R. & M. No. 3472
e,l
ee~
MINISTRY
OF TECHNOLOGY
AERONAUTICAL RESEARCH COUNCIL
'
REPORTS AND M E M O R A N D A
~0,
a
,".
" .
.
.
.
~
~.,'7"<
~.
Computer Calculations of Relaxation Regions and
Equilibrium Conditions for Shock Waves with
Tables for CO2 and N20
~r,
By T. Rees
L O N D O N : H E R MAJESTY'S S T A T I O N E R Y O F F I C E
1968
PRICE 5s. 6d. NET
Computer Calculations of Relaxation Regions and
Equilibrium Conditions for Shock Waves with
Tables for CO2 and N20
By T. Rees
Reports and Memoranda No. 3472*
December, 1965
Johannesen 2 has discussed a Rayleigh-line method for the analysis of vibrational relaxation regions
in shock waves. In this method the flow of the real gas in the relaxation region is treated as the flow
of an ideal gas (termed the ~-gas by Johannesen) with heat extraction. The Rayleigh-line method is readily
adapted for use on a computer; programs based on this method have been used to deal with experimental
data for C O 2 - H e mixtures and N 2 0 , and also to provide tables of equilibrium conditions for pure
C 0 2 and N 2 0 .
Conditions (a) immediately behind the diffusion resisted part of the wave are found using the normal
shock relations for the ideal gas and the known conditions (1) in front of the shock wave. The Mach
number m, of the a-gas is related to the Mach number Ma of the real gas by
rnl = MI (Tx/y~)~ •
(1)
The total temperature, To1, which does not vary through the diffusion-resisted part of the shock wave,
is given by,
T o , = TI(1 + ~ y = - 1)mi2).
(2)
"Fhe flow variables for the a-gas can be expressed in terms of the corresponding quantities at sonic conditions, i.e. at m = 1. This gives the well-known Rayleigh-line relations,
T
(V~+1)2 m 2
Tb -- (1 +v~m2) 2 '
TO 2(y~+ 1)m2 (1 + ~ y ~ - 1)m2)
To---~=
(1 + 7~m2)2
'
Pb _ v _ (y~+l)m z
p
Vb
p
Pb
*Replaces A.R.C. 27 581.
1 + v~m 2 '
~,+ 1
1 + v~m2 "
(3)
(4)
(5)
(6)
where suffix (b) denotes sonic conditions, and the symbols have their usual meanings. The conditions
(b) can be found from the known conditions (1) using the relations (3) to (6).
The increase in vibrational energy at any point in the relaxation region is equal to the energy that has
been extracted from the c~-gas, i.e.
~-~
= cp, ( T o - Tol)
(7)
where ¢r denotes the vibrational energy and ff denotes its local equilibrium value.
For the purpose of the computer program it is necessary to obtain an analytic expression for the equilibrium value of the vibrational energy. The simple harmonic oscillator approximation is suitable for
most gases. For example, for C02 and N20 we have
2c _
c2
c3
cp, -- cp, ( e x p ( C 1 / T ) - 1 ~ exp(C2/T)-1 ~ exp(Ca/T)-1
t
(8)
where x is the mole fraction of oscillators present and C1, C2, and C3 are the characteristic temperatures
of the various modes. For CO2 these are 959, 1920, and 3380 °K respectively, while for N20 the corresponding values are 847, 1850, and 3200 °K. It was felt that such approximations were sufficiently
accurate for evaluating experimental data, the difference between published value (Hilsenrath et aP,
McBride, Heimel, Ehlers and Gordon 4) of the vibrational energy and the values given by the harmonic
oscillator approximation increasing with temperature to 3 per cent at 5000 °K in the cases of COz and
NzO. However, a correction was applied in the preparation of the Tables of equilibrium conditions.
It was found that the difference could be expressed as a linear function of temperature, so that
~T = OHO (1 +aT+b).
(lO)
Here suffix T denotes values obtained from tables and HO those given by the harmonic oscillator expression. Values of a and b were found for CO2 and N20 in the interval 300 to 5000 °K by the method
of least squares, and these values are given with the corresponding tables of equilibrium conditions.
Where it is necessary to calculate the ratio of the specific heats of the real gas, the contribution, crib, of
the vibrational modes to the specific heats was obtained by differentiating equation (10) with respect to
T.
