# MATHEMATICAL ANALYSIS

In document A study of radiating argon flows at high opacities (Page 73-81)

## 54 This chapter is divided into three sections:-

### 5.1 MATHEMATICAL ANALYSIS

5.2 Experimental Results 5.3 Conclusion

5.1 MATHEMATICAL ANALYSIS

As could be inferred from the preceding discussion this section falls into two parts. First, a discussion on the ideal theory in which there is a uniform gas slug between the shock front and the contact surface. Second, a discussion of the modification of the ideal theory to explain the real situation in which the gas slug is not longitudinally uniform,

5.1.1 Ideal Refracted Shock Theory

As with the incident shock theory, an analysis of the oblique shock is given in most textbooks on compressible fluid mechanics (e.g. Liepmann and Roshko [I960]). However for this

investigation the analysis needs to be slightly modified for the higher electron densities encountered behind the oblique shock. Rather than describing all the equations in detail, only the

differences between the incident shock (Chapter 4) and the refracted shock theories need be discussed.

The speed of the gas, moving into the shock front comes from the discussion in section 4,2,

V2 " -

### T>1

The three conservation equations (equations 4.2.1 - 4.2.3) are used with the components of the gas speeds (perpendicular to the shock front) on either side of the shock i.e. Vj0 and V j^.

The enthalpy, state and Saha equations will be the same as for the incident shock. An extra equation is needed as the

and the geometry. This yields

tan (8 - 0) = e tan 8

where 6 is the wedge angle and e ( = P2/Pr) is the inverse of the densit}' ratio across the shock.

The computorial method of solution was developed which made use of the fact that if e and 0 are known, then from tan

(8 0) = c tan 8 the solution of interest is

(1/e - 1) - /0 /e - l)z- 4 tan 0/e tan 8

2 tan 0

An iterative procedure similar to that used with the incident shock was employed i.e. the density ratio, Pr/P2 l/e)> was manipulated to obtain the thermodynamic conditions behind the oblique shock.

Figures 5.3, 5.4, 5.5 show the theoretical calculations for the electron temperature, the electron density and the shock angle respectively. The range of incident shock speeds used in the calculations corresponding to the conditions encountered in the low performance shock tube T1 (initial pressure 5 torr). The wedge angles used were 5°, 10°, 20° and 30°. Table 5.1 lists all thermodynamic conditions behind the oblique shock at a single incident shock speed

(U 'vö.'l x 105cm/sec) and the same wedge angles as with figure 5.3 - 5,5.

5.1.2 Refracted Shock Theory for a Real Gas

As discussed in Chapter 4, due to radiation loss there will be varying thermodynamic conditions between the shock front and the contact surface. Each of the gas parameters will be varying

continuously, with the electron density and ionization fraction having the largest variation. To account for this variation, the incident

56

slug of shock heated gas is divided into a large number of parallel homogeneous strata (Figure 5.6). Each stratum has different thermo­ dynamic conditions. Each of these strata are "passed" over the wedge as if it were part of a uniform flow and hence results can be obtained from the analysis of 5.1.1 . By "passing" each stratum in turn over the wedge, an approximation to the time dependent behaviour of the shock angle and the thermodynamic conditions behind the shock can be obtained. The full thermodynamic conditions calculated at various times for a 20° wedge are given in Table 5.2 .

The implications of these two analyses (5.1.1 and 5.1.2) will be discussed in the next section with the experimental results. 5.2 DISCUSSION OF THEORETICAL AND EXPERIMENTAL RESULTS

The graphs and table presented in section 5.1 (figures 5.3 - 5.5 Table 5.1) show certain interesting features.

Due to the energy loss behind the incident shock the gas slows down in the shock fixed coordinates, and this is equivalent to an increase of speed in laboratory coordinates. Hence when the gas passes through the oblique shock, the increased incident speed partially offsets the energy loss behind the incident shock.

