Real-Time Laser Holographic Interferometry for …...Real-time laser holographic interferometry for...

11
STi F NASA Technical Memorandum 89462 Real-Time Laser Holographic Interferometry for Aerodynamics George Lee (YASA-TH-69462) hEAL-31BE LAZE6 BCLOGRAEHIC 887-22956 I&TEhEEkC?!EIBY ECE AELCCYLALJC5 (MASA) 11 F Avail: HXIS CC AOZ/EIP A01 CSCL 14E Unclas H1/35 00760E2 June 1987 National Aeronautics and Space Administration https://ntrs.nasa.gov/search.jsp?R=19870013523 2020-07-01T04:41:24+00:00Z

Transcript of Real-Time Laser Holographic Interferometry for …...Real-time laser holographic interferometry for...

Page 1: Real-Time Laser Holographic Interferometry for …...Real-time laser holographic interferometry for aerodynamics George Lee NASA Ames Research Center Mail Stop 260-1, Moffett Field,

STi F

NASA Technical Memorandum 89462

Real-Time Laser Holographic Interferometry for Aerodynamics George Lee

(YASA-TH-69462) h E A L - 3 1 B E L A Z E 6 BCLOGRAEHIC 887-22956 I&TEhEEkC?!EIBY E C E A E L C C Y L A L J C 5 (MASA) 11 F Avai l : HXIS CC AOZ/EIP A01 CSCL 14E

Unclas H1/35 00760E2

June 1987

National Aeronautics and Space Administration

https://ntrs.nasa.gov/search.jsp?R=19870013523 2020-07-01T04:41:24+00:00Z

Page 2: Real-Time Laser Holographic Interferometry for …...Real-time laser holographic interferometry for aerodynamics George Lee NASA Ames Research Center Mail Stop 260-1, Moffett Field,

NASA Technical Memorandum 89462

Real-Time Laser Holographic Interferometry for Aerodynamics George Lee, Ames Research Center, Moffett Field, California

June 1987

National Aeronautics and Space Ad ministration

Ames Research Center Moffett Field, California 94035

Page 3: Real-Time Laser Holographic Interferometry for …...Real-time laser holographic interferometry for aerodynamics George Lee NASA Ames Research Center Mail Stop 260-1, Moffett Field,

Real-time l a se r holographic interferometry for aerodynamics

George Lee

NASA Ames Research Center Mail Stop 260-1, Moffett Field, California 94035

Abstract

Recent developments i n thermoplastic recording holograms and advancements i n automated image d ig i t i za t ion and analysis make real-time l a se r holographic interferometry feas ib le fo r two-dimensional flows such a s a i r - f o i l flows. Typical a i r f o i l measurements would i n c l u d e a i r f o i l pressure d i s t r ibu t ions , wake and boundary layer p ro f i l e s , and flow f i e l d density contours. T h i s paper addresses some of the problems and requirements of a real-time l a se r holographic interferometer.

Introduction

Recent developments i n thermoplastics, automated d ig i t i za t ion , and expert systems make real-time l a se r holographic interferometry feas ib le . developed fo r steady and unsteady two-dimensional flows. Measurements include a i r f o i l pressure d i s t r ibu t ions , wake and boundary layer p ro f i l e s , 1-5 and flow f i e ld density contours. measurements can be made w i t h more e f f ic iency a n d cos t savings and w i t h higher qua l i ty data. I n the pas t , t h e holograms taken i n the wind tunnel were reconstructed i n t h e laboratory, which was ine f f i c i en t and resulted i n poorer data.

During the p a s t decade, laser holographic interferometry has been

With real-time interferometry,

A prototype real-time interferometer must satisfy a number of requirements: ( 1 ) minimize wind-tunnel- induced vibrations, ( 2 ) process and reconstruct holograms i n s i t u , and ( 3 ) perform online image processing and fringe ana lys i s . Although a ruby or Nd:YAC l a se r would be su i t ab le , improvements i n l a se r beam pointing s t a b i l i t y and beam f i l t e r i n g would grea t ly improve the qua l i ty of the holograms. T h i s is c ruc ia l for f a s t reconstructions. In s i t u processing and reconstruction w i l l require either a wet photographic process done i n place or a dry pro- cess u s i n g thermoplastic holograms. past year. required. Laboratories for dual p l a t e interferometry. image processors. Ames has developed a system that is based on a DeAnza image processor and a V A X 11/780 computer. I t is capable of semi-automatic d ig i t iza t ion , image enhancement, fringe-coordinate extraction. numbering, interpolation and extrapolation of fringe functions, and conversion of f r inge data into aerodynamic data. fo r fringe analysis. shocks.

