N A S A C O N T R A C T O R

R E P O R T

CROSS-SPECTRAL FUNCTIONS BASED ON VON KARMAN'S SPECTRAL EQUATION

by John C. Houbolt and Asim Sen

Prepared by

AERONAUTICAL RESEARCH ASSOCIATES OF PRINCETON, INC.

Princeton, NJ. 08540

f o r Langley Research Center'

N A T I O N A L A E R O N A U T I C S A N D S P A C E A D M I N I S T R A T I O N W A S H I N G T O N , I?? C.'>$, e> .- ,MA.RtM'' 1972.'

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TECH LIBRARY KAFB, NM

I 1. Report No. I 2. Government Accession No. I 3. Recipient's Catalog No.

4. Title and Subtitle 5. Repor t Date

CROSS-SPECTRAL FUNCTIONS BASED ON VON KARMAN'S SPECTRAL EQUATION I March 1972 6. Performing Organization Code

7. Author(s) 8. Performing Organization Report No.

John C . Houbolt and Asim Sen 159 10. Work Unit No.

9. Performing Organization Name and Address

Aeronau t i ca l Research Assoc ia tes o f P r ince ton , I nc .

50 Washinqton Road 11. Contract or Grant No.

P r ince ton , NJ NAS 1-9200 13. Type of Report and Period Covered

12. Sponsoring Agency Name and Address

Nat iona l Aeronaut ics and Space A d m i n i s t r a t i o n

Washington, DC 20546 14. Sponsoring Agency Code

15. Supplementary Notes

16. Abstract

C r o s s - s p e c t r a l f u n c t i o n s f o r t h e v e r t i c a l and l o n g i t u d i n a l components o f t u r b u l e n c e o f a

t w o - d i m e n s i o n a l g u s t f i e l d a r e d e r i v e d f r o m t h e p o i n t c o r r e l a t i o n f u n c t i o n f o r t u r b u l e n c e

due t o von Ka.rman. C l o s e d f o r m s o l u t i o n s i n t e r m s o f B e s s e l f u n c t i o n s o f o r d e r 5 /6 and 11/6

are found. An asympto t i c exp ress ion f o r l a rge va lues o f the frequency argument, and series

r e s u l t s f o r s m a l l v a l u e s o f f r e q u e n c y , a r e a l s o g i v e n . These r e s u l t s now fo rm the base fo r

s t u d y i n g t h e e f f e c t o f s p a n w i s e v a r i a t i o n s i n t u r b u l e n c e f o r a turbulence env i ronment

wh ich i s cha rac te r i zed by t he von Karman i s o t r o p i c s p e c t r a l r e l a t i o n s . P r e v i o u s s t u d i e s

were based main ly on the Dryden- type spect ra l representat ion.

". " ~ ~

17. Key Words (Suggested by Author(s)) Cross-Spectra l Funct ions, Von Karman's Spectral

18. Distribution Statement

U n c l a s s i f i e d -- U n l i m i t e d Equat ions, Atmospher ic Turbulence Representat ion

20. Security Classif. (of this page) 22. Price' 21. No. of Pages

U n c l a s s i f i e d U n c l a s s i f i e d 17 $3.00

For sale by the National Technical Information Service, Springfield, Virginia 22151

p'

I

CROSS-SPECTRAL FUNCTIONS BASED ON V O N - K ~ R M ~ N 1s SPECTRAL EQUATION

by John C. Houbolt and ' A s i m Sen

Aeronautical Research Associates of Pr inceton, Inc.

ABSTRACT

C r o s s - s p e c t r a l f u n c t i o n s f o r t h e v e r t i c a l and l o n g i t u d i n a l components of turbulence of a two-dimensional gust f ie ld are der ived f rom the po in t cor re la t ion func t ion for tu rbulence due t o von K&m&. Closed form solut ions in terms of Besse l func t ions of order 5/6 and 11/6 are found. An asymptot ic expression for large values of the frequency agreement, and s e r i e s r e s u l t s f o r small values of f requency, are a lso given. These resul ts now form the base for s tudying the e f fec t o f spanwise var ia t ions in tu rbulence for a turbulence environment which i s charac te r ized by the von K&m&n i s o t r o p i c s p e c t r a l r e l a t i o n s . P r e v i o u s s t u d i e s were based mainly on the Dryden-type spectral representat ion.