In the program m is decreased in steps of 0-01 from m,. At each value of m, Tand To are found from
(3) and (4) and hence 0 and r/from (10) and (7). This process is continued until a value of m, re(n), is reached
where ( 4 - q) becomes negative. A linear interpolation using re(n) and the previous value re(n- 1) gives
an approximate value of m2 for which q = ~/. Station (2) is the point at which equilibrium is reached.
Values of ~/and ~ corresponding to this value of m2 are calculated and a second interpolation, or extrapolation as the case may be, with re(n- 1) yields a better value for m(2). This is repeated until successive
values of m2 differ by less than 0.0001. Equilibrium values of the pressure, temperature, and density are
then obtained by substituting m2 into the Rayleigh-line relations. Examples of Tables prepared in this
way are given together with h flow diagram of the calculation. Also included in the Tables are values
of M* the equilibrium Mach number of the flow of the real gas produced by a shock moving into a gas
at rest. It should be noted that the method does not make any use of the relaxation equation.
The program was also used to calculate relaxation frequencies at points in the relaxation region from
experimental data using the method due to Johannesen, Zienkiewicz, Blythe and Gerrard a. The simple
relaxation equation
da
dt
P ~ ( ~ - ~)
(11)
2
was used, where • is the relaxation frequency. It was shown that • .may be written
tlp = vb Tb
" "A(p2-- p)
Pb cp~ l.l,m) pb(ff_ tr------~"
(12)
Here it is assumed that a logarithmic plot of the difference between final and local values of the. density
against distance, x, behind the shock-wave is a straight line, i.e.
(13)
log(p2-- p) = -- Ax d- B.
The function L(m) is defined by
L(m) = ~
(1-m2).
(14)
Thus, for a given?value of ml, all the quantities on the right-hand side of equation (12) can be found,
except for the value of A which is provided by experiment. Usually • was calculated for five values of
m in the interval m, to m2, and hence a plot of O versus local temperature was obtained instead of a single
overall value.
Acknowledgement.
The author is indebted to Professor N. H. Johannesen and Dr. H. K. Zienkiewicz for many helpful
discussions.
LIST OF SYMBOLS
MI
Mach number of the shock in the real gas
ml
Mach number of the shock in the a-gas
TI
Temperature in front of the shock
Pi
Pressure in front of the shock
Pi
Density in front of the shock
m2
Value of the equilibrium Mach number relative to the shock in the a-gas
M2
Equilibrium Mach number in the real gas of the flow produced by a shock wave moving into
a gas at rest
T2
Equilibrium temperature
P2
Equilibrium pressure
P2
Equilibrium density
REFERENCES
I
Hilsenrath et al
:.
Tables of thermal properties of gases.
National Bureau of Standards Circular 564. 1955.
2
N.H. Johannesen
....
Analysis of vibrational relaxation regions by means of the
Rayleigh-line method.
J. Fluid Mech., Vol. 10, pp. 25 to 32. 1961.
N. H. Johannesen. . . . .
Experimental and theoretical analysis of vibrational relaxation
H. K. Zienkiewicz, P. A. Blythe
regions in carbon dioxide.
and J. H. Gerrard
J. Fluid Mech., Vol. 13, pp. 213 to 224. 1962
B. J. McBride, S. Heimel,
..
J. G. Ehlers and S. Gordon
Thermodynamic properties to 6000 deg K for 210 substances
involving the first 18 elements.
NASA SP-3001. 1963.
Equilibrium Faluesfor Normal Shocks in
Carbon Dioxide for T1 = 295 °K.
