Multiple paths to subharmonic laminar breakdown in a boundary layer

NASA ICASE

Contractor Report

Report No.

181948

89-78

ICASE MULTIPLE PATHS TO SUBHARMONIC LAMINAR BREAKDOWN IN A BOUNDARY LAYER

Thomas A. Zang M. Yousuff Hussaini

Contract

No.

October

1989

Institute NASA Hampton, Operated

NAS1-18605

for Computer Langley

Applications

Research

Virginia

in Science

and Engineering

Center

23665-5225

by the Universities

Space

Research

Association N90-13722

(NAeA-CR-181948) MULTIPLE PATHS TO SU6HARMONIC LAMINAR B_EAKOOWN IN A BOUN_ARY LAYER Final Report (ICASF) 15 p CSCL 2OD

G3/34

National Aeronaulics and Space Adminislration Langley Research Center Hampton, Virginia 23665-5225

unclas 0251708

MULTIPLE

PATHS

BREAKDOWN

TO

SUBHARMONIC

IN

A BOUNDARY

Thomas

LAMINAR LAYER

A. Zang

Computational

Methods

Branch

NASA

Research

Center

Langley Hampton,

VA 23665 and

M. Yousuff

Hussaini

_

ICASE NASA

Langley Hampton,

Research

Center

VA 23665

ABSTRACT

Numerical duced

by the

dimensional shear

simulations

layer

secondary subharmonic

demonstrate instability disturbances

that

laminar

of two-dimensional need

not

take

breakdown

in a boundary

Tollmien-Schlichting the

conventional

layer

waves lambda

in-

to three-

vortex/high-

path.

1This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NASl-18605 while the second author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. J_

The classicalboundary-layer transitionexperiments of the early 1960%

[I],[2],[3]led

to the perception that the laminar-turbulent transitionprocess in a B1asius boundary layer may

be

called

viewed

as a sequence

the

primary

understood.

The

and

the

of instabilities.

secondary

primary

The

instabilities,

instability

first

two

are linear

in an incompressible

instabilities at onset

flat-plate

in the

and

sequence,

are now very

boundary

layer

well

is known

as a Tollmien-Schlichting (TS) wave. The TS wave, as it equilibrates nonlinearly, becomes susceptible to what is known as the secondary instability. This is truly three-dimensional in character

and

essentially

of three

instability, the

is thoroughly

and

secondary

on the

vortices

fundamental tertiary

vortex

the

much

secondary

wave

tices

which

primary

later

stabilities

The

detuned waves

To date,

the

shorter

longer

wavelengths

Numerical shear

layers

instability which

and

address

This

instability

of the

stage

suffers

peak

is possibly

imminent.

classical

a

in the

this stage

what

becomes

secondary

so-called

with

through

formation

system

the

Also associated appear

and

evolution

layers.

all of the

that

layers

are

and

This

experiments,

streamwise

the resulting

involves

to yield

their associated detuned

with

spikes

the

of the

of lambda

combination

vor-

modes,

the

wavenumber

have

not been

observed

Instead,

these

of the spectrum,

formation

exper-

wavelength

breakdowns.

broadening

observed

pattern

so-called

combine

by a rapid

associated

the

was

of "spikes",

but

of the

not

in-

just

to

also to even

[7].

of the

of subharmonic details

and

or the

of boundary-layer

spikes

case,

wave, instability

subharmonic

wavelengths

the

vortex

and an associated

dominated

wavenumbers

to breakdown

simulations

stage

primary

in which

nonlinear

are

secondary

the

a streamwise

The

spikes

both

ensues

a "spike".

subharmonic

secondary whose

the

to lead

streamwise

and

layer

spot

to have

of the primary

high-shear

for either appear

called

turbulent

case,

high-shear

multiple

a subharmonic

In the

that

of secondary

wave.

and

from

[5]-[6].

is double

experimentally

usually

is now known

arising

is staggered.

i.e., a pair

to detached

Thereafter,

the subharmonic

splitting") direction.

instabilities

breakdown.

breakdown

imentally

leads

secondary

first

wavelength,

streamwise

oscillogram,

K-type

In the

by a kink in the high-shear

of instabilities,

scenario,

is called The

instability

elements.

sequence

streamwise

known

instability,

instability.

