Experiments in Fluids 9, 85-91 (1990) Experiments in Fluids �9 Springer-Verlag 1990

Instantaneous density measurement in two-dimensional gas flow by high speed differential interferometry

J.M. Desse

Office National d'Etudes et de Recherches A~rospatiales, Institut de M6caniques des Fluides de Lille, 5, Boulevard Paul Painlev6, F-59000 Lille, France

Abstract. The aim of this paper is to present an experimental set-up using a Wollaston prism differential interferometer producing up to twenty successive short exposure white light interferograms at a high framing rate. It is shown that, through optical component calibra- tion, the interferograms can be analysed to yield the instantaneous density field. This method has been successfully tested in the two- dimensional unsteady flow generated by the interaction of a mixing layer and a cavity.

List of symbols

h H M t At N 7C

3p Cp

~o

Oy R L

E E~ d~ d~ Q ~o Q~

height of the downstream edge of the cavity height of backward facing step Mach number time time interval between two successive frames frequency double-prism median plane birefringence angle pressure fluctuation pressure coefficient biprism abscissa corresponding to any colour biprism reference abscissa corresponding to background colour deviation of light rays radius of curvature of spherical mirror virtual distance from the middle of the test section to the spherical mirror optical thickness optical thickness corresponding to background colour difference of optical thickness abscissa difference gas density stagnation gas density gas density of background colour

1 Introduction

For the observation of high speed unsteady flows different optical techniques have already been developed such as high speed schlieren or shadowgraph. Through them one can reach not only qualitative but also quantitative informa- tions such as propagation velocity of shock waves or vortical

structures (Kazandzhan and Sukhorukikh 1971; Goldstein 1983).

Previous methods do not work when quantitative infor- mations about the density field are required. In this case interferometry must be used. The problem of the analysis of monochromatic light interferograms lies in the identification of the fringes and of their shift across flow discontinuities. When the history of an unsteady flow is required, pulsed lasers can be used: but with this device only few successive interferograms can be recorded.

When the density gradients are relatively weak the differ- ential interferometry in polarized white light is very well adapted. It produces coloured interferograms, the analysis of which yields the density field after calibration of the whole set-up (Dewey et al. 1983). This paper shows an extension of this technique to the high speed visualization of rapidly varying flows.

2 Optical set-up

The optical apparatus which we used is constituted of three components: a I M F L strio-interferometer (SD200),.a light source and a recording device (Fig. 1).

The principles of differential interferometry have been discussed in details by Gontier (1957). Briefly, a Wollaston biprism decomposes a polarized light vibration into two orthogonal vibrations emerging along rays separated by a small birefringence angle e = e (2). As a consequence, two rays cross the phase object at slightly different locations. Then they are made to interfere in an analyser. The difference in optical path, which is put into evidence in this way, is char- acteristic of the body under observation as long as the set-up is autocompensated.

The initial path difference between the two rays can be made to take an arbitrary value thus yielding, in a perfectly set-up apparatus, a uniform colour on the screen simply by translation of the biprism. Regions of equal phase shift will thus appear as identically coloured on the observation plate.

86 Experiments in Fluids 9 (1990)

I!/ Strio-interferometer ( Windows

' ' - ~ IF[ash unit [ ! b / / M o O ~ t . I ~ .............. , , : '+~ 15oo.,s l / I / / , Wollasmn/ ~ : ~ : ~ ~ / P

~ . F ~ ...................... i prism, ~J I i .... { ~ ~_. ~ t _ ~ ._ , :~.:=...=.~:~.~i.1 ........................... ~< ........ ......... ~ : ~ : ~ : ~ - - ~ I ' I . . . . . . . . . . . . . . . . ,

~<~] . . . . . . . . . . . . . . . . . . . . . . . . . . i: Polariz-er ql : ~ 7i-':--i J I h

@ I t

/ Ray issuing from flash source--~

P, "

o ~ ~ , - �9 ~-

' ~ l ~ . . . . . ? . . ~ ' - 2_2 . - - - 2

["Mirror Analyser ~ l-

Fig. 1. Differential interferometry apparatus and light ray path

-0.5 j '0.6'25'0 0:5 1:5 ' 2 ' :3 ~ f ~ = x/~'~

7, . ~/ ~

-1.5 ~ ' \ # ~ . . �9 h / H = O

'~ h / H : O . 5 0 h / H = 0.75

�9 h/H= 1.00 -2.0 ~ #

i

Fig. 2. Evolution of pressure coefficient for every case

In IMFL ' s SD 200 a diverging light beam is sent towards the object to be observed. After crossing it, the beam is focused back towards the emitter, thus crossing it a second time, so that the sensitivity is increased. But, if a strictly uniform background colour is required (no phase shift), the Wollaston biprism should be crossed by a parallel beam. This is not so in our case and one could expect to have continuously varying colours in the field of observation. However some means can be used to reasonably compensate for this defect (Frangon and Sergent 1955) and obtain a large field birefringent compensator.

The apparatus including the SD 200 and the mirror is easy to set up. It has a high sensivity and it is much smaller and much less subject to vibrations than the conventional separated beams interferometers such as the Mach-Zehnder interferometer.

