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8/8/2019 Gas Dynamics Solver
1/18
isentropic flow
Page 1
1.4isentropic flow calculations
Given M, find the property ratio
Given:
0.9400 0.5658 0.8498 0.6658 1.0031
Given the property ratio, find M
Given: 0.4919 0.8278 0.6658 1.5069
M 1.0600 1.0199 0.9399 0.4278 M1
b 3.0000
subsonic solution supersonic solution
Iteration F(M) dF/dM Iteration
1 0.2000 -0.2913 1.3658 0.4133 1 4.0000
2 0.4133 -0.0173 1.1999 0.4277 2 3.2242
3 0.4277 -0.0001 1.1878 0.4278 3 2.61394 0.4278 0.0000 1.1877 0.4278 4 2.1830
5 0.4278 0.0000 1.1877 0.4278 5 1.9440
6 0.4278 0.0000 1.1877 0.4278 6 1.8675
7 0.4278 0.0000 1.1877 0.4278 7 1.8601
8 0.4278 0.0000 1.1877 0.4278 8 1.8600
9 0.4278 0.0000 1.1877 0.4278 9 1.8600
10 0.4278 0.0000 1.1877 0.4278 10 1.8600
p/p0
T/T0
/0 A/A*
p/p0
T/T0
/0 A/A*
Mold
Mnew
Mold
8/8/2019 Gas Dynamics Solver
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isentropic flow
Page 2
c 0.8333
F(M) dF/dM
-36.8474 -47.4931 3.2242
-12.0342 -19.7198 2.6139
-3.7306 -8.6587 2.1830-1.0221 -4.2762 1.9440
-0.2032 -2.6552 1.8675
-0.0165 -2.2300 1.8601
-0.0001 -2.1910 1.8600
0.0000 -2.1906 1.8600
0.0000 -2.1906 1.8600
0.0000 -2.1906 1.8600
Mnew
8/8/2019 Gas Dynamics Solver
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normal shockwaves
Page 3
1.4 0.17normal-shockwave calculations
Given M, find the property ratio
Given:
3.0000 10.3333 3.8571 2.6790 0.3283 12.0610 0.4752
Given the property ratio, find M
Given: 1.3495 3.0000 2.0000 0.8000 N/A 0.6000
M 1.1400 2.2361 2.3574 1.8284 N/A 1.8778
0.01n
1 3.0000 3.0100 2.9900 -0.4717 -0.4745 -0.4688
2 1.3301 1.3401 1.3201 0.1738 0.1718 0.1757
3 2.2136 2.2236 2.2036 -0.1780 -0.1826 -0.1735
4 1.8212 1.8312 1.8112 0.0032 -0.0013 0.00775 1.8284 1.8384 1.8184 0.0000 -0.0045 0.0045
6 1.8284 1.8384 1.8184 0.0000 -0.0045 0.0045
7 1.8284 1.8384 1.8184 0.0000 -0.0045 0.0045
8 1.8284 1.8384 1.8184 0.0000 -0.0045 0.0045
9 1.8284 1.8384 1.8184 0.0000 -0.0045 0.0045
10 1.8284 1.8384 1.8184 0.0000 -0.0045 0.0045
p2/p
1
2/
1T
2/T
1p
02/p
01p
02/p
1M
2
p2/p
1
2/
1T
2/T
1p
02/p
01p
02/p
1M
2
(M1)old
M1+ M
1- F(M
1) F(M
1+) F(M
1-)
8/8/2019 Gas Dynamics Solver
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normal shockwaves
Page 4
-0.2824 1.3301
-0.1967 2.2136
-0.4537 1.8212
-0.4484 1.8284-0.4498 1.8284
-0.4498 1.8284
-0.4498 1.8284
-0.4498 1.8284
-0.4498 1.8284
-0.4498 1.8284
dF/dM1
(M1)new
8/8/2019 Gas Dynamics Solver
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oblique shockwaves
Page 5
1.4 2.4 0.4
Weak-Shock Solution
1.4000 2.0000 30.0000 Err:502 Err:502 Err:502 Err:502
