0

-.4

-. 6

-. 8

-1.0

-1.2

-132-

0 7r/2 r 3/2 27r 57r/2

Figure 44 Development of higher order modes for an initial

finite-amplitude disturbance with a-0.5,

Re -1000.0

0

-1

-0

-20

n-I

n- 2

n-3

n-4

n-5

n-6

ILY·�\

-133-

the highly unstable nature of the boundary layers. For the

above simulation convergence has been verified by repeating

the simulation with 2N-16, P-65, which gave nearly identical

results.

-13 4-

3.3.2 Three-Dimensional Instability

When only two-dimensional disturbances are allowed the

flow, as we have seen, evolves to a quasi-equilibrium state

of travelling waves that decay over viscous time scales. If

no further interactions with other disturbances take place,

this flow remains well ordered at all times and shows no

sign of the chaotic small-scale structure of real turbulent

flows. However, the same two-dimensional state may undergo

further instabilities if subjected to infinitesimal three-

dimensional disturbances [ Herbert and Morkovin (1980)

Orszag & Patera (1983) ].

Let A be the amplitude of the two-dimensional wave and

B that of the infinitesimal three-dimensional disturbance.

With A>>B, the development of the two-dimensional wave is

not influenced by the three-dimensional disturbance, but the

amplitude of B depends on A according to Stuart(1962) i,

d(lnB) 2= bo + bAI +

dt

With bO, there are two possibilities; if b is negative,

the presence of the two-dimensional wave causes stronger

damping of the three-dimensional disturbance. However if bl

is positive, the two-dimensional wave feeds energy into the

three-dimensional disturbance; leading to an exponential

growth of the three-dimensional wave. The exponential growth

-135-

of the three-dimensional disturbance does not continue

indefinitely, as in a short time the amplitude of B becomes

comparable to A and the problem has to be reformulated.

Nevertheless, this secondary instability of two-dimensional

finite amplitude states to infinitesimal three-dimensional

disturbances seems to explain universally many features of

the early stages of transition in a wide class of "steady"

shear flows as shown by Orszag & Patera [1983]. We

therefore, seem ustified in investigating the stability of

the two-dimensional finite-amplitude states of the

oscillatory channel flow (discussed in the previous section)

to secondary three-dimensional disturbances.

Once again, we employ direct numerical simulations of

the full time-dependent Navier Stokes equations for our

investigation. In particular we will study the time

evolution of flows resulting from initial conditions,

v(x,y,z,t=o) - U + 2D + V3D

Here U is the basic laminar flow, v2D is the initial finite

amplitude two-dimensional disturbance with wave vector (a,O)

which is specified to be in the form of the least stable

eigenmode of the Orr-Sommerfeld equation for the frozen

initial profile and is normalized to have a total energy E2D

of 0.04; and V3D is the initial infinitesimal three-

dimensional disturbance which has the form of a streamwise

vortex with wave vector (0,) and has a total energy of

-136-

10 -8 . Only one mode is kept in the spanwise direction owing

to linearity and separability. N modes (2N-32) are kept in

the streamwise direction because of nonlinear effects.

As the instability depends critically on the

persistence of the two-dimensional finite amplitude state

down to transitional Reynolds numbers (since in the absence

of the two-dimensional wave the three-dimensional

disturbance has to die away according to linear theory), we

will choose for a, the most stable of the three wavenumbers

studied in the previous section, namely a-0.5 .

The stability of this flow to disturbances with various

spanwise wavenumbers , simulated numerically at Re 61000,

is illustrated in Figure 45. To begin with, the simulations

verify that there is, indeed, a strong three-dimensional

instability. The three-dimensional disturbance grows in

energy by a factor of 10 in the time it takes the fluid to

travel 5 channel widths at maximum bulk velocity. This

corresponds to a convective (and not viscous) growth rate,

and suggests that the instability mechanism is inviscid.

Secondly, the instability is most effective at P>0(1),

emphasizing the three-dimensional nature of the instability.

There is a weak maximum in the magnitude of the growth rate

at P 1, but beyond that the preference in is very weak.