Wind t u n n e l vibrations can b e minimized w i t h r i g id opt ica l t ab les and w i t h pulse l a se r s .

A thermoplastic device.(TPD) was successfully tested a t NASA Ames t h i s The TPD processing was dry and f a s t . Reconstruction was done i n s i t u but more development work is

The TPD is a commercial product from Newport Corp. and was modified by Spectron Development Online image processing can be done by a number of commercial

To make t h i s system f a s t enough for online processing, some type of expert system needs t o be designed A t a minimum, the ana lys i s m u s t determine d e n s i t y gradient d i rec t ion and m u s t recognize

Wake and boundary layer ana lys i s would require a grea te r e f f o r t .

Airfoil flow analysis

Many problems a r e encountered i n performing a i r f o i l ana lys i s w i t h l a se r holographic interferometry. problems a re : ( 1 ) reconstruction to i n f i n i t e fringe, ( 2 ) noise, ( 3 ) re f rac t ion , and (4) boundary layers determination.

Some

Figure 1 shows an interferogram t h a t was reconstructed t o the i n f i n i t e fringe mode. I n t h i s mode and for two-dimensional flows he f r inges become l ines of constant density which makes analysis ea s i e r . Most f r inge d ig i t i za t ion programs'-' have been writ ten for t h i s mode. To reconstruct to i n f i n i t e fringe mode, the idea i s t o obtain one in f in i t e ly wide f r inge i n the freestream region where the flow is uniform. flows, the region ahead of the nose shock wave is uniform and can be reconstructed to one or two f r inges . For subsonic flows, t h e process is much more d i f f i c u l t and somewhat subjective a s well a s very tedious and time consuming. from the a i r f o i l approach uniform flow.

For supersonic

This is because the flow is not completely uniform i n any region. A t bes t , the regions f a r away

Noise a l s o contributes to the d i f f i c u l t y of reconstructing i n f i n i t e f r inge interferograms. Noise causes wiggles i n t he f r inges , merges some f r inges , and adds extraneous f r inges . Note tha t i n Figure 1 , wiggles and

1

Page 4: Real-Time Laser Holographic Interferometry for …...Real-time laser holographic interferometry for aerodynamics George Lee NASA Ames Research Center Mail Stop 260-1, Moffett Field,

merging a r e more apparent i n regions f a r the r away from the a i r f o i l . suf fe r la rger e r rors than the region nearer the a i r f o i l where there a r e many f r inges . The noise sources a re v ibra t ion , poor qua l i ty optics, misaligned op t i c s , l a se r i n s t a b i l i t i e s , and ambient a i r currents. T h i s l a t t e r e f f e c t can be s ign i f i can t because beam paths i n wind tunnels can be of t h e order of tens of meters. shows an ambient condition interferogram i n a wind tunnel. occurred.

T h i s lower signal-to-noise region w i l l

Figure 2 Even under controlled conditions, a few fringes

Refraction is inherent i n many a i r f o i l flows. The bending of the l i g h t beams by refraction causes e r rors Examples a r e shock waves and

Fortunately, these regions a re usually small i n the f r inge data and, i n t h e worst cases, can r e su l t i n the lo s s of fringes. regions of rapid expansion such as on t h e nose of the a i r f o i l . and schemes can be devised to work around the problem. Refraction in shock waves w i l l r e su l t i n a broad dark l i n e . Obviously, fringe data is l o s t . More important i n flow f i e l d ana lys i s , the f r inge order across the shock wave is l o s t ; i . e . , one cannot count across t h e shock. In the case shown i n Figure 1 , one can t race a f r inge around the shock and then continue t h e d ig i t i za t ion of the flowfield. T h i s process is slow unless new f a s t software can be developed. A prac t ica l and f a s t way to handle shock waves would be t o have o r i f i c e s ahead and behind the shock to give reference fringes. Refraction near the nose w i l l cause a bright or dark spot w i t h a loss of fringes. Extrapolation of f r inges in to t h i s region works qu i t e well. Usually there a re many well defined fringes tha t can be extrapolated. However, t h i s w i l l slow down the analysis process.