0.5236 Err:502
radians
Strong-Shock Solution
1.4000 2.0000 30.0000 Err:502 Err:502 Err:502 Err:502
0.5236 Err:502
radians
1.4000 2.0000 45.0000 14.7436 2.1667 1.7143 1.2639
radians 0.7854 0.2573
1.0000
1.0000
1.4000 10.0000 20.0000 4.7347 2.8927 2.0642 1.4014
0.1745 0.3491 radians
1.2000 2.7475
Collar's MethodA B C B-AC
3.0000 11.0851 3.3486 1.0392 64.67 22.97
Iteration
1 1.7321 0.9045 -0.2095
2 0.9045 0.6274 0.1233
3 0.6274 0.4605 0.3030
4 0.4605 0.2997 0.4636
5 0.2997 Err:502 Err:502
6 Err:502 Err:502 Err:502
7 Err:502 Err:502 Err:502
8 Err:502 Err:502 Err:502
9 Err:502 Err:502 Err:502
10 Err:502 Err:502 Err:502
11 Err:502 Err:502 Err:502
12 Err:502 Err:502 Err:502
13 Err:502 Err:502 Err:502
14 Err:502 Err:502 Err:502
15 Err:502 Err:502 Err:502
+1 -1Option One: M
1and are given
M1 (deg) (deg) p2/p1 2/1 T2/T1
cot
M1 (deg) (deg) p2/p1 2/1 T2/T1
cot
Option Two: M1
and are givenM
1 (deg) (deg) p2/p1 2/1 T2/T1
cot ()
M12sin2-1
Option Three: and are given (deg) (deg) M1 p2/p1 2/1 T2/T1
(+1)/2cot ()
max
(deg) max
(deg)
xold
xnew
xII
8/8/2019 Gas Dynamics Solver
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oblique shockwaves
Page 6
16 Err:502 Err:502 Err:502
17 Err:502 Err:502 Err:502
18 Err:502 Err:502 Err:502
19 Err:502 Err:502 Err:502
20 Err:502 Err:502 Err:502
8/8/2019 Gas Dynamics Solver
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oblique shockwaves
Page 7
Err:502 Err:502
Err:502 Err:502
0.9546 1.4563
0.8879 3.8163
61.49 22.71
p02/p
01M
2
p02/p
01M
2
p02/p
01M
2
p02/p
01M
2
sonic
(deg) sonic
(deg)
8/8/2019 Gas Dynamics Solver
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oblique shockwaves
Page 8
8/8/2019 Gas Dynamics Solver
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Prandtl_Meyer flow
Page 9
Prandtl-Meyer values
M
4.0000 1.1482 65.7848 0.2527 14.4775
1.4 30.0000 degrees
6 0.1667 0.01
Iteration F(M)
1 4.0000 0.6246 0.6269 0.6223 0.2305 1.2908
2 1.2908 -0.4203 -0.4155 -0.4250 0.4742 2.1770
3 2.1770 0.0198 0.0244 0.0152 0.4560 2.1336
4 2.1336 -0.0001 0.0045 -0.0048 0.4624 2.1339
5 2.1339 0.0000 0.0046 -0.0046 0.4623 2.1339
6 2.1339 0.0000 0.0046 -0.0046 0.4623 2.1339
7 2.1339 0.0000 0.0046 -0.0046 0.4623 2.1339
8 2.1339 0.0000 0.0046 -0.0046 0.4623 2.13399 2.1339 0.0000 0.0046 -0.0046 0.4623 2.1339
10 2.1339 0.0000 0.0046 -0.0046 0.4623 2.1339
(rad) (deg) (rad) (deg)Inputs: (deg) and
+1/-1 -1/+1 MUses the Newton-Raphson method to find M for a given value of
Mold F(M+M) F(M-M) F/M Mnew
8/8/2019 Gas Dynamics Solver
10/18
Fanno flow
Page 10
Fanno flow calculations Given M, find the property ratio