a = 7.8 x 10-6(°K) -1
b = - 6-042 × 10- a
M1
ml
m2
T2
P2/Pl
P2/Pl
Pl/P2
M~
1.1
1.2
1.3
1.4
1-5
1.6
1-7
1.8
1-9
2.0
2"1
2.2
2.3
2.4
2.5
2.6
2"7
2"8
2.9
3.0
3.1
3.2
3.3
3.4
3-5
3.6
3.7
3"8
3'9
4.0
1.056
1.152
1.248
1.344
1.440
1-536
1.632
1.728
1.824
1.920
2.016
2.112
2.208
2"304
2.400
2.496
2.592
2"688
2"784
2-880
2"976
3.072
3.168
3.264
3.360
3.456
3.552
3.648
3.744
3.840
0.8726
0-8025
0-7455
0-6980
0.6579
0.6236
0.5939
0.5680
0.5453
0.5251
0-5072
0-4910
0.4765
0.4633
0.4514
0.4404
0.4304
0.4212
0.4127
0.4048
0.3975
0.3906
0.3843
0'3783
0.3728
0.3675
0.3626
0"3580
0.3536
0.3495
309-6
323.3
336.8
350.2
363.7
377.4
391.5
405-9
420.7
435.9
451.6
467.8
484.4
501-4
518.9
536.9
555"3
574"1
593"4
613"2
633"4
654"0
675.1
696.6
718.6
740.9
763.7
786.9
810.6
834'6
1.240
1-503
1-789
2.098
2-430
2.786
3.165
3.568
3.994
4.444
4-918
5.416
5.937
6.482
7.051
7"645
8.262
8"903
9"568
10.257
10.971
11.708
12.470
13.256
14.066
14"901
15.759
16.642
17.549
18.481
1.181
1.371
1.567
1.767
1.971
2.178
2.385
2.593
2.801
3.007
3.212
3.415
3.616
3.814
4.009
4.201
4"389
4.574
4.756
4.935
5.110
5.281
5.449
5.614
5.775
5.933
6.087
6.239
6.387
6.532
0.8465
0.7294
0.6383
0.5659
0.5073
0.4592
0.4193
0"3856
0.3570
0.3325
0.3113
0.2928
0.2765
0.2622
0.2494
0.2381
0.2278
0.2186
0.2102
0.2027
0"1957
0.1894
0.1835
0.1781
0.1732
0.1686
0.1643
0.1603
0.1566
0.1531
0.1653
0.3117
0.4431
0.5628
0.6727
0.7745
0.8692
0.9577
1.0408
1.1190
1"1928
1.2626
1.3288
1.3917
1.4515
1.5085
1"5629
1.6149
1"6646
1.7122
1.7578
1.8015
1.8435
1-8839
1"9228
1.9601
1.9962
2-0309
2.0644
2.0967
i
5
Equilibrium Valuesfor Normal Shocks in
Carbon Dioxide for T1 = 295 °K.
a = 7.8 x 10- 6(°K)- 1
b = - 6.042 x 10- 3
Mi
ml
m2
T2
P2/PI
P2/Pl
Pi/P2
M~
4"1
4.2
4"3
4.4
4.5
4.6
4.7
4-8
4.9
5.0
5-1
5.2
5.3
5-4
5.5
5.6
5.7
5.8
5-9
6-0
6.1
6.2
6"3
6.4
6-5
6.6
6-7
6.8
6-9
7.0
7.1
7.2
7.3
7.4
7.5
3"936
4-032
4.128
4.224
4.320
4"416
4.512
4.608
4.704
4.800
4-896
4.992
5.088
5.184
5.280
5.376
5.472
5.568
5.664
5.760
5.856
5.952
6.047
6.143
6.239
6"335
6.431
6.527
6.623
6.719
6.815
6.911
7-007
7-103
7-199
0"3456
0.3418
0-3383
0.3350
0.3318
0.3287
0.3258
0.3231
0-3204
0.3179
0-3155
0-3132
0-3110
0-3088
0.3068
0-3048
0.3029
0.3011
0-2994
0.2977
0.2961
0.2945
0.2930
0.2916
0.2902
0.2888
0-2875
0.2862
0.2850
0-2838
0.2827
0.2816
0.2805
0.2794
0-2784
859-1
884.0
909"3
935.0
961-2
987"8
1014-7
1042-1
1070.0
1098.2
1126.9
1155-9
1185.4
1215.4
1245.7
1276-5
1307.7
1339.3
1371.3
1403.8
1436.7
1470.1
1503.8
1538:0
1572.7
1607.8
1643.3
1679.2
1715.6
1752.4
1789.7
1827.4
1865.5
1904.1
1943.2
19.436
20.416
21.420
22"449
23.501
24.578
25.680
26.805
27.955
29-129
30"327
31.550
32.797
34.068
35-364
36.683
38-027
39-395
40-788
42.205
43.645
45.111
46.600
48.114
49.651
51.214
52.800
54.410
56.045
57.704
59.387
61.094
62.826
64-581
66.361
6.674
6.813
6.949
7.082
7.213
7"340
7.465
7.588
7.707
7.825
7"939
8.052
8.162
8.269
8.375
8.478
8.579
8.677
8.774
8.869
8.962
9.052
9.141
9.228
9.313
9.397
9.479
9-559
9.637
9"714
9-789
9.863
9.935
10.005
10.075
0"1498
0.1468
0"1439
0.1412
0"1386
0.1362
0-1340
0-1318
0.1297
0.1278
0.1260
0.1242
0.1225
0.1209
0.1194
0.1180
0.1166
0.1152
0.1140
0.1128
0"1116
0.1105
0.1094
0.1084
0.1074
0.1064
0.1055
0.1046
0.1038
0.1029
0.1022
0.1014
0.1007
0.0999
0.0993
2"1279
2.1581
2"1872
2.2154
2.2427
2.2690
2"2946
2.3194
2.3434
2.3666
2.3892
2.4110
2.4323
2.4529
2.4729
2.4923
2.5112
2.5295
2.5473
2.5646
2.5815
2-5978
2.6138
2.6293
2.6444
2-6591
2.6734
2-6873
2-7009
2.7141
2-7270
2-7395
2-7518
2-7637
2.7753
Equilibrium Values.for Normal Shocks in
Nitrous Oxide for T1 = 295 °K.