INTRODUCTION

In r e f e rences 1 and 2 de r iva t ions a r e g iven fo r t he c ros s - s p e c t r a f o r a two-dimensional gust f ield based upon the po in t c o r r e l a t i o n and s p e c t r a l f u n c t i o n s which were o f t en u sed i n gus t s t u d i e s a t t h a t t i m e ; s p e c i f i c a l l y , the po in t func t ions due t o Dryden and which y i e l d a 2 behavior a t high frequency for the

spectrum were employed. More r ecen t gus t s tud ie s i nd ica t e that a c o r r e l a t i o n and s p e c t r a l f u n c t i o n due t o von K&m& - one which i n d i c a t e s a behavior a t h igh f requencies - seems t o be a

b e t t e r model for s tabi l ized a tmospheric turbulence. Study of t h i s func t ion revea led that i t too could be extended to the two- d imens iona l gus t f ie ld case in c losed form fash ion . The purpose of t h i s r e p o r t i s t o p r e s e n t t h e d e r i v a t i o n and the c r o s s - s p e c t r a l f u n c t i o n s t h a t a p p l y f o r t h i s case.

0

&

I .. . . .

SYMBOLS

r

t

U

v

X

Z

v

0

modif ied Bessel funct ion of the f i r s t kind

modified Bessel function of the second kind

i n t e g r a l s c a l e of turbulence

c o r r e l a t i o n d i s t a n c e

p o i n t c o r r e l a t i o n f u n c t i o n

c r o s s - c o r r e l a t i o n f u n c t i o n f o r w component

c r o s s - c o r r e l a t i o n f u n c t i o n f o r u component

separat ion dis tance between paths 1 and 2

v e l o c i t y

c o r r e l a t i o n d i s t a n c e V t

genera l var iab le

reduced frequency - uiil v

nondimensional separat ion dis tance

rms value of w turbulence component

S

power s p e c t r a

c ros s - spec t r a

angular f r e que ne y

2

CROSS-SPECTRA FOR TWO-DIMENSIONAL GUST STRUCTURE BASED ON THE VON KARMAN SPECTRAL FUNCTION

A s i n r e f e r e n c e 1, c o n s i d e r t h e g u s t f i e l d t o b e homogenous and i so t rop ic and momentarily frozen (Taylor hypothesis). As such, the cor re la t ion func t ion should be independent of any f l i g h t tra- verse path taken. With re ference to the fo l lowing ske tch ,

1 2

co r re l a t ion be tween gus t ve loc i t i e s a long pa ths 1 and 2 should

thus be associated w i t h t he co r re l a t ion d i s t ance r =/s2+ V 2 t 2 i n p l a c e of t h e c o r r e l a t i o n d i s t a n c e x of t he po in t co r re l a t ion func t ion .

The von K&m& c o r r e l a t i o n f u n c t i o n f o r v e r t i c a l g u s t ve- l o c i t i e s i s g i v e n by

where u = , and where L i s t h e i n t e g r a l s c a l e l e n g t h . To de r ive t he c ros s - spec t r a be tween t he ve r t i ca l ve loc i t i e s a long pa th 1 and t h e v e r t i c a l v e l o c i t i e s a l o n g p a t h 2,we rep lace x by r and hence def ine the cross-correlat ion funct ion as

X

1 3391;

where u = - 1 339 (5

S ( J = - L

By means of

For l a t e r u s e , t h e f a c t i s a l s o n o t e d t h a t

The c r o s s - s p e c t r a l f u n c t i o n f o l l o w s from equation (4 ) as