(a "peak-valley

in the

manifested

perturbation

particular

same wave

aligned

secondary

hairpin

a rapid

the

[4]. The

secondary

secondary

primary

are

instability

streamwise are

have

of the

by Herbert

- the fundamental

detuned

wave

scale

lambda

kinds

the

reviewed

K-type

transition

of the

the characteristic

[s],[g],[10],[ii] and the

breakdown

transition

have produced

[12]. In this paper

laminar

breakdown

of boundary-layer

transition

we report

induced

by

early

high-

secondary

numerical

simulations

subharmonic

secondary

instability. Numerical

simulations

and

Hussaini

[9]. In the

the

evolution

is tracked

point not

z0 and necessarily

present in time

the approximation in t),

the

base

simulations rather is that flow

have

the parallel

than

in space.

the displacement is strictly

in the

been

reviewed

flow approximation The

focus

thickness streamwise

is on the is constant direction

recently

by Zang

is employed vicinity

and

of some

in x (although and

it is given

by ub(y,t) = (us(y%/'i'_), 0, 0), where us(r/) is the Blasius velocity profile which follows from the similar boundary-layer equations, and X = zo + eft; c! is the speed of a moving computational

frame.

Lengths

are

scaled

by

the

displacement

thickness,

6_, at z0,

and

velocitiesby the free-streamvelocity, U_.

In these

Although

to be parallel,

the

boundary

layer

[12], with the speed of the The numerical simulations the

algorithm

described

is assumed

units,

X/zo

it is permitted

computational frame, c1, equal reported here were performed in [13], with

the

nonlinear

= 1 + R_--;cft, where to thicken

d = 1.72. in time

to the phase speed of the 2D wave. with the boundary-layer version of

terms

treated

in skew-symmetric

form

[14]. If a and rections, L, and

fl denote

the

respectively,

= 27r/a

and

representations

the

L_ = 2_r/19.

s_ are integers

direction. al numbers

fundamental

then which

(In the cases k, = _,/s_ with

wavenumbers

fundamental

The

specify

imposed the

in the

wavelengths periodicity

number

streamwise

in these lengths

of subharmonics

are

and

directions s_L,

that

are

and

spanwise

di-

are

given

by

where

s_

s,L_,

permitted

in each

presented in this paper s, = 1 and s, = 1, 2, or 9.) The rationand /c, = _/s, label the Fourier wavenumbers in the numerical

respect

to the

fundamental

wavenumbers

a and

t9.

Case KKVF KKVS

Re 1100

wave TS

a 0.250

/3 0.000

1100

TS

0.250

1100

TS

0.250

0.209 0.000

.08624

+ .00333i

0.018

0°

2

- .01946i

0.008 36 °

2

36 °

2

36 °

9

0.209 0.000

KLSL

730 730

SQ TS

0.127 0.254

0.241 0.000

KLD

730 730

SQ TS

0.127 0.254

0.241 0.000

730

SQ

0.113

0.241

Table

.03385

.09292 + .00160i .03925 - .02254i .09292

where

U2D(y)

Sommerfeld

simulations

normalized

so that

the

velocity.

oblique

the

least

The

parameters

in mind.

numerical

recommended

of Kovasznay, corresponding amplitudes shows wave cases has

the

two

modes

of the

a and

amplitudes

the

points

both normal

Zang

phase

ft.

relevant These

are 1, and,

shift

and actual

types shear,

undergone

whether

between

equation

when

the

(Orr-

eigenfunctions used

are

in the form

e3D measure

relative

the to the

two-dimensional

and

the

with

(However,

initial

wave

with

for

resolution

to meet

required

grids

on the

symmetry

particular

component

w, £s included

sufficient spanwise

were chosen

the

order

was

is a TS

reference.

The

criterion

of 192 × 96 ×

exploited

to halve

direction.)

were

performed

Vasudeva laminar

and

detached

symmetry

and after 4_ periods layer is present and (The

figure

KKVS

of the K-type simulations with

1 demonstrates

strong

spanwise

conditions

respectively,

Figure exhibit

in the

the

KKVF

breakdowns,

of breakdown

two roll-ups.

under

[2]. The

experiment.

