An electron beam technique was used to measure density profiles of strong shock waves in argon with high accuracy. The experimental results are compared with the results of theoretical models. Of the models that are available in enough detail for comparison, the best agreement with experiment is shown by the direct simulation Monte-Carlo method (Bird 1968), assuming an inverse 12th power molecular interaction force law. It is shown that the density maximum slope thickness is not sufficient for a detailed description of the shock wave structure

Schmidt, B.

English

Caltech Authors - Main

J .  Fluid Mech. (1969), wol. 39, part 2, pp .  361-373 
Printed in &eat Britain 
361 
Electron beam density measurements in 
shock waves in argon 
By B. SCHMIDT? 
California Institute of Technology 
(Received 29 August 1968) 
An electron beam technique was used to measure density profiles of strong shock 
waves in argon with high accuracy. The experimental results are compared with 
the results of theoretical models. Of the models that are available in enough detail 
for comparison, the best agreement with experiment is shown by the direct 
simulation Monte-Carlo method (Bird 1968), assuming an inverse 12th power 
molecular interaction force law. It is shown that the density maximum slope 
thickness is not sufficient for a detailed description of the shock wave structure. 
1. Introduction 
Shock wave structure has attracted a considerable amount of interest because 
the normal shock wave is the simplest flow in which large departures from 
equilibrium occur. The flow is one-dimensional, boundary effects are absent, and 
initial and h a 1  states are equilibrium states connected by simple laws. 
In  order to make a detailed comparison of shock wave structure with the pre- 
dictions of various theoretical models, more accurate experiments than have 
been previously reported are required. In  the present work sufficiently accurate 
profiles are presented for shock waves in argon at shock Mach numbers between 
2.8 and 8. 
Shock wave thickness fails to give sufficiently detailed information about 
structure; calculated and measured profiles may differ considerably, but the 
thickness can be the same. It has been found that, judged by thickness, the 
Mott-Smith method gives good agreement with experiment. However com- 
parison of the profiles shows that the symmetric result of the Mott-Smith model 
disagrees with the asymmetric result of the experiment for strong shock waves. 
Unfortunately, enough information for comparison is available for only a 
limited number of theoretical models. Of the models that have been compared 
with the present experiments the best agreement is given by a direct simulation 
Monte-Carlo method developed by Bird (1968). 
2. Apparatus and experimental procedure 
The experiments have been performed in the 17in. shock tube at California 
Institute of Technology. The tube is made of 43.43 cm (17.1 in.) stainless steel 
t Present address : University of Karlsruhe, Karlsruhe, Germany. 
362 B. Schmidt 
tubing and consists of a 3-82 m (8.78 diameter) long driver section and a 21-3 m 
(49 diameter) long driven section. The electron beam is located 48.5 diameters 
downstream of the diaphragm and 20cm from the end wall. 
Figure 1 shows the arrangement of the electron beam apparatus. The main 
concepts in the design of the apparatus have been, first, a rigid, self-supporting 
U 
0.25 in. o.d Itcage 2.8 mm i.d. 
f !  
Needle signal 
Deflexion pulse 
GND 
FIGURE 1. Electron beam apparatus. 
Electron-beam density measurements in shock waves 363 
structure that is uncoupled from the shock tube and, second, an alignment pro- 
cedure that is guided by current measurements only, without visual observation 
of the beam. The design benefits from experience obtained with a similar appara- 
tus designed by Russell (1965, pp. 265-76). 
The electron beam is generated by a commercial T.V. picture tube gun? and 
is injected into the shock tube through a 04mm hole in the cone tip (‘needle’ 
in figure 1). A small Faraday cage with 2.8 mm i.d., located opposite to the cone, 
collects the beam. 
The cone tip and the Faraday cage are arranged symmetrically about the shock 
tube axis and protrude into the shock tube far from the walls. Thus, the beam 
traverses only the least bowed part of the shock wave. The gap between cone tip 
and cage (sampling length) is variable between 50 and 118mm. 