One of the principal measurements of a i r f o i l s a r e pressure d i s t r ibu t ions from which forces a re derived. T h i s requires d ig i t i z ing the f r inges on the edge of the boundary layer which i s a l i ne near the a i r f o i l sur- face. The determination of t h i s l i n e is straightforward i n some cases and somewhat subjective i n other cases.

The straightforward case is shown i n Figure 3 . The boundary layer cons is t s of a s e r i e s of closely spaced f r inges tha t a r e nearly parallel t o t h e upper surface. Above the boundary layer , the f r inges a re spaced f a r the r apar t and a r e nearly normal to t h e a i r f o i l . For t h i s case, i t is easy t o define the boundary layer. I t is the locus of the points a t which each fringe takes a sharp turn.

A d i f f i c u l t case is shown i n Figure 4 . On t h e upper surface of t h i s a i r f o i l , the boundary layer is t h i n and the outer f r inges a re almost pa ra l l e l to the surface. layer fringes and outer flow f r inges .

I t is d i f f i c u l t t o d i s t inguish between boundary

An intermediately d i f f i c u l t case, Figure 1 , is to determine the edge of the shear layer i n the I ' Lion behind the shock wave. Note tha t the f r inges a re much more closely spaced i n the shear layer than i n t h e inviscid flow. Note also tha t t h e outer inviscid flow fringes can vary from being pa ra l l e l to being normal to those i n the shear layer . For t h i s case, the determination of the edge is not c lear cu t .

For a l l three cases, it probably requires a trained operator to define the edge for a real-time interferom- e t ry system. However research i n a r t i f i c i a l intell igence for t h i s problem is being pursued.

Real-time interferometer requirements

The three general requirements fo r real-time interferometry a re : ( 1 ) eliminating v ibra t ion , ( 2 ) i n s i t u processing and reconstruction, and ( 3 ) online image processing and f r inge ana lys i s .

The wind tunnel is a high vibration, noisy environment. Every e f f o r t m u s t be made to eliminate o r reduce

Most important, a l a se r w i t h a pulse width t h a t is very shor t (compared t o any vibra- vibration, so opt ics should be mounted on vibration-isolated tab les . Machines should be isolated from opt ics and the t e s t section. tion frequency) m u s t be used. Finally the l a se r should be s t ab le ; e .g . , the beam should not wander and the in tens i ty should remain f a i r ly constant.

Another requirement for real-time interferometry is i n s i t u recording, processing, and reconstruction. The usual process of taking the holograms a t t h e wind tunnel and then developing and reconstructing them a t the laboratory, requires hours. than a minute. The dry process uses a thermoplas- t i c material tha t is developed by heat. t es t ing and a re commercially ava i lab le . Last year a thermoplastic holocamerJ was modified by Spectron Devel- opment Laboratories fo r double-plate aerodynamic interferometry. pensation can be done by mechanically moving one of the thermoplastic p la tes . Ames Research Center, and t h e time required for processing was l e s s than a mir:ute. remaining i n place d u r i n g recording, processing, and reconstruction, i n f i n i t e fringe interferograms should be attained quickly. ca l ly . the s i l v e r halide fi lm i s f l ex ib l e , which could be a source of noise. compared to a r ig id thermoplastic) would be d i f f i c u l t .

In s i t u systems using wet or dry processing can process the holograms i n l e s s The wet process has a self-contained f i l m developing u n i t .

Both types of processes were i n i t i a l y developed for nondestructive

One of its fea tures is tha t vibration com- I t was successfully tested a t

With the two holograms

In the ideal case of no vibration or noise, i n f i n i t e f r inge should be a t ta ined automati- The wet process holocameras have not yet been used i n wind tunnels. One an t ic ipa ted problem is that

Reconstruction with f l ex ib l e f i l m ( a s I\ var ia t ion of double-plate, real-time interferometry

2

Page 5: Real-Time Laser Holographic Interferometry for …...Real-time laser holographic interferometry for aerodynamics George Lee NASA Ames Research Center Mail Stop 260-1, Moffett Field,

was shown by Basler7 i n which an ambient condition holographic p l a t e was carefu l ly superimposed i n the object beam. The disadvantage of t h i s method is the lack of vibration compensation.