Given:
0.5000 2.1381 1.1429 1.8708 1.3398 1.0691
Given the property ratio, find M
Given: 2.1381 1.1000 2.0000 20.0000 0.8000
M 0.5000 0.6742 0.4663 0.0289 0.5378 M1
M1 Solution
Iteration F(M)
1 4.0000 37.1250 -29.0000 5.2802
2 5.2802 -58.9665 -138.5678 4.8546
3 4.8546 -10.8418 -90.0515 4.7342
4 4.7342 -0.6860 -78.8234 4.7255
5 4.7255 -0.0034 -78.0501 4.7255
6 4.7255 0.0000 -78.0463 4.7255
7 4.7255 0.0000 -78.0463 4.7255
8 4.7255 0.0000 -78.0463 4.7255
9 4.7255 0.0000 -78.0463 4.7255
10 4.7255 0.0000 -78.0463 4.7255
M1, the largest fLmax
/D is
([(+1)/2]ln[(+1)/(-1)]-1/p
0/p
0* calculation by Newton-Raphson method
Mold dF/dM Mnew
Mold dF/dM Mnew
fLmax
/D calculation by Newton-Raphson method
Mold dF/dM Mnew
8/8/2019 Gas Dynamics Solver
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Fanno flow
Page 11
6 0.5367 0.0066 -6.2184 0.5378
7 0.5378 0.0000 -6.1704 0.5378
8 0.5378 0.0000 -6.1702 0.5378
9 0.5378 0.0000 -6.1702 0.5378
10 0.5378 0.0000 -6.1702 0.5378
M>1 SolutionIteration F(M)
1 4.0000 -0.1669 0.0797 6.0940
2 6.0940 -0.0675 0.0271 8.5887
3 8.5887 -0.0250 0.0104 10.9913
4 10.9913 -0.0073 0.0051 12.4241
5 12.4241 -0.0012 0.0036 12.7558
6 12.7558 0.0000 0.0033 12.7693
7 12.7693 0.0000 0.0033 12.7693
8 12.7693 0.0000 0.0033 12.7693
9 12.7693 0.0000 0.0033 12.7693
10 12.7693 0.0000 0.0033 12.7693
Mold dF/dM Mnew
8/8/2019 Gas Dynamics Solver
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Fanno flow
Page 12
1.4
2.4
0.4
0.7143
0.8571
6.0000
0.1667
0.8215
8/8/2019 Gas Dynamics Solver
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Fanno flow
Page 13
8/8/2019 Gas Dynamics Solver
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shock tube
Page 14
1.4 1.4 1
0.14286 50
0.85714
7.0 0.01
f df/dp2.00000 21.03105 20.87650 21.18696 -15.52281 3.35485
3.35485 7.72947 7.66697 7.79230 -6.26662 4.58828
4.58828 1.84580 1.80926 1.88246 -3.65990 5.09261
5.09261 0.15297 0.12219 0.18385 -3.08291 5.14223
5.14223 0.00119 -0.02911 0.03159 -3.03503 5.14263
5.14263 0.00000 -0.03030 0.03039 -3.03466 5.14263
5.14263 0.00000 -0.03030 0.03039 -3.03466 5.14263
5.14263 0.00000 -0.03030 0.03039 -3.03466 5.14263
5.14263 0.00000 -0.03030 0.03039 -3.03466 5.14263
5.14263 0.00000 -0.03030 0.03039 -3.03466 5.14263
5.14263 0.00000 -0.03030 0.03039 -3.03466 5.14263
5.14263 0.00000 -0.03030 0.03039 -3.03466 5.142635.14263 0.00000 -0.03030 0.03039 -3.03466 5.14263
5.14263 0.00000 -0.03030 0.03039 -3.03466 5.14263
5.14263 0.00000 -0.03030 0.03039 -3.03466 5.14263
1
4 a1/a4c
1p
4/p
1
c2
c3 p
pold f(p+p) f(p-p) pnew
8/8/2019 Gas Dynamics Solver