-00

For tuna te ly , th i s equat ion, wi th R represented by equat ion (4) has a closed form integrat ion ( t ran&?orm 941 i n r e f e r e n c e 3 ) . Af te r some manipulation and use of equat ion (5), t h e r e s u l t f o r %2 for the case under cons idera t ion i s found t o be

4

2 = + (1.339v). where 1.339

( J = - S L

Because

equat ion (7) may a l s o be w r i t t e n i n t e r m s of the frequency argu- ment v as

This i s t h e b a s i c r e s u l t of the p resent paper . I t s eva lua t ion i s encumbered because of the uncommon f r a c t i o n a l o r d e r s for the Bessel funct ions. This eva lua t ion and assoc ia ted curves a re d i s - c u s s e d i n a subsequent sect ion.

A s a check, equation (8) should reduce t o the one-dimensional s p e c t r a l r e s u l t when 0 = 0 . This reduct ion i s as follows. For z small

K n ( z ) = 2 ( n - l)! z ; n f o n-1 -n

With t h i s r e s u l t , e q u a t i o n (8) becomes

5

L 1

From t he known r e l a t i o n s

n! = r ( n + 1) = n r ( n )

= 1.339

th i s equa t ion reduces , as i t should, t o the fo l lowing wel l known r e s u l t f o r the one-dimensional case

6

EVALUATION O F EQUATION (8)

Since no tables were found for the Besse l func t ions of order 6 and cont inuous property of Besse1 , func t ions wi th respec t t o t h e i r o rder . The known closed form half-order funct ions were introduced as follows:

5 11 , evalua t ion was made by cons ide r ing t he ana ly t i ca l ly

- e z ~ 5 / 2 ( z ) = 1 + 2 + - ~ 5 / 2 ( ~ ) - 2 3 3

Z

Similar expressions were w r i t t e n fo r t he func t ions of order 1 and 2. Then f o r a given value of z , a p l o t of y ( z ) versus n was made, and f rom the resu l t ing curve values o? yn(z ) a t n = 5 and n = were read by interpolation. The values of the Bessel funct ion needed in equat ion (8) then followed from the r e l a t i o n s

11

Spec i f i c numer i ca l r e su l t s found fo r t he c ros s spec t r a a r e shown i n f i g u r e 1. These may b e u s e d d i r e c t l y i n s t u d i e s where va r i a t ion i n gus t ve loc i t i e s ac ross t he span i s be ing t r ea t ed . The fn te rpola t ion technique as used here for evaluat ion was aided by cons ider ing the asymptot ic express ion for equa t ion (8) and t h e r e s u l t s a t very low z , as d i scussed i n t he nex t s ec t ion .

Asymptotic - .~ r e s u l t -~ and expres s ion fo r low z .- For l a r g e z

7

8

-1 -r t

I I -

t 1-

I

Y OL v -

I .

Figure 1. Cross-Spectra for Treatment of Nonuniform Spanwise Gusts

With this r e su l t , equa t ion (8) reduces to the fol lowing asymptot ic r e s u l t , a p p l i c a b l e a t l a r g e z

)8’3[8 (1.339) 2 0 5/3 z -413 3

Figure 2 gives a comparison of results as obtained from t h i s equat ion and from equation (8) . It i s seen that the asymptotic r e s u l t s e r v e s q u i t e w e l l f o r v > 1 .

A f a c t of i n t e re s t ing phys i ca l s ign i f i cance can be brought out by consider ing the behavior of t he c ros s - spec t r a l func t ion a t l a r g e frequency argument i n a par t icular nondimensional manner . In response s tud ies i t i s of ten convenient or i n s t r u c t i v e to express r e s u l t s i n t e r m s of t h e r a t i o of t he c ros s spec t r a to t he po in t spec t ra , that i s , equat ion (8) divided by equation (10) . With the use of equat ion (11) ins t ead of equat ion ( 8 ) , i t can be shown that for high f requencies th i s r a t i o i s simply

It i s noted that the r a t i o i s a func t ion of the frequency argument

V i t can be reasoned that ce r t a in r e sponse phenomena may be expected t o peak at a c e r t a i n - Us r ega rd le s s of the sca le of turbulence. For example, i n t h e c a s e o f r o l l i n g r e s p o n s e due. to spanwise v a r i a t i o n s i n v e r t i c a l g u s t v e l o c i t i e s , maximum r o l l i n g power may be expected, as e x p r e s s e d i n t h e s p e c t r a l r a t i o f o r m , when

i s near T , where s i s taken as the wing span.