Cage and cone are supported by a 30.48 cm (12 in.) i.d. ring, which is arranged 
concentric to the shock tube. The ring is wedge-shaped on the upstream side. 
A stiff tube, the stem, connects the 12 in. ring with the gun chamber block located 
outside of the shock tube. The gun chamber can be shut off from the shock tube 
by a ball valve inside the stem, operated from outside the shock tube. The entire 
apparatus is supported independently of the shock tube and the only connexion 
between the two is a very flexible bellows. The low pressure required inside the 
gun chamber is maintained by an oil diffusion pump. In the worst case (pl = loop ,  
ball valve open) the pressure is below 0 . 1 ~ .  
The gun generates an electron beam$ of roughly l-Omm diameter that is 
reduced by the injection hole to 0.8mm diameter. A beam spreading over the 
sampling length has not been observed. 
Faraday cage and injection cone are insulated electncally and connected to 
ground through a l O O O Q  resistor. Therefore, the currents to cage and cone 
generate voltage signals which are fed separately over preamplifier (Philbrick 
P25AH) into an array of oscilloscopes (Tektronix 555 and 535). 
Before every shot the shock tube is pumped to 0.1 ,u (1 p = lO-aTorr), flushed 
with the test gas at  50 p, and again pumped to 0.1 p. Then the test gas is released 
into the shock tube from a calibrated volume of one thousandth of the driven 
section volume. The pressure inside the calibrated volume is controlled by a pre- 
cision pressure gauge (Wallace & Tiernan, 0-50Torr abs.) and the test section 
pressure is checked occasionally with a McLeod vacuum meter. 
The shock wave velocity is measured with thin film heat-transfer gauges 
arranged 50 cm apart, with the downstream gauge 1.45 diameters upstream 
of the electron beam. The signals from the gauges are amplified and fed into 
a time interval meter (Hewlett Packard counter 5233 L). 
To record the zero current level for the cage signal, and the beam intensity at 
the point of injection, the electron beam is deflected electrostatically. Approxi- 
mately 50ps after passage of the shock wave a square wave signal is applied t o  
deflexion plates inside the injection cone (cf. figure 3). 
A few milliseconds after the shock wave passes the measuring station, the beam 
t Superior Electronics Laboratories, N.J., type SE 110 E. 
$ The measurements are made with 15 kV acceleration voltage and 90-120 pa beam 
current. 
364 B. Xchmidt 
high voltage is switched off automatically and the ball valve isolates the gun 
chamber from the shock tube. These measures increase the lifetime of the gun 
from only a few runs to over eighty. 
3. Calibration and data reduction 
In  order to obtain a calibration under static conditions the beam is modulated 
with square wave that is just strong enough to  deflect it from the injection 
hole. The subsequent cage and cone signals are displayed on a synchronized 
dual beam oscilloscope and recorded photographically. The traces are two syn- 
chronous square waves with amplitudes proportional to (i) the beam current I, at 
the injection point and (ii) the beam current I at the cage after attenuation by 
scattering over the sampling length. If the gain ratio of the two channels is unity 
the ratio of the square wave amplitudes is equal to the current ratio IJI. 
331 
FIGWEE 2. Electron beam Calibration. Argon, 3 in. gap, 15,000 V. 
If the exponential scattering law I/Io = exp ( - Kp) holds, where p = density 
and K = proportionality factor, thenlog Io/I plotted verswp should be a straight 
line. The curve shown in figure 2 is the average of several calibrations. Sur- 
prisingly, it consists of two straight lines with different slopes, the transition 
occurring near p = 90p. A reasonable explanation for this behaviour is not 
known yet. 
It has been found unnecessary to use the static calibration to reduce the shock 
Electron-beam density measurements in shock waves 365 
wave data. For, since the values, p1 ,pa, Il and 12t corresponding to the equilibrium 
states far upstream and far downstream of the shock wave are known, the 
measured current can be converted to density directly using the scattering law. 