The effect of the speeding up is seen in figure 5.7 . This shows the theoretical decay of the electron density (as observed in the laboratory frame) behind an incident shock (2) and an oblique shock (r) (not in the same flow field). Both have been drawn with their peak values at 1.0, and the peak electron densities are

18 3

comparable for these two cases (Ne^ ^ 1.59 x 10 /cm Ne^ 1.15 18 3

x 10' / cm ) . The decay behind the oblique shock is much slower, making interpretation of time resolved observations much simpler

than with the incident shock. As with the incident shock studies, the temperature behind the oblique shock could not be measured to sufficient accuracy for a good comparison to be made between

experimental results, there was agreement.

With the electron density measurements, the same method was used as with the incident shock, though the integration was not as critical, as the electron density is only slowly varying in time. Within the experimental error, agreement was obtained between experimental and theoretical values (Table 5.3). A time integrated spectrum showing the incident and refracted flow is seen in figure 5.8.

The wedge flow takes some time to become established and this has been discussed by Sandeman et al (1971). In the present investigation for the high electron density flows, the time required for the passage of the system of waves which precede establishment of the oblique shock is about 4ysec. However as this system of waves occupies a relatively small part of the flow field during its passage, it makes only a small contribution to the time integrated luminosity pattern, and the edge of the luminous region corresponds to the shock.

As a result of the fact that the luminosity photograph recorded integrated luminosity and that high contrast film was used, the luminosity photograph is expected to yield an angle a little smaller than that predicted by the ideal theory (section 5.1.1). Sandeman et al (1971) used a low contrast film, with the intention of recording the position of the shock wave early in the flow. Their results showed good agreement between the experimental shock angles and those predicted from the ideal theory. Figure 5.9 shows the theoretical angles for an ideal gas and the measured angles. The experimental values are significantly lower as was expected and in approximate agreement with that predicted by the theory of 5.1.2.

The experimental angles of the shock on the wedge are only accurate to ±1 degree due to the shot to shot variations and the errors in

58 measurement of the film. A typical photograph is seen in figure

5.8. Furthermore the theoretical values need to be integrated over the time of the exposure. However within their limitations, there Is agreement between theory and experiment. This is seen on figure 5.9 where an approximate "corrected theoretical" curve has been added. The agreement between theoretical and experimental values of electron densities, temperatures and shock angles means

that the thermodynamic conditions behind the oblique shock are as expected from the theory proposed.

The monochromator (Chapter 3) recorded the continuum radiation from an element just behind the oblique shock. A typical graph of the theoretical continuum is seen in figure 5.10. Two typical traces are shown in figures 5.11A and 5.11B for flow over a 26° wedge.

The second case (U^ = A.4 x 10^cro/sec into 30 torr) is a significantly weaker shock. The radiation loss behind the incident shock will be less pronounced and hence there is a longer period of near constant intensity.

The intensity and the uniformity of the continuum radiation from the gas behind the oblique shock means the flow over the wedge can be used as a spectroscopic standard. As discussed in the review paper on the photographic process (Logan 1971 B) at the moderately short exposure times encountered in this investigation,

reciprocity failure is likely to be important. Hence it is necessary to calibrate the emulsion with an exposure time comparable with that used in the actual experiment. Thus the continuum radiation proved a very useful light source because the continuum intensity as seen in figure 5.11, was nearly a step function. Hence very suitable for film calibration especially in the near ultra violet and infra red where continuous standard light sources are difficult to obtain.