In our experiments the object under observation is the test section of a transonic wind tunnel. The analysis of the interference fringes yields a measure of the optical path dif- ference across the crystal, the test section and the surround- ing atmosphere. However the optical set-up is autocompen- sating, as the deviations which occur on the outgoing and returning light rays in the atmosphere and in the biprism yield optical path variations which are opposite and thus cancel each other. The measured phase shift is then charac- teristic of the object under observation. From this, in the case of two-dimensional flow, one can infer the component of the gas density gradient along a direction normal to the interference fringes. The gas density itself is obtained by integration.

The light source is specially aimed at getting instanta- neous interferograms. It is an electronic flash unit (HAD- L A N D HL500), equipped with a xenon bulb, delivering 1000 J is 500 ps. The rise time is about 20 gs and the power is constant within 5% during the whole duration. This is important in order to obtain successive interferograms of the same illumination. The colour temperature of the emitted light is very close to that of the sunlight, so that the various colours observed behind the Wollaston prism are nearly those found in the Newton scale.

The high speed recording device is a rotating drum and rotating prism C O R D I N 350 camera. The framing speed can reach 35,000 frames per s. The pictures are 10 x 8 mm 2 in size and are recorded on a 35 mm film. The exposure duration is 1.5 ~ts, and in our case the time interval separat- ing two successive frames has been chosen to be 28.5 ~ts.

3 Experimental application

3.1 Set-up

This new technique has been used to try and obtain quanti- tative information about the unsteady flow in the vicinity of a cavity located close to the base of a blunt body. Indeed, previous studies seem to indicate that significant drag reduc- tion is associated with the emission of high frequency vorti- cal structures from the cavity. However the formation mech- anism of these structures is not well known.

A large scale representation of this type of flow has been made on a two-dimensional model of a backward facing step of height H = 40 mm. The reattachement floor is 250 mm long, thus avoiding the formation of a strong Karman vortex street. An edge of variable height, h, and thickness 4 mm has been located 40 mm downstream of the step. Four values of h have been chosen, namely 0, 20, 30 and 40 mm, the first of which being the reference case. The model spans the wind tunnel test section which is 42 mm wide and 200 mm high.

J. M. Desse: Instantaneous density measurement in two-dimensional gas flow 87

0 -4-k~,,0. 2 0.4 "-"~f0.6 t (ms)

1 5 7 9 11 13 15 17 19 21

i illll llli 0[ t 11'0'2 [04 I1 06,

2 4 6 8 10 12 14161820 Corresponding interferogroms of flow

~P (mb)

1.5

1.0

0.5

2240Hz

I

0

x/H=0.5

2000 ' 4000 ' 60'00N(Hz) Fig. 3. Analysis of signal and synchroni- zation with interferograms

6P (rob)

25

20-

15.

10.

5.

0 -0.12 0

A h/H=0.50 N=22AOHz

h/H=0.75 N=4869Hz �9 h /H= l .00 N=7017Hz

v

A ~A A *

" 0:5 1:5 31s x m Fig. 4. Unsteady pressure amplitude

mixing layer is intercepted by the edge: the flow downstream of the edge is close to that observed downstream of the step alone.

The situation is different when the edge is under the mix- ing layer or when it only partially intercepts it, as for h/H = 0.75 and 0.5. In the former case, the edge partially inter- cepts the mixing layer, the pressure is decreased in the cavity and increased downstream of the edge. Reattachment occurs much closer, and on the front part of the model the pressure continuously decreases due to the sucking action of the cav- ity. In the latter case the edge only affects the recirculation domain. Upstream of separation, the results are close to those obtained for h = 0, whereas downstream they are sim- ilar to those for h = 0.75 H.

The upstream Mach number has been kept constant for all tests at a value of 0.42. The model is thick compared to the test section height. The value of Mach number measured above the separation step (---0.62) is close to that which is obtained for one-dimensional flow in a converging nozzle (0.57), the difference being accounted for by viscous displace- ment on the wind tunnel walls. The model is equipped with pressure taps distributed on the center line and a few un- steady pressure transducers (KULITE) in order to associate fluctuating pressure and instantaneous interferograms.

The field of view has a diameter of 250 mm.

3.2 Mean pressure

Figure 2 shows the mean pressure coefficient Cp for different values of h.

When h = 0, Cp is constant up to about 3.5 H where the mixing layer begins to reattach. One can note the intensity of the pressure gradient on the front part of the model, as well as a neat increase of pressure just ahead of separation.

For h = H the distribution of Cp is basically the same. The behaviour is representative of all configurations where the

3.3 Unsteady pressure

The signal from the pressure transducers is recorded on analog magnetic tape for later conversion to digital and processing. Up to 7,000 samples per channel can be obtained from the conversion system at a rate of 160 KHz. Figure 3 shows the signal from the transducer located on the side of the cavity together with the corresponding Fourier spec- trum. Also included are ticks indicating the times at which interferograms were taken, some of which are shown in Fig. 6. The amplitude of the strongest component of the spectrum is plotted in Fig. 4 versus the position of the trans- ducer for the 3 non-zero values of h, with an indication of the corresponding frequency. We can see that N increases with h. However, the pressure fluctuations are maxima for h = 0.75 H. In fact this value corresponds to the cavity which leads to the best drag reduction.