It has been found that, within the scatter and for pl < 25p, the dynamic calibra- 
tion constant checks the lower range static calibration constant. 
FIGURE 3. Typical oscilloscope traces. M ,  = 6-07, pl = 25 p ,  argon. 
Figure 3 shows typical oscilloscope traces for one run. The cage and the cone 
signals are marked by C and N (N is shifted half a division horizontally to avoid 
signal overlapping). The l ops  long distorted square waves on the top picture 
are used to determine the gain ratio of the cage and the cone circuits. The bottom 
picture shows the shock wave part of the cage signal enlarged. The bright dots 
t The usual shock notation will be used throughout this report. 1 = equilibrium state 
upstream of the shock wave, 2 = equilibrium state downstream of the shock wave. 
366 B. Schmidt 
on the trace are time marks from a quartz oscillator controlled time mark 
generator. 
Figure 4 shows how the data are obtained from the bottom picture of figure 3 
(the zero current level is obtained from the top picture). Because all pictures show 
that the beam current decreases nearly linearly for at least two shock thicknesses 
downstream of the shock wave (cf. error estimation), the end of the wave and 
therefore the location of point ' 2 ' is indistinct. In  this work point 2 has been 
assigned to the point at which the curved trace merges into the linear decrease 
(figure 4). This method results in much less scatter than using lo and I;, determined 
on the top picture of figure 3, to calculate l2 and therefore the position of point 2; 
the tolerance 6 in figure 4 indicates the error in l2 that would result if an error 
of only k 2.5 % is made reading I. from figure 3. 
Conversion of the time t to the distance z for the purpose of plotting density 
profiles is accomplished by multiplying t with the measured shock speed u,. 
0 L 
E 
1 
4. Error estimation 
An error estimate indicates that, of the quantities derived from the measure- 
ments, the largest errors are associated with the mean free path A,,? the shock 
thickness 8,s and the normalized density R = (p-p1)/(p2-p1). In the worst 
case, A, can be wrong by 5 4.6 %, 6, by k 3-3 % and R by k 3 yo. The main con- 
tributions to these errors are connected with the determination of p1 and the 
reading error of At (cf. figure 4) and cage current. 
Other factors that introduce errors are the finite rise time r of the cage circuit 
(r = O.lps,  measured), the finite ratio of the electron beam diameter d to the 
t A, = ~ , / ( y / Z ~ ) p ~ / p , a ~  where p = viscosity, p = density and a = speed of sound 
A, = 5.49 cm for y = Q, T, = 300 "K and p1 = 1 p. 
$ & = u, At where u, = shock wave speed and At is defined in figure 4. 
Electron-beam density measurements in shock waves 367 
shock thickness S, the measurement of shock speed, and shock curvature. All 
these errors amount to less than 1 yo. 
As already mentioned, a continuous but small density rise downstream of the 
shock wave is observed in all measured profiles. This is partly caused by the 
boundary layer on the shock tube wall behind the shock wave, whose displace- 
ment velocity causes the density to increase downstream of the shock wave and 
the shock wave to bulge forward. 
Calculating the density rise behind the shock wave using the measured curva- 
ture (Liepmann & Bowman 1964) gives a smaller density rise than measured, 
but with the same dependency; the density rise is nearly independent of shock 
strength and becomes smaller with increasing pl. 
Furthermore, it is observed that the deviations behind the shock wave depend 
(inversely) on the sampling length. These distortions may have been caused by 
aerodynamical disturbances generated by the cone and the cage, or by a changing 
inflow pattern at  the injection hole. They show the same tendency with varying 
p I  and it(, as those caused by the boundary layer. A reasonable explanation for 
effects which depend on sampling length is not known yet. 
5. Results 
The results of the measurements are shown in figure 6 .  The profiles are the 
averages of the results of at  least nine runs at various p1 and fixed H,. The 
extent of scatter at one Mach number is indicated on figure 6. The profle for 
H, = 2.8 is not shown on figure 5 ;  except for a shorter foot on the low density 
side it coincides with the profile for M, = 6. 
