**where I (x) is the modified Bessel function of the first kind, o**

**6.4.2.4 Theoretical Results**

**The thermodynamic conditions obtained using this analysis and the**
**usual operating conditions in T2 are given in Table 6.2.**

theoretical results are presented here in order to assess more fully the implications of radiation loss from the reservoir region. The initial radial distribution of the rate of energy loss is seen in

Figure 6.7. This shows that, whereas at the centre of the gas slug 95% of the energy lost by emitted radiation is regained by absorption, at the side wall only 45% of the energy is regained by absorption, Yet within 0.8mm from the side wall 90% of the emitted energy is being

regained by absorption. The effect of this greater energy loss at the wall is to cool down the gas near the wall more quickly. This results

in a radial distribution of temperature as seen in Figure 6.8.

The calculations (of Table 6.2) were repeated with different shock tube radii (from 0.1cm to 9 cm). The time dependence of enthalpy is seen in Figure 6.9. This shows that the larger the shock tube radius the slower the decay of enthalpy. However it should be noted that, with a larger shock tube, longer test times will generally be expected,

6.4.2.5 Limitations of a Reflected Shock Tunnel

Due to the fast decay of the enthalpy in the reservoir region under the usual conditions (Mach 16.6 into 50.8 torr) it would seem that the radiation loss is much greater than can be tolerated. In fact, 45% of the enthalpy is lost in 50ysec and 80% is lost in 300ysec (an approximate value of the expected test time).

Using the criterion that less than one third of the enthalpy should be lost during the test time one can determine upper limits on the operating conditions of a reflected shock tunnel. With a particular electron density and temperature, an effective optical depth of the test gas after shock reflection can be estimated. After relating this to the size of the shock tube one then can use the decay curve of Figure 6.8

**77**

300ysec. This gives a critical value for the electron density and temperature. As the electron density is the most important factor for both the radiation loss and optical depth, its critical value was

considered to be the most important condition. From this condition, a critical initial shock velocity can be determined that produces the critical electron density behind the reflected shock.

In shock tunnel T2, with an initial pressure of 50.8 torr of argon, the critical shock velocity is "3.9 x 10~cm/sec (Mach 12). With 50.8 torr of air the critical velocity is "7.4 x 10^cm/sec (Mach 22).

6.4.3 Test Slug and Nozzle Flow

If one uses the test slug as a reservoir for a hypersonic nozzle, one has now a particle loss mechanism to apply to the slug. The mass flow from the reservoir is restricted by the throat and hence the conditions at the throat are important. To determine the conditions at the throat,in the present case situated very close to the end wall of the shock tube, an isentropic expansion is used. An extra condition is required to solve for the throat conditions and it is provided by the fact that at the throat the flow velocity V Ä must be the local speed of sound a^.

Using the standard nomenclature of a starred subscript (*) for the throat, one has the following equations

Conservation of Energy

One also has the enthalpy, state and S a h a ’s equations at the throat and the speed of sound relations

Isentropic Condition S** = S* = 4 Ra + 2R£n + *—* *■* I

* 7 2 1 -a Tm

a

/y*(l + a*)RT*

where v. = -- . is the effective value of y at the throat,

he calculated from S a h a ’s state and entropy (dS = 0) equations. (N.B. the entropy equation used in this analysis was derived ignoring the differential of ionization potential, hence to be consistent the

differential again has been ign o r e d ) .

Y* **1 +**
dT , da

### "t + T+S'

~ dT 2~a da h 2 "r J / kT T ~ 1-c■a a , da where ~ 1(1 + a)**1 m a**

### (5

{2 (1-a) 2R , I ) Tm } aOne is concerned with the alteration of the test slug due to

particle loss through the nozzle. Stratifying the test slug and passing

each element through the throat, one has the relationship where is the

effective cross sectional area of the element of interest in the shock tube. The rate of loss of mass through the throat is

dm

dt p *U*A.V

and the mass in the element is p^Aydx, hence the approximate time for

the element to pass through the nozzle is

*Py * *Ay* dx p 7 d 7 dx

At* = ■— • ■■ ■ — = —■ • . . .

p* A* p*

P 7 ~ A **y**

where d is the diameter), for typical values / 1.3, **/ ^ ~ b 5 ,**

dx " 3 cm, U Ä "2.6 x 105 ; At ~ 680psec.