Ou/Oy,

1. They

frequency,

[15]; this typically stage.

subharmonic

in Table

temporal performed

in that

simulations

to the

indicates

for the KKVF simulation, a strong detached high-shear already

e_[(o,ls®)_-_z] },

linear

are given

The were

and

Komoda, and

table

breakdown

of grid

first

fundamental

1

at x = 0 and z = 0. Thus, e2D and of the 2-D and oblique disturbances

8 measures

wave.

simulations

laminar

number The

The

(SQ)

by Krist

256 by the the

form

wavenumbers

streamwise

of five simulations

or a Squire

direct

1.78 x 10 -5

waves.

experiments wave

angle

- .02082i

(1)

stable

prescribed

maximum

The

0.00622

Simulations

+

given in Eq. (1), these maxima occur maxima of the streamwise fluctuations freestream

4.5 x 10 -s

+

for the

their

- .02254i + .00160i

are of the

.+

U_D(y ) are

and

or Squire)

0.00622

.03630

i: Nonlinear

0.00622 1.78 x 10 -5

+ .00160i

.03925 .09292

u(x,0) = 1

1

0.018 0.008

0.125 0.254

for the

8 x

+ .00333i - .01263i

SQ TS

conditions

0

.08624 .09396

ii00 730

initial

C

0°

KLSH

The

W

that high-shear

plane

after

experiment addressed

the former

the

simulation

at these

high

initial

layers.

The

figure

3_ periods

of the

2-D

for the KKVS simulation. In both by the time of the illustrations each

corresponds

to the two-spike

stage.)

Three-

dimensionalflow-field visualizations of both casesreveal that the transition is associated with the formation of a large-scalelambda vortex/detached high-shearlayer for which the breakdownoccurs first in the symmetry plane and then spreadsoutwards in the spanwise direction. (The vortex lines for the KKVS caseshownthe left frame of Figure 2, for example, indicate the presenceof a strong large-scalelambda vortex.) The subharmonicinstability evolvesmore slowly in this case,as is typical of secondary instabilities at large amplitudes of the primary wave. Nevertheless,the KKVS simulation doesfurnish unmistakeableevidencethat strong detachedhigh-shearlayers are not exclusively the province of K-type breakdowns. Continuation of these two simulations to later times showsthat the strong similarity of the breakdownsremains,except,of course,that in the former casethe patterns are aligned in the streamwisedirection, whereasin the latter casethey are staggeredwith twice the streamwisewavelength. We shall refer to this type of breakdown,which in this caseis commonto both the fundamental and subharmonicinstabilities, as a primary-scalebreakdown in deferenceto the scaleof the vortical patterns (relative to the scaleof the primary wave)which ultimately roll-up. The next two simulations were under the conditions of the Kachanov and Levchenko [6] experiments, which focusedon subharmonic instabilities. The initial amplitudes here, particularly for the three-dimensionalcomponents,were much smaller than those for the other

experiment.

until

after

KKVS KLSL

3-D

the

cases

Indeed, primary

the

cases

the amplitudes

wave

had

breakdowns

differed

only

passed

occurred

in the

Figure 3 illustrates the components for-t}le-_e

were

so low that

Branch

II of the

well before

amplitude

time

immediate

down

in these

down

in the

contains

only vortex

smaller

scale

breakdown

but

detached it originates

plane familiar

or "peak"

plane

and

spreads

places,

scale

the

roll-up,

such

as the

breakdown,

than

KLSH

at many

the

primary

simulation KLSL although

and

KLSH

and

in the

4.

right

The

frame

KLSL

in the

does

in the which

near-wall

such

this

much

with

some

in the

sym-

subharmonic

This

in detached

break

that

commences

region.

eventually

break

visualizations,

direction.

places

not

plane

develop

which

spanwise

in some

does

break-

symmetry

2, reveal

5. Thus,

of a breakdown

plane,

case

are associated

shear

in Figure

laminar

The

of Figure

These

strong

locations,

ultimate

previously.

plane.

as shown

the vortices

of the

wave

breakdown shear

We shall

down

in the layers refer

to

are on a much

wave.

exhibits

case.

KKVF

The

energy [9] in the 2-D and II is reached at t = 675,

three-dimensional

symmetry

outward

symmetry

since

and

paradigm

spanwise

kinetic Branch

discussed

layer,

wall,

(In the

perturbation.

character

Eventually,

the

classical

occurring

the

layers.

near the

diffuse,

than

shear

follow

in other

The

from

not

it as a small-scale smaller

shear

case shown

away

but does

symmetry,

KLSL

strong

3-D

in Figure

two simulations

detached

form

is the

is documented

as the

weak

vortices

plane,

is more

a very

initial

did not occur

primary wave. However, the primary from the subharmonic waves [16].

for this paper

This

manner

lines for the

moderately metry

two cases. same

as the

interest

curve.

II was reached.)

evolution of the normalized simulationsl On tl_is scale

two

breakdown

neutral

Branch

of the

as evidenced by the subsequent decay of the eventually resumes growth because of feedback Of more

laminar

The

smaller

a stronger shear

layer

scale

features

detached of the form

high-shear

latter off the

case

does

symmetry

layer

in the

eventually plane

symmetry exhibit

as well.

