Numerical Simulation of Boundary-Layer Transition

NASA Technical Memorandum 85984

NASA-TM-85984 19840020677

Numerical Simulation of Boundary-Layer Transition P.R. Spalart

June 1984

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NASA Technical Memorandum 85984

Numerical Simulation of Boundary-Layer Transition P R Spalart, Ames Research Center, Moffett Field, California

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National Aeronautics and Space Administration

Ames Research Center Moffett Field California 94035

NUMERICAL SIMULATION OF BOUNDARY-LAYER TRANSITION P. R. Spa1art NASA Ames Research Center Moffett Field, CA 94035 USA I•

FORMULATION The transition to turbulence in boundary layers is investigated by direct numeri-

cal solution of the nonlinear, three-dimensional, incompressible Navier-Stokes equations in the half-infinite domain over a flat plate. x-direction (streamwise) and in the from 0 to

z-direction (spanwise)j the

y-coordinate extends

"".

The simulation is periodic in the

x-direction, unlike the experiments in which

the boundary-layer thickness grows in the the

Periodicity is imposed in the

x-direction.

A forcing term is added to

x-momentum equation that approximates the convection terms associated with this

spatial growth.

The approximation is based on boundary-layer assumptions (applied

only to the mean flow) and the self-similarity of the mean-velocity profile.

With

this forcing applied, the laminar velocity profile, instead of becoming an error function and thickening without bounds, is a Blasius profile.

Thus, the stability

characteristics are very close to the experimental characteristics.

Furthermore, the

equation can be written in a moving reference frame, so that the boundary-layer thickens in time while retaining a Blasius profile.

The procedure of adding a forc-

ing term allows the disturbances to extract energy from the mean flow, and is much preferable to a procedure in which the mean-velocity profile would be imposed. The spatial representation is spectral in all directions [1].

The basis func-

t10ns that are used represent divergence-free velocity f1e1ds and satisfy the boundary conditions as suggested by Leonard and Wray [2].

Leonard and Wray app11ed a weak

formu1at10n, wh1ch e1im1nates the pressure and allows an accurate and straightforward t1me-advance scheme.

Leray's weak formulation 1S used here [3].

An advantage of

th1s formulat10n over Leonard's is that 1t keeps the numer1cal Stokes operator real, symmetr1c, and negative-definite.

On the other hand, Chebyshev polynom1als cannot be

used. The

x- and z-d1rections are treated by Four1er ser1es.

In the

y-d1rect10n,

the veloc1ty f1eld 1S f1rst split 1nto "irrotat10nal" and "vort1cal" components 1n order to better accommodate two d1fferent length scales. vort1cal component 1S the

~rrotational

s~ngle

6, the th1ckness of the boundary layer.

components is

n1ficantly larger than exponent1al

The length scale of the

6.

funct~on

Th1s

The length scale of

A, the wavelength 1n the (x,z) plane, Wh1Ch 1S S1girrotat~onal

component can be represented by a

for each horizontal wave-vector.

To represent the vort1-

cal component, an exponent1a1 mapping 1S app11ed from [O,oo[ 1nto [0,1[, and sh1fted Jacob1 polynom1als are used in the transformed coord1nate.

The vort1cal component

1nfin1tely d1fferent1able over the closed interval [0,1], so that the the polynom1al method w111 be faster than algebra1c.

~onvergence

~s

of

The cost of the transforms from

real space to Jacobi space is of the order of few basis-functions versus

y.

N2 •

Figure 1 is a plot of the first

All the functions decay exponentially as

y

~ ~

but

the first function, which includes the irrotational component, decays much more slowly than the other ones. The time-advance scheme is hybrid and second-order accurate.

The convection

terms are treated by a Runge-Kutta scheme which is explicit, third-order accurate, and conditionally stable; the Stokes terms are treated by the Crank-Nicolson scheme. II.

RESULTS In order to check the convergence of the method, the Orr-Sommerfeld equation

was solved for a Blasius profile and for a real wave-number.

This problem is known

Cr = I, Ci < 0 Figure 2 is a contour plot of the error in the principal discrete eigen-

to produce a few discrete eigenvalues and a continuous spectrum on the aX1S [4].

value as a function of 1n the

y-direction.

Yo, the length scale of mapping, and The convergence as

Ny

~ ~

with

Yo

Ny, the number of points fixed is very fast.

expected to be faster than algebraic, but not as fast as exponential [5]. also indicates the optimum value of

Yo:

about 20*.

