Appendix N Compressible Flow Tables for an Ideal Gas with k

639

APPENDIX

Volume and Area Formulas:

hxx xx

rr

y

b

h

3

b

h

2

A = bh

2

Ixx = bh3

36

Ixx = r4

8

Ixx = 0.110r4

A =

π

r2

r

2

3

4

π

FIGURE A.1

Centroids and moments

of inertia of plane areas

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640 APPENDIX

TABLE A.1 Compressible Flow Tables for an Ideal Gas with k  1.4

M or M1  local number or Mach number upstream of a normal shock wave; p/pt  ratio of static

pressure to total pressure; /t  ratio of static density to total density; T/Tt  ratio of static

temperature to total temperature; A/A*  ratio of local cross-sectional area of an isentropic stream

tube to cross-sectional area at the point where M  1; M2  Mach number downstream of a normal

shock wave; p2/p1  static pressure ratio across a normal shock wave; T2/T1  static pressure ratio

across a normal shock wave; pt2/pt1  total pressure ratio across normal shock wave.

Subsonic Flow

Mp/pt/tT/TtA/A*

0.00 1.0000 1.0000 1.0000 

0.05 0.9983 0.9988 0.9995 11.5914

0.10 0.9930 0.9950 0.9980 5.8218

0.15 0.9844 0.9888 0.9955 3.9103

0.50 0.8430 0.8852 0.9524 1.3398

0.52 0.8317 0.8766 0.9487 1.3034

0.54 0.8201 0.8679 0.9449 1.2703

0.56 0.8082 0.8589 0.9410 1.2403

0.58 0.7962 0.8498 0.9370 1.2130

0.60 0.7840 0.8405 0.9328 1.1882

0.62 0.7716 0.8310 0.9286 1.1657

0.64 0.7591 0.8213 0.9243 1.1452

0.82 0.6430 0.7295 0.8815 1.0305

0.84 0.6300 0.7189 0.8763 1.0237

0.86 0.6170 0.7083 0.8711 1.0179

0.88 0.6041 0.6977 0.8659 1.0129

0.90 0.5913 0.6870 0.8606 1.0089

(Continued)

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APPENDIX 641

TABLE A.1 Compressible Flow Tables for an Ideal Gas with k  1.4 (Continued)

Supersonic Flow Normal Shock Wave

M1p/pt/tT/TtA/A*M2p2/p1T2/T1pt2/pt1

1.00 0.5283 0.6339 0.8333 1.000 1.0000 1.000 1.000 1.0000

1.01 0.5221 0.6287 0.8306 1.000 0.9901 1.023 1.007 0.9999

1.02 0.5160 0.6234 0.8278 1.000 0.9805 1.047 1.013 0.9999

1.13 0.4511 0.5663 0.7966 1.013 0.8892 1.323 1.084 0.9978

1.14 0.4455 0.5612 0.7937 1.015 0.8820 1.350 1.090 0.9973

1.15 0.4398 0.5562 0.7908 1.017 0.8750 1.376 1.097 0.9967

1.16 0.4343 0.5511 0.7879 1.020 0.8682 1.403 1.103 0.9961

1.17 0.4287 0.5461 0.7851 1.022 0.8615 1.430 1.109 0.9953

1.18 0.4232 0.5411 0.7822 1.025 0.8549 1.458 1.115 0.9946

1.19 0.4178 0.5361 0.7793 1.026 0.8485 1.485 1.122 0.9937

1.20 0.4124 0.5311 0.7764 1.030 0.8422 1.513 1.128 0.9928

1.21 0.4070 0.5262 0.7735 1.033 0.8360 1.541 1.134 0.9918

1.40 0.3142 0.4374 0.7184 1.115 0.7397 2.120 1.255 0.9582

1.45 0.2927 0.4158 0.7040 1.144 0.7196 2.286 1.287 0.9448

1.50 0.2724 0.3950 0.6897 1.176 0.7011 2.458 1.320 0.9278

1.55 0.2533 0.3750 0.6754 1.212 0.6841 2.636 1.354 0.9132

1.60 0.2353 0.3557 0.6614 1.250 0.6684 2.820 1.388 0.8952

(Continued)