When the edge interacts with the mixing layer the large vortices which can be seen in Fig. 7 are formed more quickly. For h = H the vortices have a higher frequency but a smaller size and they have nearly disappeared when they are con- vected over the edge. Furthermore, on the interferograms of

88 Experiments in Fluids 9 (1990)

Fig. 5. Double prism calibration - variation of colours with ~ - 4o

Fig. 9 one can see that the behaviour of the mixing layer behind the edge is similar to that corresponding to h = 0.

3.4 Analysis of the interferograms

During the tests the biprism is located at the center of curva- ture of the spherical mirror in order to obtain a uniform background colour (4o). In these conditions we realized a calibration of the observed tint versus transverse position of the biprism ~ - ~ o , as indicated in Fig. 5. The eolour spec- trum is symmetrical with regard to the white fringe.

When analysing interferograms one starts at a position where the background colour is uniform and, moving along a path normal to the piprism, one notes the observer colour. F rom it and from the calibration of Fig. 5, the difference in optical thickness is deduced: dE=d x .(~--~o)/(R--L').

The relation between density and refractive index then leads to the component of the density gradient. This analysis is visually conducted on a film reader, with a magnification of about 30. Density is obtained by integration along the path.

Figures 6 - 9 show interferograms obtained on each con- figuration with horizontal infinite fringes. One can see that vortices are represented as a succession of centered rings.

For instance, in Fig. 6 one can see a series of very small vortices close to the separation point. The vortices shed acoustic waves which can be seen in Fig. 8 and 9. These waves move upstream in the flow. In this case the external field is no more uniform and the analysis of the interfero- grams cannot be realized.

A full examination of one interferogram of each case h = 0 and 0.5 H has been achieved. For this the gas density has to be known at one point in the field, which has been chosen in the steady isentropic uniform flow domain above the model, where pressure probes could be inserted, which, as previous- ly stated, yield a value of 0.61 for the Mach number.

At first we considered h = 0.5 H and analysed all the inter- ferograms along the x = H line. Figure 10 shows the evolu- tion of the deviation 0y and the difference in optical thickness (E-- Eo) versus y for different instants of time. This reveals a periodic evolution in time of (E -Eo) , indicating mass ex- change between the cavity and the outside.

Fig. 6. Interferograms h/H=O, At=28.5 gs

Fig. 7. Interferograms h/H =0.5, At = 114 p.s

Fig. 6

Fig. 7

Fig. 8

Fig. 9

J. M. Desse: Instantaneous density measurement in two-dimensional gas flow 91

3 5 7 9 11 13 15 17 19 21 Corresponding inferferograms of flow At=57ffs

-50

y/H! 50

1060Y 0.5

- ~ 0 ~

-O.5

y/H l 1000 -1000

Fig. 10. Evolution of Oy and E - E o at x / H = l for h/H=0.5

Figure i1 shows the result of the analysis of interfero- grams 5 and 1 of Figs. 6 and 7, respectively: the instanta- neous gas density field is represented on a 3D plot.

Across the mixing layer the density increases and shows little variation up to x = H. In the case h = 0, the same can be seen downstream of that location. On the other hand, for h = 5 H, the density shows a variation which represents 15% 20% of the undisturbed flow density 4e" For instance, along the reattachment plate 4 is close to 0.65 40, whereas in the middle of the second vortex it reaches a value of 0.9 4o-

4 Conclusion

The new technique of high speed interferometry provides good quality interferograms. As it is shown, the density field can be reconstituted as long as an overall calibration of the set up is performed. In our experiments quantitive informa- tions have been obtained concerning the formation of vorti- cal structures and their evolution.

This example gives a general idea of the large possibilities offered by the technique, the exploitation of which should be made much easier by some kind of automatic processing of the interferograms.

Fig. 8. Interferograms h/H=0.75, At=57 gs

Fig. 9. Interferograms h/H= 1.0, At=28.5 gs

Fig. 11. Reconstitution of gas density field for h/H=O and 0.5

References

Dewey, J. M.; Heilig, W.; Reichenbach, H.; Walker, D. K. 1983: The analysis of coloured interferograms of shock waves. In: Flow visualization (ed. Jang, W.-J.). Vol. 3, pp. 478-482. Washington: Hemisphere

Francon, M.; Sergent, B. 1955: Compensateur bir6fringent/t grand champ. Opt. Acta, 182-184

Goldstein, R. J. 1983: Fluid mechanics measurements, pp. 377-422, Washington: Hemisphere

Gontier, G. 1957: Contribution fi l'6tude de l'interf6rom6rie differen- tielle fi biprisme de Wollaston. Publications scientifiques et tech- niques du ministere de Fair, Paris

Kazandzhan, E. P.; Sukhorukikh, V. S. 1971: Interference measure- ments in gas dynamics. Soviet Physics- Doklady 15, 939-942

Received August 15, 1989