It is

The plot

Figure 3 shows that the numeri-

cal spectrum includes a string of eigenvalues that becomes denser and tends to the Cr = I, Ci < 0 aX1S. Its convergence is much slower than that for the d1screte eigenvalues; the reason is that the corresponding eigenfunctions behave like sine waves as

y

~ ~,

which makes them hard to approximate with the expansion functions

in Fig. l. The early non11near stages of transition of a Blasius boundary layer, d1sturbed by a vibrating ribbon, were then s1mulated 1n three d1mensions.

A two-dimensional

Tollmein-Schlichting (TS) wave of f1n1te ampl1tude was 1ntroduced 1n the 1n1t1al field, as well as three-dimensional wh1te n01se of much lower energy.

The streamwise

per10d was tW1ce that of the TS wave; the spanw1se per10d was chosen much longer to av01d constra1n1ng the spectrum.

Spanw1se lines of part1cles were 1ntroduced, near

the cr1t1cal layer, to s1mulate the smoke l1nes used 1n exper1ments. F1gure 4 summar1zes the t1me-evolut10n of the flow.

The energy of the funda-

mental TS wave and the energy carr1ed by all the other wave-vectors are plotted separately.

The TS wave grows from branch I to branch II of the TS stab1lity d1agram,

then starts decaY1ng.

The energy of the other modes rema1ns small unt1l after the

flow crosses branch II; then it grows very rap1dly, and non11near 1nteract10ns take place.

The shape factor

H of the boundary layer rema1ns at the Blas1us value of

2.6 unt1l trans1tion occurs; then 1t rapidly decreases.

The agreement w1th Kachanov's

experiments is excellent [6]. S1mulat10ns were conducted w1th the same background n01se, but d1fferent values for the TS wave ampl1tude. ary layer. occur.

F1gure 5 conta1ns top V1ews of the part1cles 1n the bound-

If the maX1mum TS wave amplitude 1S less than 0.3%. trans1t1on does not

In Fig. 5(a), with an amp11tude of 0.9%, three-d1mens10nal breakdown occurs

and 1S of the subharmon1c or "H" type (the lambda-shaped part1cle l1nes are staggered).

In Fig. 5(b), with amplitude 5%, the lambda patterns are not staggered. indicating a Klebanoff-type breakdown.

The patterns appear ''broken,'' a result of the randomness

of the initial three-dimensional disturbance.

The qualitative agreement with Saric's

experiments is good [7]. Figure 6 is a plot of the spectrum in an (x,z) plane at the beginning of an H-type breakdown.

The fundamental TS wave still dominates the spectrum; its higher

harmonic is also present.

The growing three-dimensional subharmonic component is

obvious; the wave number and the broadband character of the instability agree very well with Herbert's small disturbance theory [8]. III. 1.

2. 3. 4. 5. 6. 7. 8.

REFERENCES Gottlieb, D. and Orszag, S. A., ''Numerical Analysis of Spectral Methods," NSF-CMBS Monograph No. 26, Society of Industrial and Applied Mathematics, Philadelphia, Penn., 1977. Leonard A. and Wray, A., "A New Numerical Method for the Simulation of ThreeDimensional Flow in a Pipe," NASA TM-84267, 1982. Temam, R., ''Navier-Stokes Equations and Nonlinear Functional Analysis," NSF-cMBS Monograph No. 41, Society of Industrial and Applied Mathematics. Philadelphia. Penn., 1983. Grosch. C. E. and Salwen, H., "The Continuous Spectrum of the Orr-Sommerfeld Equation. Pt. 1. The Spectrum and the Eigenfunctions," J. Fluid Mech., Vol. 87. Pt. 1, 1978, pp. 33-54. Boyd, J. P., '~e Optimization of Convergence for Chebyshev Polynomial Methods in an Unbounded Domain." J. Compo Phys., Vol. 45. No. I, 1982. pp. 43-79. Kachanov, Y. S. and Levchenko. V. Y., "The Resonant Interaction of Disturbances at Laminar-Turbulent Transition in a Boundary Layer," J. Fluid Mech., Vol. 138. 1984, pp. 209-247. Saric, W. S., Kozlov, V. V.. and Levchenko, V. Y.. "Forced and Unforced Subharmonic Resonance in Boundary-Layer Transition," AIAA Paper 84-0007, 1984. Herbert, T •• "Analysis of the Subharmonic Route to Transition in Boundary Layers," AIAA Paper 84-0009, 1984.

