THE DIFFUSIVITY OF NONLINEAR WAVE PROPAGATION IN PIPES

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THE DIFFUSIVITY OF NONLINEAR WAVE PROPAGATION IN PIPES

Cornelius Shih

To cite this version:

Cornelius Shih. THE DIFFUSIVITY OF NONLINEAR WAVE PROPAGATION IN PIPES. Journal

de Physique Colloques, 1979, 40 (C8), pp.C8-53-C8-55. �10.1051/jphyscol:1979811�. �jpa-00219516�

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JOURNAL DE PHYSIQUE Colloque C8, supplement au n°ll, tome 40 novembve 1979, page c 8 - 53

THE DIFFUSIVITY OF NONLINEAR WAVE PROPAGATION IN PIPES.

Cornelius C. SHIH

Meehanieal Engineering Department The University of Alabama in Huntsville Huntsville, Alabama 25807 Résumé. - Les distributions de vitesse dans le temps et dans l'espace le long de tuyaux de 7.62 cm et 2.54 cm de diamètre et d'une longueur de 64 m produites par des ondes nonlinéaires de pression d'intensités différentes ont été mesurées et comparées avec les résultats théoriques unidimension- nels obtenus par la méthode des caractéristiques avec ou sans le terme de diffusion dans le but de déterminer les caractéristiques d'atténuation de l'onde de propagation. Les données ont été ana- lysées afin d ' é t a b l i r la relation entre le rapport de flux d'énergie et la distance dans le tuyau exprimée en fonction du coefficient d'atténuation a et de la d i f f u s i b i l i t é . On a montré que ces parètres caractéristiques sont fonction de la géométrie du tuyau : diamètre, longueur, et rugosité r e l a t i v e .

Abstract. - Temporal and spacial velocity distributions along the pipes of 7.62 cm and 2.54 cm in diameter and 64 m in length produced by non-linear pressure waves of various intensities were measured an compared with onedimensional theoretical results obtained by the method of characteristics with and without the diffusion term for the purpose of determining the attenuation characters!"tics of the wave propagation. Data were analyzed for the relationship between the energy f l u x ration and pipe distance i n terms of attenuation coefficient a and d i f f u s i v i t y 6. These characteristic para- meters are determined to be the function of pipe geometries such as diameter, length and relative roughness.

1. INTRODUCTION. - Experimental studies / l / , / 2 / , / 3 / of attenuation characteristics of non-linear pressure waves of progressive type propagating in pipes of 7.62 cm and 2.54 cm in diameters and of various lengths up to 64 m, were measured in temporal and spacial velocity profiles and compared with theoretical results obtained by the method of characteristics with and without the isentropic assumption.

Non-linear attenuation of the pressure waves were presented in terms of attenuation coef-

f i c i e n t a and d i f f u s i v i t y 6, based on the experimental results. The energy flux was determined from the velocity profiles measured at various sections along the pipes, and the decay of the total energy flux was related to the space coordinate and pipe size.

Experimental Data Presentation. - Experimental data of velocity profiles were measured at the center of each of f i v e sections located 3.60 ; 16.62 ; 29.45 ; 45.44 ; 55.40 m from the pipe i n l e t , respectively, in the pipes of 7.62 and 2.54 cm in diameter. Two pressure waves of progressive type applied to each pipe for the time- dependent measurement of velocity profiles with hot-wire anemometers were presented for comparison in this paper.

Figures 1 to 4 shows time-dependent velocity profiles measured at the f i v e pipe sections in the 7.62 cm ( 3 - i n . ) and 2.54 cm ( 1 - i n . ) pipes for two pressure waves produced with nominal burst pres- sures of 28.10 and 56.20 g/mm . Also presented are 2

Article published online by EDP Sciences and available at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979811

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JOURNAL DE PHYSIQUE

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ISENTROPIC f

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... EXPERIMENTAL

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Section ... EXPERIMENTAL

Section

TlhlI. MILLISFCOYDF. RLLATIVC TO IhITIAL PISTON POSITION

TI^, MILLIFF.CONDS, RELATIVE TO INITIAL PISTO?( POSITION P = 5 6 . 2 0 g / m m f , 2 . 5 4 cm p l p e .

Figures 1

-

4 : Comparison o f Numerical C a l c u l a t i o n s With Experimental Data f o r B u r s t Pressure.

p i s t o n v e l o c i t y p r o f i l e a t t h e p i p e i n l e t , theore- t i c a l r e s u l t s o f wave propagation analyzed i n terms o f v e l o c i t y p r o f i l e s f o r i s e n t r o p i c process and f r i c t i o n a l processes w i t h f o u r assumed values o f f r i c t i o n a l c o e f f i c i e n t i n each o f t h e f i g u r e s .