a = 15x 10 -6 (°K)-i
b = -6"493 x 10 -a
M1
ml
m2
T2
P2/Pl
1.1
1.2
1"3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2-7
2"8
2-9
3.0
3.1
3.2
3-3
3.4
3-5
3.6
3.7
3.8
3"9
4.0
1.050
1.146
1.241
1"336
1.432
1.527
1"623
1.718
1.814
1.909
2.005
2-100
2.196
2-291
2.387
2.482
2.577
2.673
2.768
2.864
2.959
3.055
3.150
3.246
3-341
3.437
3.532
3.628
3.723
3.819
0"8680
0.7984
0.7416
0.6943
0.6543
0"6201
0"5905
0-5646
0.5419
0"5218
0.5039
0.4878
0.4733
0.4602
0.4482
0.4373
0.4273
0-4181
0-4096
0.4018
0.3945
0-3877
0-3814
0.3755
0.3700
0.3648
0-3599
0.3553
0.3509
0.3468
308-9
322-1
335.0
347.8
360.7
373"9
387.4
401.3
415.6
430.3
445.4
461.0
477.0
493.5
510-4
527.8
545.7
564.0
582-7
601-9
621.6
641.6
662.1
683"0
704.4
726.2
748.4
771"0
794.0
817"4
1.238
1-499
1.783
2.090
2.420
2.773
3.150
3.549
3.972
4.419
4.889
5.382
5.899
6.440
7.004
7.592
8.204
8.839
9.498
10.181
10.888
11.619
12.374
13.152
13:955
14.781
15.632
16.506
17.404
18.327
P2/Pl
1.182
1"373
1.570
1.773
1.979
2.188
2.398
2.609
2.820
3.030
3.238
3.444
3.648
3.849
4.048
4.243
4.435
4.623
4.808
4.990
5.168
5.342
5.513
5.680
5.844
6-005
6.162
6.316
6.466
6-614
Pl/P2
M~
0.8458
0-7282
0.6367
0.5640
0.5053
0.4571
0.4170
0.3833
0.3546
0.3301
0.3088
0.2903
0.2741
0.2598
0.2470
0-2357
0.2255
0.2163
0.2080
0-2004
0-1935
0.1872
0-1814
0.1760
0.1711
0.1665
0.1623
0.1583
0.1546
0.1512
0.1661
0"3134
0.4458
0"5664
0.6773
0.7799
0.8755
0.9648
1.0487
1.1277
1-2022
1-2728
1.3396
1.4032
1.4636
1.5211
1.5761
1.6285
1.6787
1-7267
1.7727
1.8169
1.8593
1-9000
1.9392
1.9769
2"0132
2"0482
2.0820
2"1145
Equilibrium Valuesfor Normal Shocks in
Nitrous Oxide for T1 = 295 °K.