8.00

~.......,----,-~,.......---:::o--:::>",--:~

4.00 2 00 Yo

1.00 .50

o

2

a

8

O::::::I .251LO--2LO-=3l:0==4CO===5b ao Ny

Fig. 1

First four basis-functions.

F~g.

2

Error

e~genvalue.

~n

Orr-Sommerfeld

.."

0

u-

t·: a:

« Il. > a:

-1.0

C)

-1.5

« z «

• • •

~

-2.0

• ...• •

•

-.5

0

ut..:

«

~

Il.

.2

.4

> a:

-1.0

C)

-1.5

« z « ~

Ie 0

6

.8

• •

-20 . 0

1.0

2

REAL PART, Cr (a) Ny

1/

• • : ...• •• •

•

-.5

a:

.4

6

8

REAL PART, Cr

= 50.

= 100.

(b) Ny

Fig. 3

Numerical spectrum:

Yo

= 1.5.

H (II)

~

25

TS

H

..... "

.' ...... OTHER MODES

1000

2000

3000

4000

TIME

Fig. 4 60

Time-evolut~on

.. -',

x 30

'"

.

of the disturbance energy and of the shape factor.

......

...-....

"

.r. ,-

..,...~I

0

50

."'"

..

100

150

200

z

(a) Amplitude

60

o

••.• .1

, .. : ' ,

~~.:a

1" :~" ,•• , , J>1 ,:: .". ~ ..'It:' "" ,,': "......, .

","

~ \. ,,-~~,

","

.'

-•• "

"',-

,.',

••••

~ . l~'\: ...~/ \ . ~~~:.;,..J .\,. :~.,.~::I!::\ i 50

100

150

z

.,

.1.

300

t..,

",_' :-.

j, ;..\ .-'. . ,l · .f , .. ____ ... ~ ..~-_.~.

5

Smoke

l~nes

1n

J '\,

. .-.. . . , .... . •........ .,' ••• "," ~' . . . . . . ,.;..,; • •'

",4

....... •

., /. ': ':...;: i"r~.. :; .. _/'.':2..\

200

250

300

(b) Amplitude = 5%. F~g,

350

0.9%.

, .

I-

x 30

=

250

-.

trans~tion~ng

boundary layer.

350

10

MEAN FLOW SUBHARMONIC

o

Fig. 6

Two-d1mens10nal spectrum.

1 Report No

2 Government Ace_Ion No

3 Recipient's Catalog No

NASA TM-85984 4 Title and Subtitle

5 Report Date

June 1984

NUMERICAL SIMULATION OF BOUNDARY-LAYER TRANSITION

6 Performing Organization Code

7 Author(s)

8 Performing Organization Report No

P. R. Spa1art

A-9796 10 Work Unit No

9 Performing Organization Name and Address

T-6465

Ames Research Center Moffett Field, CA 94035

11

Contract or Gra nt No

13 Type of Report and Period Covered 12 Sponsoring Agency Name and Address

Technical Memorandum

Nat1ona1 Aeronautics and Space Administration Washington, DC 20546

14 Sponsoring Agency Code

505-31-01

15 Su pplementary Notes

Point of Contact: P. R. Spa1art, Ames Research Center, MS 202A-1, Moffett Field, CA 94035 (415) 965-6667 or FTS 448-6667. 16 Abstract

-

The transition to turbulence in boundary layers was invest1gated by direct numerical solution of the nonlinear, three-dimensional, incompress1b1e Navier-Stokes equations in the half-infinite domain over a flat plate. PeriodicHy was imposed in the streamwise and spanwise directions. A body force was applied to approximate the effect of a nonparallel mean flow. The numer1ca1 method was spectral, based on Four1er series and Jacobi po1ynom1a1s, and used divergence-free bas1s funct1ons. Extremely rap1d convergence was obta1ned when solving the linear Orr-Sommerfeld equat1on. The early nonlinear and three-d1mensiona1 stages of transition, in a boundary layer d1sturbed by a vibrating ribbon, were successfully simulated. Excellent qua11tative agreement was observed with either experiments or weakly non11near theor1es. In particular, the breakdown pattern was staggered or nonstaggered depending on the disturbance amp11tude.

17 Key Words (Suggested by Author(s))

18 Distribution Statement

Boundary layer Transition Numerical method

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Unclassified

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End of Document