- 1 37-

ar .5 ,.1 .0

X -10

-20

ir/2 3r / 4r

Fiure 45 Spanwise wavenumber dependence of the three-

dimensional growth rate at Re 1000.0, ca-0.5

-2 .0

-138-

We may characterize the transition process by three

parameters; the transitional Reynolds number, the time scale

of the instability and the spatial dmensionality of the

perturbation responsible for the instability. We have so far

shown the instability described above is three-dimensional

and grows on convective time scales. It still remains to be

shown that the instability cuts off at Re 6 500, which is

the value observed experimentally for transition. This point

is demonstrated'in Figure 46 where the evolution of the two-

dimensional quasi-equilibrium state and the three-

dimensional perturbation are followed at Reynolds numbers of

200, 500 and 1000 for a=0.5 and P-1.0 . Note that a Reynolds

number on the order of 500 is singled out as the critical

Reynolds number below which three-dimensional perturbations

decay. This statement may require some clarification

considering the observation that the three-dimensional

disturbance shows positive growth at all three Reynolds

numbers. The point is, that the three-dimensional

disturbance needs the two-dimensional finite-amplitude wave

to continue its growth. For Reynolds numbers below 500

before the three-dimensional disturbance has grown to finite

amplitudes, the level of the two-dimensional wave has

dropped to such small amplitudes that it can no more

maintain the instability process. As a result the three-

dimensional wave never grows to finite amplitudes and the

instability is turned off. This argument has been verified

by continuing the simulation at Re -200 to longer times, and

-139-

U

X -10

-20

r/2 3r/4

Figure 46- Reynolds-number dependence of the three-

dimensional growth rate at a0.5, = 1.0

Observe that disturbances decay for Re <500.

7r

-140-

the three-dimensional disturbance is indeed seen to

ultimately decay as a result of the decay of the two-

dimensional wave. Note also that the the rate of decay of

the two-dimensional base state is roughly the same in all

three Reynolds numbers, indicating that the cutoff is due

mainly to viscous damping of the three-dimensional

perturbation.

-14 1-

3.3.3 Secondary Instability and Transition

We would like to determine to what extent the secondary

instability mechanism discussed in this section relates to

the transition process that has been observed in

experiments.

With respect to transition, the single most important

parameter is the Reynolds number. The critical Reynolds

number predicted by the secondary instability has already

been shown to be in good agreement with the value of 500

that is observed in experiments.

The next question is whether the secondary instability

presented here saturates in an ordered state when it reaches

finite amplitudes or whether it results in chaotic behavior;

and in general to what degree does the resulting flow

resemble turbulent flow structures that are observed in

experiments. To answer this question we have carried out a

large numerical simulation at a Re of 1000. and with

initial disturbances with wavenumbers a=0.5, -1.0 .

The initial development of an infinitesimal three-

dimensional disturbance ( with wave vector (0,6) and initial

-14 2-

energy-10-8 ) in the presence of a finite-amplitude two-

dimensional wave ( with wave vector (a,0) and initial

energy-4xl0- 2 ) was first simulated using a run with

resolution P128, 2N-32, 2M-2 and starting at time t-r/2. At

the time t9r/10 when the three-dimensional disturbance had

grown to an energy of .015 (and with En2 still equal to

.0475) the run was restarted with resolution P-128, 2M-32,

2N=16.

The results of the full simulation are shown in Figures

47 and 49. In Figure 47, the overall energy of the three-

dimensional disturbance field (integrated over the channel

cross-section) is plotted as a function of the phase of the

cycle. For comparison the overall energy of the disturbances

(integrated over the pipe cross-section) from the

experimental run at Re =1080, A=10.6 is also shown in Figure

48. Note the excellent agreement between the results of the

simulation for 3/2<wt<5r/2 and experimental results. (The

results of the simulation for r/2wt3r/2 are not in as good

an agreement because of the effect of the artificial initial

conditions). In particular the disturbances can be seen to

decay during the acceleration phase (3r/2<wt<2r) and

explosively reappear with the start of the deceleration

(wt-2r) in accord with experiments. Furthermore, the plots

of instantaneous average velocity from the numerical

simulations show striking similarity to the experimental

profiles (Figure 6), in that for the acceleration phase the

-143-

. 1

.16

.14

.12

.10

.08

.06

.04

.02

0

r/2 3r/ 2 57r/2

wt

Figure 47Z Phase variation of the overall energy of three-

dimensional disturbances from the numerical

simulation (Re6-1000. H-10.0)

. A

-144-

.10

.16

" .14l-

elC.4

.12

+ .10

C%1(.

' .06

.04

.02

n

-/7r/2 3r/2/2wt

Figure .8. Phase variation of the ove'rall energy of

disturbances from experiments (Re -1000.,A-10.0).

Here the value of w' is estimated to be equal to

V' .

,,

-145-

wt - 3/2 4 r/8 +7r/4

2r' +7r/8 +r/4 +32r/8

+3r /8

U/U0

5r/2

U/U0

Figure 49. Average velocity profiles for various phases

from the numerical simulation at Re -1000.,

A-10.0 (curve B) compared to the laminar

solution (curve A).

10

a

4

22

a

,0

a

S

2

0

0-1

"I

1P.

--

-146-

profiles are "full", while during the deceleration phase

they are inflectional outside the boundary layer.

-147-

IV. SUMMARY AND CONCLUSIONS

In this work we have attempted to bridge some of the

gaps between theory and experiment regarding transition and

turbulence in bounded oscillatory Stokes flows. Earlier work

on this subject consist of, on the one hand, experiments

that show the flow undergoes transition at a Re6500; and on

the other hand, theoretical studies that are either in the

form of Floquet analyses of the "linearized" equations and

predict all disturbances would decay, or are in the form of

various quasi-steady treatments of the "linearized"

equations which in turn are subjects of dispute because of

the assumption of quasi-steadiness.