S p e c i f i c comparison made between t h e two p i - pes i s t h e r e l a t i o n s h i p among a t t e n u a t i o n c o e f f i - c i e n t s a, p i p e d i s t a n c e

x

and t h e energy d i f f u - s i o n expressed i n terms o f energy f l u x r a t i o n , E/Eo as shown i n F i g u r e 5. E denotes t h e energy f l u x on

as a f u n c t i o n o f X. Values o f E/Eo were c a l cula- t e d from v e l o c i t y p r o f i l e data as f u n c t i o n s o f t i m e i n Figures 1 t o 4 using Equation (1). Slopes o f t h e l i n e s averaged through p l o t t e d p o i n t s i n F i g u r e 5 y i e l d v a r i e d a t t e n u a t i o n c o e f f i c i e n t s , a f o r t h e two pipes. The average values o f a f o r 7.62 and 2.54 cm pipes are r e s p e c t i v e l y 0.0102 m-' and 0.0173 m-I f o r t h e f i r s t 43 m o f t h e pipes.

However, a appears t o be t h e f u n c t i o n o f X, s i - g n i f y i n g t h a t t h e a t t e n u a t i o n i s non-1 i n e a r . a time average contained i n t h e pressure wave i n Comparison and Discussion o f Data.

-

^Since

propagation process and Eo i s t h e energy f l u x on t h e energy f l u x r a t i o i s suggested /4/ t o be a t i m e average contained i n ' t h e progressive wave

a t t h e o r i g i n o f propagation.

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and

w26

a =

3

_a ⁽³⁾

where w i s t h e angular o r r a d i a n frequency, 6 t h e d i f f u s i u i t y . Values o f 6 c a l c u l a t e d from Equation ( 3 ) w i t h w = 78.5 rad/sec and a = 350.52 m/

sec are noted t o be 71.29 m /sec f o r 7.52 cm p i p e 2

o and 121.11 m /sec f o r 2.54 cm pipe. 2

x (m)

F i g u r e 5 reveals t h a t t h e t h e o r y may be v a l i d F i g u r e 5 : Energy F l u x R a t i o Vs. Pipe Length. i n a s i g n i f i c a n t p o r t i o n o f t h e p i p e except t h e

s e c t i o n n e x t t o t h e p i p e e x i t . The experimental The energy ratio may be approximated by evidence seems t o demonstrate t h a t t h e a t t e n u a t i o n

t e n d t o increase along t h e pipe. The t h e o r e t i c a l

-

(1) a n a l y s i s was developed s i m p l y f o r t h e plane pro- g r e s s i v e wave propagating i n a one-dimensional plane w i t h o u t c o n s i d e r a t i o n o f the f r i c t i o n a l e f - f e c t along t h e p i p e w a l l . Therefore, any p r e d i c t i o n

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C.C. SHIH

o f t h e e f f e c t s o f p i p e s i z e and w a l l f r i c t i o n was n o t p o s s i b l e .

However, the experimental a n a l y s i s e v i d e n t l y shows, through graphical presentation, t h a t t h e ef- f e c t s o f p i p e geometry are pronounced on t h e a t t e - n u a t i o n c h a r a c t e r i s t i c s f o r t h e type o f wave propagation i n t h e pipes studied.

The values of a and 6 are considered r a - t h e r h i g h due t o h i g h h u m i d i t y o f about 80 % r e l a - t i v e i n t h e atmosphere a t t h e time o f experimenta- t i o n . But t h i s f a c t o r o f n a t u r e cannot be e a s i l y c o n t r o l l e d , so i s t h e temperature o f t h e atmosphere. However, i t might be decuced t h a t t h e a t t e - n u a t i o n c o e f f i c i e n t as w e l l as d i f f u s i v i t y may n o t be a constant value i n t h e process o f n o n - l i n e a r wave propagation, and t h e exponential decay o f energy f l u x i n t h e wave seems t o e x h i b i t some non- 1 in e a r i t y .

REFERENCES

/1/ Shih C., "Non-linear Pressure Wave Propagation i n Pipes", Proc., Interagency Symposium o f Uni- v e r s i t y Research i n T r a n s p o r t a t i o n Noise, Stanford, C a l i f .

,

March 1973, pp 583-598.

/2/ Shih C., " A t t e n u a t i o n C h a r a c t e r i s t i c s o f Non- l i n e a r Pressure Wave Propagation i n Pipes", Proceedings, Fourth I n t e r n a t i o n a l Symposi um on Non-1 i n e a r Acousti cs-IPC Science and Techno1 ogy Press Ltd.

,

August 1973.

/3/ S h i t C.

,

" I n v e s t i g a t i o n s o f S c a l i n g Characteris- t i c s f o r D e f i n i n g Design Environments due t o T r a n s i e n t Ground Winds and N e a r - f i e 1 d Non- l i n e a r Acoustic F i e l d s " , Report NAS8-28249, UAH, UAH, Sept. 1973.

/4/ L i n g h t h i 11 M. J.

,

"Surveys i n Mechanics", E d i t e d by G. K. B a t c h e l o r and R. M. Davis, Cambridge U n i v e r s i t y Press, pp 250-351, 1956.