The technique has been used f o r qua l i ta t ive ana lys i s of the unsteady buffet flow over an a i r f o i l .

The th i rd requirement for real-time interferometry is f a s t image processing and ana lys i s . Several methods can be used to analyze the interferogram. over the e n t i r e flow f i e l d . These methods have been used for measuring deformations of opaque so l id surface i n nondestructive t e s t ing and for surface measurements i n op t ica l qua l i ty t e s t ing . require simultaneous recording of multiple phase-shifted images or multiple reference beam holography.

O n e is d i r ec t phase measurement using heterodyne techniques8-''

Phase sh i f t i ng techniques

Other methods use image-processing systems to enable a computer-aided evaluation of the fringe pattern. Many f a s t image processors can convert a p ic ture to d i g i t a l form for computer processing. a l inear photodiode array camera and o thers use a h i g h resolution camera w i t h a frame grabber for A to D con- version. Some systems a re "stand alone" and use small computers and some a r e connected to a larger host com- puter. capab i l i t i e s . grammed fo r the computer,

Some processors use

Picture enhancement and res tora t ion to improve blurred and noisy p ic tures a re standard processing Software tha t converts pictures into maps and quant i f ies the required properties can be pro-

Several semi-automatic d i g i t a l image processing codes' ' - 1 3 have been developed for aerodynamics. NASA Ames Research Center has been using Becker's code fo r the past few years. T h i s system has the following capabili- t i e s : The frame is averaged t o improve the signal-to-noise r a t i o , and the f r inge pattern is d ig i t ized . Each fringe is numbered and given a coordinate; however, the user m u s t assign the d i rec t ion of the fringes and change s igns a s the f r inge pattern bends i n the opposite d i rec t ion (Figure 5 ) . f r inge pattern w i t h f r inge numbers on the CRT. I t i s easy to check t o see i f any fringes a r e missed o r m i s - numbered. data from an interferogram tha t require zooming because of the lack of resolution. 5 sec t ions , 3 sections and no magnification showed t h a t zooming improved t h e accuracy of t h e r e su l t s . Becker's code has interpolation and extrapolation routines for regions where re f rac t ion has destroyed the f r inges .

The system a l so plays back the

Another feature of Becker's program is zooming and merging of the magnified views. Figure 6 shows The da ta l2 w i t h zooming of

F i n a l l y

Current research in to f u l l y automated systems i s being investigated. This system w i l l u t i l i z e expert systems knowledge to de tec t f r inges , determine fringe orders, enhance p ic tures where necessary, and eliminate noise. knowledge, r e l a t e t o past experience, use information from each ana lys i s s t e p to a s s i s t i n se lec t ing the n e x t s t ep , and make in t e l l i gen t guesses.

Work is progressing on developing an expert system ru les base tha t can draw on domain-specific

Prototype real-time interferometer

Current work toward development of real-time interferometry a t NASA Ames Research Center w i t h an applica- Figure 7 shows a sketch of a proposed t ion toward t h e s t u d y of o sc i l l a t ing a i r f o i l s w i l l be presented here.

real-time system. tab le , and an image processor using a minicomputer. a t 5300 A with a pulse width of the order of 100 nsec. qu i t e r e l i ab le .

I t cons is t s of a pulsed l a s e r , a thermoplastic recorder mounted on a vibration-isolated The pulse ND:YAC l a s e r is made by Quanta Ray and operates

This laser has been used for over 5 years and has been One improvement would b e t o redesign the f i l t e r i n g opt ics t o improve t h e beam wandering.

Figure 8 shows a vibration-isolated op t i c s system. A l l the op t i c s , the l a s e r , and the thermoplastic recorder a r e mounted on a r ig id table. t es ted . perform online realignment. t ion can be done w i t h t h e Nd:YAC l a s e r . Past reconstructions were done with a He-Ne l a se r , which requires realignment and a more complicated procedure.

The effectiveness of t h i s system fo r f i l t e r i n g vibrations w i l l be The thermoplastic holographic recorder has provisions fo r double-plate interferometry. I t a l so can

A major e f f o r t w i l l be t o determine i f realignment is necesary and i f reconstruc-

Current work i n image hardware and software is progressing. Both desktop and minicomputer-based image systems are being developed. wind tunnel. minicomputer system is being developed for more general and more complicated flows and fo r f u l l y automatic operation. online experimental and theore t ica l integration.