Our experiments for oscillatory flow in a pipe, for the

range of parameters 230<Re <1710, 3<A<10 , are in good

agreement with earlier investigations and show the flow

undergoes transition at a Re'-500 independent of the value

of A. The resulting turbulent flow is characterized by the

sudden explosive appearance of turbulence with the start of

the deceleration phase of the cycle, and reversion to

"laminar" flow during the acceleration phase. Instantaneous

profiles of ensemble-averaged axial velocity provide a

strong 'clue' to the mechanism of instability. During the

acceleration phase, these profiles are "full" and resemble

-148-

the laminar profiles that would be obtained for sinusoidal

flow starting from initial conditions u(r,O)=O. During the

deceleration phase, however, the velocity profiles are

inflectional ust outside the boundary layers as a result of

momentum exchange between the low speed fluid near the wall

and the higher speed fluid further towards the center. These

profiles have a striking similarity to "instantaneous"

profiles that have been observed in steady boundary layers

during the ejection phase of the turbulent bursts (at

"peaks"); and suggest that the fundamental process of

generation of turbulence in the oscillatory flow may be in

the form of bursts similar in nature to those that occur in

steady boundary layers. The relaminarization of the flow

during the acceleration phase can then be attributed to

cessation of "bursting" because of the favorable pressure

gradients that are present during this period.

The starting point for a burst in steady shear flows,

is the development of streamwise vorticity as a result of

nonlinear interaction of two- and three-dimensional

disturbance waves. Furthermore the development of the three-

dimensional waves can be traced back to instability of

infinitesimal three-dimensional perturbations in the

presence of finite-amplitude two-dimensional equilibria or

quasi-equilibria. Both the nonlinear interaction and the

development of infinitesimal three-dimensional

disturbances, are independent of the shape of the mean

-149-

profiles; and are the result of interaction of two- and

three-dimensional waves in a general shear flow. It

therefore, seems reasonable to expect the same mechanism to

prevail for oscillatory flows, provided the two- and three-

dimensional waves can develop. The time variation of the

flow, in this case, would be unimportant as long as the

ratio Re6/A, representing the ratio of time scale of

oscillation to the (convective) time scale of the

instability, is large.

To substantiate these ideas, we have studied the

stability of the oscillatory flow in a channel (for A=10) to

various classes of infinitesimal and finite-amplitude two-

and three-dimensional disturbances. The results of this

study can be summarized as follows,

- For infinitesimal disturbances, Squires' theorem can

be extended to the oscillatory flow problem, with the

conclusion that only two-dimensional infinitesimal

disturbances need to be considered. The development of a

two-dimensional disturbance which initially is in the form

of the least stable eigenmode of the Orr-Sommerfeld equation

for the "frozen" initial profile is followed by direction

simulations of the Navier-Stokes equations. The results of

the simulation immediately explain the discrepancies between

quasi-steady theories and Floquet theory treatments of the

linearized equations. For the first cycle the disturbance is

-150-

seen to develop in ways not too different from quasi-steady

predictions. During this period the disturbance follows the

inflection points or critical layers of the instantaneous

profiles. Because of the intrinsic nature of the profiles,

at the end of the cycle the disturbance ends up in a radial

location too far away from the wall to be able to perturb

the same modes that were excited earlier. Instead it excites

modes that are located further away from the wall. This mode

hopping continues until the disturbance has moved all the

way to the center of the channel, at which point it settles

into a periodic steady-state and decays as predicted by

Floquet theory. It should be noted that while the

disturbance is still localized near the boundary layers, it

does amplify with inviscid growth rates. This inviscid

growth, however, can not explain the transition phenomenon

that is observed in experiments; because maximum growth for

this type of instability is observed around flow reversal

(i.e, at the start of acceleration) which is opposite in

phase to where turbulence is observed in experiments. These

disturbances, however, may explain the "weakly turbulent"

flow regime that was observed in Hino's experiments during

the acceleration phase of the cycle.

- No finite-amplitude two-dimensional equilibria were

found, however quasi-equilibria (with viscous decay rates)

were observed for two-dimensional finite-amplitude

disturbance at transitional Reynolds numbers.

-151-

-The secondary instability of the above qu,asi-

equilibrium states to infinitesimal three-dimensional

disturbances can successfully explain the transitional Re6

of 500 that is observed in experiments. The infinitesimal

three-dimensional disturbance is seen to grow explosively

(on convective time scales) to finite amplitudes. The

mechanism of instability is the result of interaction of the

two-and three-dimensional waves and is independent of the

shape (and inflectional nature) of the profiles. Here, again

the time variation of the basic flow is not of much

consequence as long as the ratio Re /A>>,,1.

- Nonlinear interaction of the two- and three-

dimensional disturbance waves, shows structures that are in

very good agreement with experiments. Disturbances are seen

to decay during the acceleration phase of the cycle, and

explosively grow with the start of deceleration. The

instantaneous average velocity profiles are inflectional

during the deceleration phase and are full during the

acceleration.

-152-

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