This

the is

an intermediate casebetweenthe distinct primary-scalebreakdownof the KKVS simulation and the small-scalebreakdownof the KLSL run. The principal factor determining the type of laminar breakdownwhich occursappearsto be the amplitudeof the primary waveat the stageof incipient three-dimensionality, i.e., when the

three-dimensional

component

reaches

a level

comparable

to that

the primary amplitude is large at incipient three-dimensionality, occurs; when it is small, a small-scale breakdown ensues; and level, the additional

breakdown numerical

boundary-layer

thickness

a primary-scale amplitude chosen

has a mixture simulations.

one.

held Here

fixed

the

for a primary-scale for the

breakdown

KLSH

To date

there

have

been

breakdowns.

as they

in the

were

the

wave

and

actual

shear

experiment

focusing

observations

behavior

When

then

reaches larger

simulation

of the

primary

KKVF

to have

amplitude is associated TS

wave

KKVS

simulation. done

past

layers

in the

been

is

primary-scale

high-shear

to identify

to the

appear

a suffÉcient

a clean

of detached

by two with the

is manifestly

KLSL

be as easy

[2] corresponding

on subharmonic

and

if an even

stability

would

breakdown

unstable

of the

eventual

layers

the

Moreover,

breakdown

the

then

e4 = 1.78 x 10 -4,

no experimental Such

value,

to occur.

primary.

This speculation is supported simulation is repeated, but

is always

with

small-scale

amplitudes

subharmonic no experiments

primary

for example,

Thus

initial

at its initial

breakdown

case,

develops.

with its small Branch II.

of both features. When the KLSL

of the

a primary-scale breakdown when it is at an intermediate

for case

However, at such

large

amplitudes. At lower ing to even KLD, wise

longer

allowed

scales

mode rapid

Note

This

type

is quickly

The breakdown although

reveal

that

however,

that

in a boundary features

of both

The authors are pleased Herbert and Y. Kachanov.

both

final simulation, for 9 stream-

The can

to illustrate basic

study

two

that types

in a given

helpful

5

has of the

for the

detuned there

are

primary

is indicative

the

(4/9,1)

initial

and

conditions,

together.

to a primary-scale which

amplitudes

be present

to acknowledge

leads those

of the

of the

allow not

Notice

also

initial

been

Flow

breakdown.

for only one yet

of the the

3 _, s "7, 13 andS. as _,

harmonics

parameter low

both

evolve

broaden-

wavenumber

in the

eventually

than

is also possible aimed

is included

simulation

complete

layer. types

The domain

streamwise

involves

difference

expensive

at sufficiently

here

the

which

modes

and

particular

more

breakdown

reported

interactions.

computational

setting

the former

sum

so a more

a small-scale simulations

and

this

by

instability

only

other

is much

harmonics,

suspect,

component

generated

in the

by spectral

wavenumber to be 4 the the streamwise 6f various harmonics. This simulation

although

of such

and

breakdown

difference

room

wave

secondary

that

of simulation

streamwise We

primary

[6], [7] suggest

sum and

by leaving

component the evolution

development

field visualizations

reports

by various

or combination,

1) modes.

latter

and

effects

of the

three-dimensional Figure 6 shows

a de-tuned, (5/9,

experimental

for both

subharmonics

initial wave.

the

amplitudes

or two

performed.

three-dimensional

instability.

are several

primary-scale

paths and

to laminar small-scale,

case.

discussions

with Thomas

Corke,

Thorwald

References [1] P.S. Klebanoff, boundary-layer

K.D. Tidstrom, and L.M. instability. J. Fluid Mech.,

Sargent. 12:1-34,

The 1962.

three-dimensional

nature

[2] L.S.C. Kovasznay, H. Komoda, and _ B.R. Vasudeva. Detailed flow field In Proc. 1962 Heat Transfer and Fluid Mechanics Institute, 1-26. Stanford Palo Alto, 1962. [3] F. 1%.Hama and J. Nutant. Detailed flow-field a thick boundary layer. In Proc. 1963 Heat 77-93. Stanford Univ. Press, Paio Alto, 1963. [4] T. Herbert. 526, 1988.

Secondary

instability

observations in the Transfer and Fluid

of boundary

layers.

Ann.