The desktop system is portable, stands alone, and could be eas i ly moved to t h e I t s speed and memory a r e probably suf f ic ien t t o handle t h e a i r f o i l case. The f a s t e r and la rger

T h i s system can be t ied t o mainframe computers f o r I t s software w i l l incorporate expert systems.

Concluding remarks

The s t a t e of the a r t i n l a se r holographic interferometry indicates that real-time interferometry for two- dimensional flows is feas ib le . i n t h i s paper.

The main problems and requirements for a real-time system have been discussed With a concerted e f f o r t , a prototype system could be demonstrated uithin a year or two.

3

Page 6: Real-Time Laser Holographic Interferometry for …...Real-time laser holographic interferometry for aerodynamics George Lee NASA Ames Research Center Mail Stop 260-1, Moffett Field,

References

1 . Lee, G. , "Application of Holography to Flow Visualizations," NASA TM 84325. 1984. i

2. Spaid, F. W . and Bachalo, W . D., "Experiments on the Flow About a Supercritical Airfoil Including Holographic Interferometry," J. Aircraft, Vol. 18, pp. 287-294. 1981.

3. Johnson, D. A. and Bachalo, W . D., "Transonic Flow Past a Symmetrical Airfoil-Inviscid and Turbulent Flow Properties," AIAA J., Vol. 18, pp. 16-24. 1980,

4 . Bachalo, W . D., Measurements of Supercritical Airflow Using Interferometry, Aerometrics, Inc., Mountain View, CA. 1982.

5. Bachalo, W. D,, "An Experimental Investigation of Circulation Control Flow Fields Using Holographic Interferometry," NASA CR 166482. 1982.

6. Umstater, H. L., Doty, J. L., and Trolinger, J. D., "Dual Thermoplastic Holography Recording System," Proc. of SPIE, Vol. 523, Applications of Holography. 1985.

7. Basler, D., Experimental Investigation of Shock Induced Buffeting Through Holographic Interferometry, AIAA-87-0123, 25th Aerospace Sciences Mtg., Reno, NV, January 1987.

8. Bruning, J. H., et al., "Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses," App. Optics, Vol. 13, pp. 2693-2703. 1974.

9. Yatagai, T. and Kanou, T., "Aspherical Surface Testing with Shearing Interferometer Using Fringe Scanning Detection Method," Opt. Eng., Vol. 23, pp. 357-360. 1984.

10. Smythe, A. and Moore, R., "Instantaneous Phase Measuring Interferometry," Opt. Eng., Vol. 23, pp. 361-364. 1984.

11. Becker, F. and Yu, Y . H . , "Digital Fringe Reduction Techniques Applied to the Measurement of Three Dimensional Transonic Flow Fields," Opt. Eng., Vol. 24, No. 3, pp. 429-434. 1985.

12. Torres, F. J., Application of Digital Holographic Interferometry to Presswe Measurements of Symmetric, Supercritical, and Circulation-Control Airfoils in Transonic Flow Fields, Paper No. 693-18, S P I E 30th Int'l. Tech. Sympos. on Optical and Optic-electronic Engrg., San Diego, CA. 1986.

13. Eecker, F., Meier, C. E. A., and Wegner, H . , Applications of Digital Image Processing, IV, Andrew G. Teseher, ed., Proc. of SPIE 359. 1983.

Page 7: Real-Time Laser Holographic Interferometry for …...Real-time laser holographic interferometry for aerodynamics George Lee NASA Ames Research Center Mail Stop 260-1, Moffett Field,

ORIGIIL'AL PACE IS

Figure 1 . Infinite fringe interferogram

Figure 2 . Ambient condition interferogram

5

Page 8: Real-Time Laser Holographic Interferometry for …...Real-time laser holographic interferometry for aerodynamics George Lee NASA Ames Research Center Mail Stop 260-1, Moffett Field,

ORIGINAL OF POOR

Figui

PAGE 1s QUALSW

re 3 . Detec

Figure 4. Detec t i

It ion of boundary layer: invisc id flow fr inges nearly normal to bounda layer f r i .nges