a = 15x 10 -6 ( ° K ) - I
b = -6-493 x 10 -3
M1
ml
m2
T2
P2/Pl
P2/Pi
PI/P2
M~
4.1
4.2
4.3
4.4
4.5
4"6
4.7
4.8
4.9
5.0
5"1
5.2
5"3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6"8
6.9
7.0
7.1
7.2
7.3
7"4
7.5
3.914
4.009
4.105
4.200
4.296
4.391
4.487
4"582
4.678
4.773
4.869
4.964
5.060
5.155
5.250
5-346
5.441
5"537
5.632
5.728
5.823
5.919
6-014
6.110
6.205
6-301
6.396
6.491
6.587
6-682
6.778
6.873
6"969
7"064
7-160
0.3429
0.3393
0.3358
0.3324
0.3293
0.3263
0.3234
0.3207
0-3180
0.3155
0.3131
0"3108
0.3086
0.3065
0.3045
0.3026
0.3007
0.2989
0.2972
0.2955
0.2939
0.2923
0-2909
0.2894
0.2880
0.2867
0.2854
0.2841
0.2829
0.28~17
0.2806
0.2795
0.2784
0.2774
0.2764
841-3
865.6
890.3
915-4
940.9
966-8
993.1
1019.9
1047.1
1074.6
1102.6
1131.0
1159.8
1189.0
1218.7
1248.7
1279.2
1310.1
1341.4
1373.1
1405.2
1437.8
1470.7
1504.1
1537.9
1572.2
1606.8
1641.9
1677.4
1713.4
1749.7
1786.5
1823.7
1861.4
1899.4
19.273
20.244
21.238
22.256
23.299
24.365
25-456
26.570
27"709
28.871
30.058
31.269
32.503
33.762
35.045
36"351
37.682
39"036
40.415
41.818
43.245
44.695
46.170
47.668
49.191
50.738
52.308
53.903
55.522
57.164
58-831
60.521
62.236
63.974
65.736
6.758
6.899
7-037
7.173
7.305
7.435
7.561
7.685
7-807
7.926
8.042
8.156
8.267
8.376
8.483
8-588
8.690
8-790
8.888
8.984
9-078
9.171
9-261
9.349
9.436
9.520
9.603
9.685
9-764
9.842
9"919
9.994
10.067
10.139
10.210
0"1480
0.1449
0.1421
0.1394
0-1369
0.1345
0-1323
0.1301
0.1281
0.1262
0.1243
0-1226
0.1210
0.1194
0.1179
0.1164
0.1151
0-1138
0.1125
0.1113
0.1102
0"1090
0.1080
0.1070
0.1060
0.1050
0.1041
0.1033
0.1024
0.1016
0.1008
0.1001
0"0993
0"0986
0.0979
2.1460
2.1764
2.2057
2.2341
2.2616
2.2882
2.3139
2"3388
2.3630
2.3864
2.4091
2.4311
2.4525
2.4733
2.4934
2.5130
2.5320
2-5504
2.5684
2.5859
2.6028
2.6194
2.6354
2.6511
2.6663
2.6812
2.6956
2.7097
2-7234
2-7368
2.7498
2.7626
2.7750
2-7871
2.7989
Flow Diagramfor Calculation of Equilibrium Values.
IREAD GAS PROPERTIESI
+
l~
°,I
Ic,~cu,A~ co,o,,,o,, ~.,o,.,,o oo. ~o, °= oo I
+
,IDECREASE m BY O'OI I, "
t
CALCULATE T, To~AND HENCE 71 AND ~ CORRESPONDING'
TO m
t
ITEST SIGNOF(n-
+ VEI
I, IF NEGAT,
fi)l
)
,F POSITtVEI
INTERPOLATE TO GIVE A FIRST VALUE
OF m2
INTERPOLATE USING VALUES OF ~ AND ~ CORRESPONDING
TO m2 AND THE PREVIOUS VALUEOF m TO GIVE A
BETTER ESTIMATE FOR m2
IF SUCCESSIVE VALUES OF
m2 DIFFER BY LESS THAN
0"0001
+
IF SUCCESSIVE VALUES OF
m2 DIFFER BY MORE THAN"
0"0001
CALCULATE EQUILIBRIUM PROPERTIES BY SUBSTITUTION
OF m2 INTO THE RAYLEIGH - LINE RELATIONS
Printed in Wales for Her Majesty's Stationery Office by Allens Printers (Wales) Limited.
Dd.129527
K5.
•
R. & M. No. 3472
@) Crown copyright 1968
Published by
HER MAJESTY'S STATIONERYOFFICE
To be purchased from
49 High Holborn, London w.c.l
423 Oxford Street, London w.l
13A Ca.4tle Street, Edinburgh 2
109 St. Mar}¢ Street, CardiffCF1 IJW
Brazennose Street, Manchester 2
50 Fairfax Street, Bristol 1
258 259 Broad Street, Birmingham 1
7-11 Linenhall Street, Belfast BT2 SAY
or through any bookseller
R° & M. No. 3472
S.O. C o d e N o . 2 3 - 3 4 7 2