59 B e f o r e c o n c l u d i n g t h i s s e c t i o n , a f ew comments s h o u l d b e made on a p o s s i b l e c o m p l e t e t h e o r e t i c a l a n a l y s i s o f t h e r a d i a t i o n l o s s b e h i n d a n o b l i q u e s h o c k . A t y p i c a l t i m e f o r a p a r t i c l e t o f l o w o v e r t h e we d ge w i l l b e a b o u t 5 y s e c and t h i s w i l l b e " p a r t i c l e t i m e " , w h e r e a s i n t h e i n c i d e n t s h o c k t h e " p a r t i c l e t i m e " r e q u i r e d t o r e a c h 5 cm b e h i n d t h e s h o c k f r o n t i s 6 0 y s e c . I n f a c t m o s t m e a s u r e m e n t s o f t h e r m o d y n a m i c p r o p e r t i e s b e h i n d an o b l i q u e s h o c k w e r e t a k e n i n t h e f i r s t 2 . 5 y s e c a f t e r t h e g a s p a s s e s t h r o u g h t h e s h o c k f r o n t . Thus t h e r a d i a t i o n l o s s w i l l n o t b e a s i m p o r t a n t a s i t i s w i t h t h e i n c i d e n t s h o c k . However a f u l l a n a l y s i s o f t h e wedge f l o w w o u l d n e e d t o i n c l u d e t h e r a d i a t i o n l o s s and i n t h a t c a s e t h e r a d i a t i o n m o d e l s w o u l d n e e d t o b e r e s t u d i e d , a s t h e s p h e r i c a l m o d e l i s c e r t a i n l y n o t a p p l i c a b l e . The m o d el t h a t s u g g e s t s i t s e l f i s a c y l i n d r i c a l m o d e l , w i t h i t s l e n g t h t h e w i d t h o f s h o c k t u b e ( o r d u c t ) an d d i a m e t e r t h e p e r p e n d i c u l a r d i s t a n c e f r o m t h e s h o c k f r o n t t o t h e we dge s u r f a c e , t h r o u g h t h e p o i n t o f i n t e r e s t . H o w e v e r , a s i t was o n l y o f m a r g i n a l ^ n t e p e s t t h e a c t u a l c a l c u l a t i o n s o n s u c h a m o d e l w e r e n o t p e r f o r m e d . F u r t h e r m o r e , r a d i a t i o n l o s s , i f a p p r e c i a b l e , wo ul d g i v e r i s e t o a c u r v e d s h o c k f r o n t , f o r i n t h e c o n t i n u i t y e q u a t i o n dA ( L i e p m an n and Roshko [ i 9 6 0 ] ) w i l l no l o n g e r b e z e r o . 5 . 3 CONCLUSION I n t h i s c h a p t e r , i t h a s b e e n shown t h a t t h e t h e o r y p r o p o s e d an d t h e e x p e r i m e n t a l r e s u l t s a r e i n a g r e e m e n t w i t h i n t h e e r r o r s i n h e r e n t i n t h e e x p e r i m e n t a l m e a s u r e m e n t s . However t h e p l a c e o f s u c h a d i s c u s s i o n i n t h e o v e r a l l p l a n o f t h e t h e s i s s h o u l d b e e m p h a s i s e d . S u p e r f i c i a l l y , i t wo ul d seem t o b e o f l i t t l e c o n s e q u e n c e t o t h e b a s i c t h em e o f t h e t h e s i s . The r a d i a t i o n m od el w h i c h ha d b e e n shown t o w o r k i n t h e e l e c t r o n d e n s i t y r a n g e o f

60 1 0 16 - IO"1 ^/cnr" was n o t u s e d t o d e s c r i b e t h e r a d i a t i o n . l o s s b e h i n d t h e o b l i q u e s h o c k . F u r t h e r m o r e , t h e e x p e r i m e n t s w e r e d o n e o n t h e l o w e r p e r f o r m a n c e t u b e T l , an d t h e maximum e l e c t r o n d e n s i t i e s o b t a i n e d w e r e o n l y ~2 x 1 0 ' 8 / c m 0 . T h i s v a l u e was o n l y s l i g h t l y a b o v e t h e e l e c t r o n d e n s i t i e s e n c o u n t e r e d b e h i n d t h e i n c i d e n t s h o c k i n T 2 . A l t h o u g h i t wo u l d h a v e b e e n i n t e r e s t i n g t o e x t e n d t h e i n v e s t i g a t i o n t o T2 i t was t h o u g h t t h a t l i t t l e a d d i t i o n a l i n f o r m a ­ t i o n c o u l d b e g a i n e d f ro m s u c h a s t u d y . H e n c e , a s t i m e was s c a r c e ,

In document A study of radiating argon flows at high opacities (Page 73-81)