.on of boundary layer: i n v i s c i d flow fringes nearly paral le l to boun

N=O +1 +2 +3-SIGNAMBlGUlTY

FRINGE BENDING TO CAUSE DOUBLE VALUE FRINGES

. dar ‘Y layer fr inges

Figure 5 . Sign ambiguity and double value fr inges

6

Page 9: Real-Time Laser Holographic Interferometry for …...Real-time laser holographic interferometry for aerodynamics George Lee NASA Ames Research Center Mail Stop 260-1, Moffett Field,

0 TRANSDUCER MEASUREMENTS

1.0 r 0 IMAGE PROCESSOR (1 SECTION)

.a

.6

.4

.2

D TRANSDUCER MEASUREMENTS

r o IMAGE PROCESSOR 13 SECTIONS)

I--

w 0

2

u. U w 8

0 IMAGE PROCESSOR (5 SECTIONS)

.8' 1.0 xtc

Figure 6. Effects of magnification. a) no magnification, b ) three times magnification, c) five times magni f ica t ion

7

Page 10: Real-Time Laser Holographic Interferometry for …...Real-time laser holographic interferometry for aerodynamics George Lee NASA Ames Research Center Mail Stop 260-1, Moffett Field,

WIND TUNNEL -

ORIGINAL PAGE IS OF POOR QUALITY

I I LASER

HOLOGRAMS

IMAGE

CAMERA VIDEO

IMAGE PROCESSOR

1 COMPUTER I +

I GRAPHICS I

Figure 7. Prototype real-time interferometer

Figure 8. V i bra t ion- i sola ted optical- table

8

Page 11: Real-Time Laser Holographic Interferometry for …...Real-time laser holographic interferometry for aerodynamics George Lee NASA Ames Research Center Mail Stop 260-1, Moffett Field,

Report Documentation Page

1. Report No.

NASA TM 89462 2. Government Accession No. 3. Recipient's Catalog No.

_____

15. Supplementary Notes

P o i n t o f Contac t : George Lee, Ames Research Center , M/S 260-1, M o f f e t t F i e l d , CA 94035, (415)694-4136 o r FTS 464-4136

P r e s e n t e d a t SPIE 1987 I n t e r n a t i o n a l Symposium S o u t h e a s t on O p t i c s , E lec t ro-Opt ics and S e n s o r s , Orlando, F l o r i d a , May 17-22, 1987.

16. Abstract

Recent developments i n t h e r m o p l a s t i c r e c o r d i n g holograms and advancements i n automated image d i g i t i z a t i o n and a n a l y s i s make real-time laser h o l o g r a p h i c i n t e r f e r o m e t r y f e a s i b l e f o r two-dimensional f lows such as a i r f o i l f lows. a i r f o i l measurements would i n c l u d e a i r f o i l p r e s s u r e d i s t r i b u t i o n s , wake and boundary l a y e r p r o f i l e s , and flow f i e l d d e n s i t y contours . some of t h e problems and requirements of a real-time laser h o l o g r a p h i c i n t e r f e r o m e t e r .

T y p i c a l

T h i s paper a d d r e s s e s

4. Title and Subtitle

R e a l - T i m e Laser Holographic I n t e r f e r o m e t r y f o r Aerodynamics

7. Author(s1

George Lee

5. Report Date

June 1987

6. Performing Organization Code

8. Performing Organization Report No

A-87205

10. Work Unit No.

505-61-01

R e a l - t i m e i n t e r f e r o m e t r y Holography

9. Performing Organization Name and Address

A m e s Research Center M o f f e t t F i e l d , CA 94035

12. Sponsoring Agency Name and Address

N a t i o n a l Aeronaut ics and Space Adminis t ra t ion Washington, DC 20546-0001

Un c 1 ass i f i e d- Unl i r n i t e d Sub j e c t C a t ego ry-35

11. Contract or Grant No.

13. Type of Report and Period Covered

T e c h n i c a l Memo ran dum

14. Sponsoring Agency Code

17. Key Words (Suggested by Author(s))

IASA FORM 1626 OCT 86

18. Distribution Statement

For sale by the National Technical Information Service, Springfield, Virginia 22 16 1

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of pages

Un c 1 a s s i f i e d Un classi f i e d 9

22. Price

A 0 1