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642 APPENDIX

1.90 0.1492 0.2570 0.5807 1.555 0.5956 4.045 1.608 0.7674

1.95 0.1381 0.2432 0.5680 1.619 0.5862 4.270 1.647 0.7442

2.00 0.1278 0.2300 0.5556 1.688 0.5774 4.500 1.688 0.7209

2.80 0.368510.946310.3894 3.500 0.4882 8.980 2.451 0.3895

2.90 0.316510.848910.3729 3.850 0.4814 9.645 2.563 0.3577

3.00 0.272210.762310.3571 4.235 0.4752 10.330 2.679 0.3283

3.50 0.131110.452310.2899 6.790 0.4512 14.130 3.315 0.2129

6.50 0.385520.364320.1058 75.13 0.4004 49.130 9.156 0.21151

7.00 0.241630.260920.92591104.1 0.3974 57.000 10.47 0.15351

7.50 0.155430.190420.81631141.8 0.3949 65.460 11.88 0.11331

†xn means x  10n.

TABLE A.1 Compressible Flow Tables for an Ideal Gas with k  1.4 (Continued)

Supersonic Flow Normal Shock Wave

M1p/pt/tT/TtA/A*M2p2/p1T2/T1pt2/pt1

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APPENDIX 643

2

3

4

6

8

1 × 10–3

1 × 10–2

2

3

4

6

8

SAE 20W

SAE 10W-30

Castor

oil

1 × 10–2

8

1 × 10–1

2

3

6

8

4

2

3

4

120100806040200

Temperature, °C

Glycerine

FIGURE A.2

Absolute viscosities of

certain gases and liquids

[Adapted from Fluid

Mechanics, 5th ed., by

1971, McGraw-Hill Book

Company, New York.

Used with permission of

the McGraw-Hill Book

Company.]

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644 APPENDIX

0 50 100 150 200 250

1 × 10–6

2

3

4

6

8

1 × 10–5

1 × 10–7

1 × 10–6

1 × 10–2

6

8

Temperature, °F

Kinematic viscosity, ft2/s

2

3

6

8

4

6

8

4

120100806040200

Kinematic viscosit

Temperature, °C

Glycerine

Benzene

Ethyl alcohol

Carbon tetrachloride

Gasoline (S = 0.68)

Mercury

FIGURE A.3

Kinematic viscosities of

certain gases and liquids.

The gases are at standard

pressure. [Adapted from

APPENDIX 645

TABLE A.2 Physical Properties of Gases [T  15C (59F), p  1 atm]

Gas

Density

kg/m3

(slugs/ 3)

Kinematic

Viscosity

m2/s ( 2/s)

R

Gas

Constant

J/kg K

( -lbf/slug-ⴗR)

cp

J

k

g

K

a

Btu

lbm-°R

b

kⴝ

c

p

cv

S

Sutherland’s

Constant

K(ⴗR)

Air 1.22

(0.00237) 1.46  105

(1.58  104)287

(1716) 1004

(0.240) 1.40 111

(199)

Carbon dioxide 1.85

(0.0036) 7.84  106

(8.48  105)189

(1130) 841

(0.201) 1.30 222

(400)

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646 APPENDIX

TABLE A.3 Mechanical Properties of Air at Standard Atmospheric Pressure

Temperature Density Speciﬁ c Weight Dynamic Viscosity Kinematic Viscosity

kg/m3N/m3N ⴢ s/m2m2/s

20C1.40 13.70 1.61  1051.16  105

10C1.34 13.20 1.67  1051.24  105

0C1.29 12.70 1.72  1051.33  105

80C1.00 9.81 2.09  1052.09  105

90C0.97 9.54 2.13  1052.19  105

100C0.95 9.28 2.17  1052.29  105

120C0.90 8.82 2.26  1052.51  105

slugs/ 3lbf/ 3lbf-s/ 2 2/s

0F0.00269 0.0866 3.39  1071.26  104

20F0.00257 0.0828 3.51  1071.37  104

40F0.00247 0.0794 3.63  1071.47  104

60F0.00237 0.0764 3.74  1071.58  104

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APPENDIX 647

TABLE A.4 Approximate Physical Properties of Common Liquids at Atmospheric Pressure

Liquid and

Temperature

Density

kg/m3

(slugs/ 3)Speciﬁ c

Gravity

Speciﬁ c

Wei ght

N/m3 (lbf/ 3)

Dynamic

Viscosity

N ⴢ s/m2 (lbf-s/ 2)