X I 4  
, Ma = 8. FI~IJFLE 5. ----, Ma = 4; -, M , - 6;-.-.- 
368 B. Schmidt 
The effect of M8 on the profiles is small, the only significant differences being 
on the low density side of the wave (the deviations observed in the high density 
part of the profdes are within experimental accuracy). Thus, increasing shock 
strength causes the profiles to become more asymmetric. 
1 -0 
0.9 
0.8 
0-7 
0-6 
7 0.5 
0.4 
0.3 
0.2 
0.1 
0 
Q <  
- - 6 - 5 - 4 - 3 - 2 - 1  0 1 2  3 4 5 6 7 
“ I 4  
FIGURE 6. M ,  = 4, argon, m, present experiments; 
x, Schultz-Grunow & Frohn (1965). 
Experimental results for comparison of full profiles are available only in 
the paper of Schultz-Grunow & Frohn (1965, pp. 250-64). As figure 6 shows, the 
scatter of their experiments is much larger than ours, and therefore important 
details are lost. 
A comparison of some theoretical results with the experiments is shown in 
figures 8 (a),  ( b )  and (c). The selection of theories is not complete because detailed 
information is available for only a limited number of theoretical models. Very 
good agreement is obtained with the direct simulation Monte-Carlo method of 
Bird (1968), using an inverse 12th power intermolecular force law (P N r-Y, 
v = 12, r = distance of colliding molecules). The difference in agreement between 
v = 12 and v = 11 was easily distinguishable. Bird gives an estimated accuracy 
for his results of 1 %. 
This is the 
same dependence that has been measured in recent heat transfer experiments 
(Matula 1967) covering the same temperature range (300-6000 O K ) .  In  this work 
the same viscosity-temperature law is used in all the theoretical models. 
The agreement with the other models is not as good, and, aside from dis- 
crepancies in predicted shock thickness, the most important area of disagreement 
v = 12 corresponds to a viscosity-temperature relation p N 
Electron-beam density measurements in shock waves 369 
is the extent of asymmetry evidenced by the profiles. A sensitive measure of 
asymmetry used here for quantitative comparison is 
wherez = "/A,, R(z) = (p(z) -p l ) / (pz -p l )  andR(0) = 0.5. Theresultsfor different 
theoretical models and the experiments are shown in figure 7. The Navier- 
Stokes equations and the BGK model (Liepmann et al. 1962) give too asymmetric 
a profile. The Mott-Smith method (Muckenfuss 1962) gives a symmetric profile 
(& = 1) so its disagreement with experiment increases with shock strength. 
1.5 
1.4 
1 -3 
Q 
1.2 
1.1 
1.0 
1 2 3 4 5 6 7 8  
M8 
FIG- 7. A, Navier-Stokes; 0, BGK; 0, Mott-Smith; 
0, Monte-Cerlo; v, present results. 
Many more experimental and theoretical results are available for shock thick- 
ness. Although the present experiments do not disagree with previous measure- 
ments of shock thickness, the data are shown for completeness in figure 9. 
The present data agree in the large with the results of Russell (1965)' Camac 
24 Fluid Mech. 39 
370 B. Xchmidt 
1.0 
0.9 
0-8 
0.7 
0.6 
f 0.5 
0.4 
0.3 
0.2 
01 
0 
Q d  
- 
FIGURE 8. (a) Ma = 2-8, p N ---.- , Navier-Stokes ; . . . . . ., BGK model ; - - - - - ., 
, Navier- Mott-Smith model;  , present results. (b )  Ma = 8, p - 2'0.88. ---.- 
Stokes; ... ..., BGK model; ------, Mott-Smithmodel; - , present results. 
Electron-beam density measurements in shock waves 371 
-9 -8 -7 -6 -5  -4  -3 -2 -.1 0 1 2 3 4 5 6 
X I 4  
FIGURE 8 ( c ) M ,  = 8.- , experiment; ---- , Monte-Carlo method, Y = 12. 
