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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

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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

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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-)

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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

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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

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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

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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)

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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

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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

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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

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Fanno flow

Page 12

1.4

2.4

0.4

0.7143

0.8571

6.0000

0.1667

0.8215

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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

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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

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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

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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

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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