[5] Yu. S. Kachanov, V. V. Koslov, and V. Ya. Levchenko. wave in a boundary layer. Izv. Akad. Nauk SSSR. Mekh. Russian). (See also Fluid Dyn. 12:383 (1978).) [6] Yu. S. Kachanov laminar-turbulent

and V. Ya. Levchenko. transition in a boundary

Rev]

in transition. Univ. Press,

transition: Mechanics

Fluid

of

process in Institute,

Mech.,

20:487-

Nonlinear development of :a Zhidk. I Gaza, 3:49, 1977 (in

The resonant interaction layer. J. Fluid Mech.,

of disturbances at 138:209-247, 1984.

[7] T. C. Corke. Measurements of resonant phase locking in unstable axisymmetric lets and boundary layers. In Nonlinear Wave Interactions in Fluids, (1%.W. Miksad, T.1%. Akylas and T. Herbert, eds.), 37-65. ASME, New York, 1987. [8] A. Wray and M. Y. Hussaini. Numerical Roy. Soc. Loudon, Series A, 392:373-389,

experiments 1984.

in boundary-layer

stability.

Proc.

[9] T. A. Zang and M. Y. Hussaini. Numerical simulation of nonlinear interactions in channel and boundary-layer transition. In Nonlinear Wave Interactions in Fluids, (R.W. Miksad, T.R.. Akylas and T. Herbert, eds.), 131-145. ASME, New York, 1987. 5_

[10]

E. Laurien and L. Kieiser. Numerical simulation Of boundary-layer sition control. J. Fluid Mech., 109: 403-_40, !980.

[11]

K. S. Yang, P. 1%. Spalart:, and J. H. FerzigerTransition in Boundary Layers. 1%eport No. Stanford University, 1988.

_

_

=

[12] P. 1%. Spalart and K. S. Yang. Numerical flow. J: Fluid Mech., 178:345-365, 1987. [13]

T. A. Zang and S. E. Krist. Numerical channel flow. Theoret. Comput. Fluid

[14]

T. A. Zang. tions. Appl.

[15]

S. E. Krist and TP 2667, 1987.

On the rotation Numer. Math., T. A. Zang.

r

tran-

Numericai Studies:on Laminar'turbulent TF-37. Dept. of Mechanical Engineering,

\

!

study "

experiments Dynamics,

Simulation

[16] J. D. Crouch. Analysis of three-dlmenslonal of NASA/ICASE Workshop on Instability Voigt, eds.) Springer, New York, 1989.

and

.....

and skew-symmetric in press. Numerical

transition

of ribbon-induced

transition

on stability and 1:41-64, 1989, forms

transition

for incompressible

of Channel

Flow

in Blasius

in plane

flow simula-

Trans{tion.

NASA

structures in boundary layers. In Proc. and Transition, (M. Y. Hussaini and R. G.

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Report

an(_

I. Report No, NASA CR-181948 ICASE Report 4. Title and Subtitle

Documentation

Page

2. Government AccessionNo. No.

MULTIPLE PATHS IN A BOUNDARY

3. Recipient'sCatalog No.

89-78 5. ReportDate

TO SUBHARMONIC LAYER

LAMINAR

BREAKDOWN

October

1989

6. PerformingOrganizationCode

7. Author(s)

8. Performing OrganizationReportNo.

Thomas A. M. Yousuff

89-78

Zang Hussaini

10. Work Unit No. 505-90-21-01

Performing Organization Name and Address 11. Contract or Grant No.

Institute for Computer Applications in Science and Engineering Mail Stop 132C, NASA Langley Research Center Hampton, VA 23665-5225 12. SponsoringAgency Name and Address National

Aeronautics

and

Space

NASI-18605 13. Type of Reportand Period Covered Contractor

Report

14. Sponsoring Agency Code

Administration

Langley Research Center Hampton, VA 23665-5225 15. Supplementa_ Notes Submitted Langley Richard

Physics

Technical Monitor: W. Barnwell

to Review

Letters

Final Report 16. Abstract

layer waves tional

Numerical simulations demonstrate that laminar breakdown induced by the secondary instability of two-dimensional to three-dimensional subharmonic disturbances need not lambda

vortex/high-shear

layer

transition,

path.

18. DistributionStatement

17. Key Words (Suggestedby Author(s)) boundary-layer mechanics

fluid

34 01

- Fluid Mechanics and Heat - Aeronautics (General)

Unclassified 19. SecuriW Classif.(of this report) Unclassified

in a boundary Tollmein-Schlichting take the conven-

_. SecuriWClassif. (of thispage) Unclassified

Transfer

- Unlimited 21. No. of pa_s 14

_. Price A03

NASA FORM 1626OCT86 NASA-Langley,

1989

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