FICXTRE 9. Al = J(6/6?r) f i / p 1 4 .  m, Camac; 0, Linzer & Hornig; x , argon, Robben & 
Tdbot; +, helium, Robben & Talbot; 8, Russell; 0, Schultz-Grunow & Frohn; v, 
present results. 
24-2 
372 B. Schmidt 
I I I 
2 3 4 5 6 7 8 9 10 
M ,  
FIGURE 10. p N TO'". I ,  Navier-Stokes; 2, BGK model; 3, Mott-Smith model; 
0, Monte-Carlo method; v, present results. 
(1965, pp. 240-49) found thicker shock waves ( - 15 %) for M, < 8; his data merge 
with the present ones a t  M, w 8. Slightly thinner waves ( -  6 yo) are reported by 
Schultz-Grunow & Frohn (1965) for M, = 4 and 5. Linzer & Hornig (1963) found 
even thinner shock waves ( -  12 yo). The results of Robben & Talbot (1966) 
agree with the present data at M, = 2.8, give thicker shock waves for argon 
( -  8 %) and thinner ones for helium ( N 12 yo) a t  M, = 3.4 and merge with 
Camac's results at M. = 10.7. 
The measured shock thickness is compared with the model solutions in 
figure 10. Disagreement with the Navier-Stokes solution and the BGK model 
are visible here, too. The Mott-Smith method agrees well with the experiment; 
the differences in the profile do not show up here. This example demonstrates that 
shock wave thickness fails to reflect important details of shock wave structure. 
6.  Conclusion 
It has been shown that a high degree of accuracy is required if the measure- 
ment of density profiles is to exhibit important details of shock structure. Of the 
theoretical models that are available in enough detail for comparison, the best 
agreement with experiment is given by the direct simulation Monte-Carlo method 
(Bird 1968), assuming an inverse 12th power molecular interaction force law 
(p - T0.68). Further, it has been shown that thickness is not sufficient for de- 
scribing details of shock wave structure. 
For the present experiments the limits in accuracy have been set by the 
pressure measurement (pl) and reading errors of the oscilloscopes. 
Electron-beam density measurement in shock waues 373 
The author is greatly indebted to Prof. D. Coles for his helpful suggestions on 
the design of the electron beam and to Prof. B. Sturtevant for his valuable criti- 
cism of the manuscript. Thanks are also due to Prof. H. W. Liepmann for his 
encouragement and continued interest in the work. 
The research employed the Graduate Aeronautical Laboratories, California 
Institute of Technology 17 in. shock tube and auxiliary equipment, which were 
constructed under NASA Grant NsG 40-60. This work was carried out while 
the author was in receipt of a NASA International University Fellowship in 
the Space Sciences. 
R E F E R E N C E S  
BIRD, G.A. 1968 Proceedings of the Sixth International Symposium on Rarefied Gas 
CAMAC, M. 1965 Rarefied Gas Dynamics, I (Ed. de Leeuw). New York: Academic. 
LIEPMANN, H. W. & BOWMAN, R. M. 1964 Php. Fluids, 7, 2013-15. 
LIEPMANN, H. W., NARASIMHA, R. & CHAHINE, M. 1962 Phys. Fluids, 5, 1313-24. 
LINZER, M. & HORNIG, D. F. 1963 Phys. Fluids, 6, 1661-68. 
MATULA, R. A. 1961 ASME Paper 67-WAIHT-3. 
MUCKENFUSS, C. 1962 Phys. Fluids, 5, 1325-36. 
ROBBEN, F. & TALBOT, L. 1966 Phys. Fluids, 9, 633. 
RUSSELL, D. 1965 Rarefied Gas Dynamics, I (Ed. de Leeuw). New York: Academic. 
SCHULTZ-GRUXOW, F. & FROHN, A. 1965 Rarefied Gas Dynamice, I (Ed. de Leeuw). New 
Dynamics (in print). 
York: Academic. 


Electron beam density measurements in shock waves in argon

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Electron beam density measurements in shock waves in argon

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