Kinematic

Viscosity

m2/s ( 2/s)

Surface

Te n si on

N/m* (lbf/ )

Ethyl alcohol(1)(3)

20C (68F) 799

(1.55) 0.79 7,850

(50.0) 1.2  103

(2.5  105) 1.5  106

(1.6  105)2.2  102

(1.5  103)

Carbon tetrachloride(3)

20C (68F) 1,590

(3.09) 1.59 15,600

(99.5) 9.6  104

(2.0  105) 6.0  107

(6.5  106)2.6  102

(1.8  103)

Glycerine(3)

20C (68F) 1,260

(2.45) 1.26 12,300

(78.5) 1.41

(2.95  102)1.12  103

(1.22  102)6.3  102

(4.3  103)

Kerosene(1)(2)

20C (68F) 814

(1.58) 0.81 8,010

(51) 1.9  103

(4.0  105)2.37  106

(2.55  105)2.9  102

(2.0  103)

*Liquid–air surface tension values.

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648 APPENDIX

TABLE A.5 Approximate Physical Properties of Water* at Atmospheric Pressure

Temperature Density Speciﬁ c Weight Dynamic

Viscosity Kinematic

Viscosity Vapor Pressure

kg/m3N/m3N ⴢ s/m2m2/s N/m2 abs

0C1000 9810 1.79  1031.79  106611

5C1000 9810 1.51  1031.51  106872

40C992 9732 6.53  1046.58  1077,380

50C988 9693 5.47  1045.53  10712,300

60C983 9643 4.66  1044.74  10720,000

70C978 9594 4.04  1044.13  10731,200

slugs/ 3lbf/ 3lbf-s/ 2 2/s psia

40F1.94 62.43 3.23  1051.66  1050.122

50F1.94 62.40 2.73  1051.41  1050.178

60F1.94 62.37 2.36  1051.22  1050.256

70F1.94 62.30 2.05  1051.06  1050.363

80F1.93 62.22 1.80  1050.930  1050.506

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APPENDIX 649

TABLE A.6 Nomenclature

Symbol Dimensions Description Symbol Dimensions Description

A L2 Area

Aj L2 Jet area

B L Linear measure

B . . . Extensive property

b L Linear measure; wing span

CT . . .  rust coeﬃ cient

Cv . . . Coeﬃ cient of velocity

c L/T Speed of sound

cf . . . Local shear stress coeﬃ cient

cp L2/T2 Speciﬁ c heat at constant pressure

d L Depth

E ML2/T2 Energy

E L Speciﬁ c energy

Ev M/LT 2 Elasticity, bulk

e L2/T2 Energy per unit mass

Fr . . . Froude number

F, F ML/T2 Force

FD ML/T2 Drag force

FL ML/T2 Li force

H L Head

h L Height

h L Piezometric head

k . . . Ratio of speciﬁ c heats

ks L Equivalent sand roughness

L L Linear measure

l L Linear measure

/

L Linear measure

n . . . Manning’s roughness coeﬃ cient

n T1 Rotational speed

ns . . . Speciﬁ c speed

nss . . . Suction speciﬁ c speed

p M/LT 2 Pressure

p M/LT 2 Change in pressure

P ML2/T3 Power

(Continued)

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650 APPENDIX

p* M/LT2 Pressure at M  1

pt M/LT2 Total pressure

pv M/LT 2 Vapor pressure

pz M/LT 2 Piezometric pressure

Re . . . Reynolds number

r L Linear measure in radial direction

S L2 Planform area

St . . . Strouhal number

S0 . . . Channel slope

U0 L/T Free-stream velocity

u L/T Velocity component, x direction

u L2/T2 Internal energy per unit of mass

u* L/T Shear velocity

u L/T Velocity ﬂ uctuation in x direction

un . . . Unit vector, normal direction

v L/T Velocity ﬂ uctuation in y direction

W ML2/T2 Work

W ML/T2 Weight

We . . . Weber number

Greek Letters

 . . . Angular measure

 . . . Lapse rate

 . . . Kinetic energy correction factor

 . . . Eﬃ ciency

 . . . Angular measure

 . . . Turbulence constant

 . . . Aspect ratio of a wing

 M/LT Dynamic viscosity

 M/LT2 Shear stress

TABLE A.6 Nomenclature (Continued)

Symbol Dimensions Description Symbol Dimensions Description

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