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Rayleigh flow
Page 15
Rayleigh flow calculations
Given M, find the property ratio Given:
0.1000 2.3669 0.0560 42.2500 0.0468 1.2591
0.5000 1.7778 0.7901 2.2500 0.6914 1.1141 1/2.0000 0.3636 0.5289 0.6875 0.7934 1.5031
Given the property ratio, find M
Given: 2.3669 0.0560 42.2500 0.7934 1.1141
M 0.1000 0.1000 0.1000 0.5774 0.4999 Subsonic
7.1429 1.9999 1.4845 Supersonic
1.1141
M1
Iteration F(M) dF/dM
1 3.0000 1.5369 1.6830 2.0868
2 2.0868 0.3679 0.8757 1.6667
3 1.6667 0.0779 0.5061 1.5128
4 1.5128 0.0102 0.3739 1.4854
5 1.4854 0.0003 0.3507 1.48456 1.4845 0.0000 0.3500 1.4845 Answer
p/p* T/T* /* T0/T0* p0/p0* (+1)(1)
(+1)/(2)(+1)/(1)
p/p* T/T* /* T0/T0* p0/p0* (1)/(+1)()/(31)
p0/p
0*
Mold
Mnew
Mold
Mnew
8/8/2019 Gas Dynamics Solver
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Rayleigh flow
Page 16
1.4
2.4
0.4
0.7143
0.8571
6.0000
0.1667
0.4375
0.845154
Mcritical
8/8/2019 Gas Dynamics Solver
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Isothermal flow
Page 17
Isothermal flow calculation Given M, find the property ratio
Given:
2.0000 0.4226 1.5750 0.4226 2.0720 0.9013 1/0.4880 1.7319 0.9167 1.7319 1.2772 0.9010
Given the property ratio, find M
Given: 0.4226 1.5750 0.4226 2.0720 0.9013
M 2.0000 2.0000 2.0000 0.2688 0.4880 Subcritical
2.0000 2.0000 Supercritical
2.0720
Subcritical
Iteration F(M) dF/dM
1 0.2000 0.1514 -2.5037 0.26052 0.2605 0.0164 -1.9973 0.2687
3 0.2687 0.0002 -1.9426 0.2688
4 0.2688 0.0000 -1.9418 0.2688
5 0.2688 0.0000 -1.9418 0.2688 Answer
Supercritical
Iteration F(M) dF/dM
1 3.0000 1.5578 2.0150 2.2269
2 2.2269 0.2714 1.3052 2.0190
3 2.0190 0.0208 1.1044 2.0002
4 2.0002 0.0002 1.0859 2.0000
5 2.0000 0.0000 1.0857 2.0000
6 2.0000 0.0000 1.0857 2.0000 Answer
0.9013
Subcritical
Iteration F(M) dF/dM
1 0.2000 -0.7321 2.1189 0.5455
2 0.5455 0.1569 2.7141 0.4877
3 0.4877 -0.0007 2.7325 0.4880
4 0.4880 0.0000 2.7325 0.4880
5 0.4880 0.0000 2.7325 0.4880 Answer
SupercriticalIteration F(M) dF/dM
1 4.0000 -28.0530 -24.7265 2.8655
2 2.8655 -7.2143 -12.3607 2.2818
3 2.2818 -1.6202 -6.9327 2.0481
4 2.0481 -0.2305 -4.9833 2.0019
5 2.0019 -0.0086 -4.6147 2.0000
6 2.0000 0.0000 -4.6000 2.0000 Answer
(+1)p/p* T0/T0
* /* p0/p0* fLmax/D (1)(+1)/(2)
(+1)/(1)p/p* T0/T0
* /* p0/p0* fLmax/D (1)/(+1)()/(31)
p0/p
0*
Mold
Mnew
Mold
Mnew
fLmax
/D
Mold
Mnew
Mold
Mnew
8/8/2019 Gas Dynamics Solver
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Isothermal flow
Page 18
1.4
2.4
0.4
0.7143
0.8571
6.0000
0.1667
0.4375
0.845154
Mcritical
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