O S - only; the scale L no longer appears. Because of this f a c t ,

V

- U S

V

A use fu l exp res s ion fo r small z i s a l so r ead i ly found through use of t he fo l lowing de f in i t i ons .

9

Through means of these express ions , equa t ion (8) may be expanded i n t o a s e r i e s i n v o l v i n g 0 and z , with t h e f o l l o w i n g r e s u l t .

1 1 / 3 + .181(;) (-1 -t - 10

- ,9770 513 (1 + - 3 z + w z +68816 22 z6 + ...) 2 9 4

- . o y j 6 0 ~ ~ / ? ( 1 + -& z2 + q m Z 9 4 +i€mTzT z6 + ...) (14)

This equat ion was used t o check t he eva lua t ion of the c ross - s p e c t r a l f u n c t i o n s a t low values of z .

10

2 Q W

Figure 2a. Cross-Spectra f o r 0 = 0.2 . 11

1.0

IO"

I o - ~

IO+ I

12

0-2 IO" 1.0 IO IO2 W L

Figure 2b. Cross-Spectra for cr = 0.4 .

I .o

IO"

I o - ~

I d 4

lo-* 10- I

i I I I

= 1. I

I .o OL V

=-

Figure 2 c . Cross-Spectra for CJ = 0.6 .

14

OL V

y =- I .o IO

Figure 2 d . Cross-Spectra f o r CJ = 0.8 .

1.0

I o-2

I o - ~

-~ I i I i i i i i i i~ i 1 i i 1 1 i I i t l i i ~ i i i i i I i i i I 1 1 1 1 c t

3- 1.0 IO IO OL V

y =-

Figure 2 e . Cross-Spectra f o r (5 = 1.0 .

CROSS SPECTRA FOR LONGITUDINAL TURBULENCE

The p o i n t c o r r e l a t i o n f u n c t i o n f o r t h e l o n g i t u d i n a l component of tu rbulence tha t cor responds t o equat ion (1) i s given by

U (15)

X where u = 1.339L . I f x i s rep laced by ' r =d- as done wi th equat ion ( 2 ) , t he fo l lowing c ros s -co r re l a t ion func t ion a p p l i c a b l e t o l o n g i t u d i n a l t u r b u l e n c e i s obta ined .

where u = ____ 0 /- and 0 = - S

1.339 L O From t h i s func t ion

t h e c r o s s - s p e c t r a l f u n c t i o n for l ong i tud ina l t u rbu lence i s found t o be

CONCLUDING REMARKS

This paper has p resen ted c losed fo rm so lu t ions fo r t he c ros s - spec t ra l func t ions based on von K&rm&nls spec t r a l equa t ion . These r e l a t i o n s and assoc ia ted curves thus form a new and more appropr ia te b a s e f o r s t u d y i n g t h e e f f e c t s of spanwise var ia t ions in tu rbulence on a i r c r a f t g u s t r e s p o n s e .

16

REFERENCES

1. Houbolt, John C. "On t4fe Response of Structures Having Multiple Random Inputs. Jahr. 1957 der WGL, Fr iedr . Vieweg & Sohn (Braunschweig) , pp 296-305.

2. Diederich, Franklin W. and Drischler, Joseph A. "Effect of Spanwise Variations i n Gust In tens i ty on the L i f t Due to Atmospheric Turbulence. NACA TN 3920, A p r i l 1957.