Thus the whole slug can be passed through the nozzle throat a stratum at a time.

6.5 CONTAMINATION OF THE TEST SLUG

As discussed in Chapter 4, the loss of test gas through the boundary

79
**as a slight reduction in the shock velocity. ** **However? previous**

**experimenters (Copper et. al. 1 1965] , Slade 11970J ) found that taking**
**these effects into account, still gave a much larger test time in shock**
**tunnels than observed. ** **For the T2 shock tube the somewhat simplified**
**analysis of 6.4.3 yielded a slug length of 2.9 cm and a resulting test**
**time of 680ysec, whereas the mass spectrometer was detecting helium**
**as early as lOOysec.**

**Two mechanisms were advanced to explain the departure from ideal**
**theory. ** **In the theory put forward by Markstein (1967), instabilities**
**may develop at the contact surface and due to the density differences**
**there, when the reflected shock hits the contact surface, any distortions**
**would give rise to an axial velocity of part of the contact surface.**

**The other theory, that of Davies and Wilson (1969), was discussed**
**in section 6.2. ** **On this theory driver gas can pass through the bifurcated**
**section of the reflected shock and then flow along the walls and through**
**the throat of the nozzle.**

**One point that should be noted is the different effects of the**
**two mechanisms. ** **Markstein’s instabilities mix the driver gas and the**
**test gas making the resulting gas mixture of no practical use. ** **However**
**in Davies and Wilson’s theory the driver gas does not contaminate the**
**test gas in the stagnation region but rather the nozzle flow. ** **The test**
**gas is still suitable for use with the nozzle if the driver gas flowing**
**along the walls could be diverted from flowing into the centre of the**
**tube at the end wall.**

**As an exploratory exercise, several spectroscopic measurements**
**were taken at the end of the shock tube. ** **This was blanked off with a**
**number of end pieces containing perspex windows. ** **In this way, the**
**time resolved spectra was obtained at various positions across the shock**
**tube. ** **As noted previously, the spectra observed comes from the outer**
**layers of the slug and hence working back from the observed spectra to**

**the gas conditions, is quite a difficult job. ** **A typical time resolved**
**spectrum is seen in figure 6.6.**

**At the centre of the shock tube, the spectra has a very bright**
**continuous flash, initially lasting about 40ysec followed by further**
**continuum but not as bright. ** **A line spectra from impurities super**
**imposed on the continuum develops at ~85psec and various of these lines**
**reverse at "190ysec. ** **(N.B. ** **Some of the spectral lines that are**
**reversed in the initial flash are reversed again at this later time.**

**These correspond to sodium and calcium etc. ** **At the later time absorption**
**lines of iron and chromium appear as well).** **The continuum radiation**
**recorded by the photographic emulsion lasts about a millisecond. ** **At the**
**edge of the shock tube there are a number of absorption lines in the**
**initial flash and the start of the impurity lines is ^SOysec, however**
**the later absorption lines do not appear and the continuum luminosity**
**disappears about 360ysec. ** **The three equally spaced intermediate**
**stations give "luminous lifetimes" of lOOOysec (2.6 mm from centre),**
**700ysec (5.3 mm) and 640ysec (8.0 mm). ** **They all have the impurities**
**coming in about 80ysec. ** **This time has to be measured off the time**
**resolved spectral photograph and is only accurate to a few microseconds**

**(± 5ysec). ** **Table 6.4 gives the onset time of the impurities and the**
**luminous lifetime of radiation at the different observation stations.**

**These show that without particle loss through the nozzle the**
**luminous region of the gas contracts slowly to the centre of the shock**
**tube. ** **In fact at 1 millisecond there is still a luminous region in**
**the centre of the shock tube (5 mm in diameter).** **However these results**
**of the luminous lifetimes are only of marginal interest to this thesis**
**which is mainly concerned with the gas behaviour when there is a nozzle**
**present. ** **Results at times after SOOysec will have little relevance**

**81**