Stagnation Properties and Mach...
Transcript of Stagnation Properties and Mach...
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AE3450Stag. Props. and Mach Number -1
School of Aerospace Engineering
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
Stagnation Properties and Mach Number• Rewrite stagnation properties in terms of Mach number for
thermally and calorically perfect gases• Stagnation Temperature
– from energy conservation:no work but flow workand adiabatic
RT
γ−γ+=
2v2
1
2o M2
11TT −γ+= (VI.6)
12o M2
11pp −γ
γ
��
���
� −γ+= (VI.7)
RTTTo
γ−γ+=
2v2
11
2v1 2
po c
TT +=
2
2v2
11a
−γ+=
1ooTT
pp −γ
γ
��
���
�=from state
eq. for isen. process
�To (and ho) constant foradiabatic flow
• Stagnation Pressure– from entropy conservation:
reversible and adiabatic����isentropic (∆∆∆∆s=0)
�po (and so) constant ifalso reversible
AE3450Stag. Props. and Mach Number -2
School of Aerospace Engineering
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
Compressible po and Bernoulli Equation• Incompressible flow, Bernoulli eqn. also gives a stagnation
pressure (static + dynamic pressure) 2v21 ρ+= ppo
( ) ...2
111122
11
12
112
2212 +��
���
� −γ��
���
� −−γγ
−γγ+−γ
−γγ+=�
�
���
� −γ+=−γ
γ
MMMppo
( ) ( ) ...x2
1nnnx1x1 2n +−++=+
...4
v21v
21 2
22 +ρ+ρ+= Mppo
higher terms negligible for small M (<0.3)Bernoulli
ργ=
pM
22 v use
• Expand compressible po in Taylor series
...v8
v2
1...222
1 22222
2 +ρ+ρ+=+���
����
�γ+γ+= Mpp
MMppo
0.32/4=0.0225
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AE3450Stag. Props. and Mach Number -3
School of Aerospace Engineering
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
Stagnation Density and Tables• Stagnation Density
– from To, po andideal gas law (ρ=p/RT)
– ρρρρo constant for isentropic flow
11
oo
TT −γ
��
���
�=ρρ
11
2o M2
11−γ
��
���
� −γ+=ρρ (VI.8)
• Tables– To/T and po/p tabulated in text (John) as function of M
listed as Tt/T and pt/p (t for total T and total p) – γ=5/3 (Table A.3): atoms (Ar, He, …) at “not too high” T– γ =1.4 (Table A.1): diatomics (N2, O2, …) at “moderate” T– γ =1.3 (Table A.2): more atoms or higher T– make your own?
AE3450Stag. Props. and Mach Number -4
School of Aerospace Engineering
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
Stagnation versus Static Properties• Static Properties
– represent the properties you would measure if you were moving with the flow (at the local flow velocity)
– always defined in the flow’s reference frame
still air
moving air
still air
• Stagnation Properties– always defined by conditions at a point– represent the (static) properties you’d
measure if you first brought the fluid at that point to a stop (isentropically) with respect to a chosen observer
– depends on observer’s reference frame
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AE3450Stag. Props. and Mach Number -5
School of Aerospace Engineering
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
Stagnation Properties: Example• Supersonic projectile (M=2) flying through still air • Static conditions: T∞=250 K, p∞=0.003 atm
shock
A B C D
streamline
M=2T∞=250Kp∞=0.003atm
• Find:1. To at A (ToA) relative to
observer on projectile 2. ToD (same observer) <, >, = ToA ?3. poB (same observer)4. poC (same observer) <, >, = poB ?
AE3450Stag. Props. and Mach Number -6
School of Aerospace Engineering
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
Stagnation Properties: Example (con’t)
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AE3450Stag. Props. and Mach Number -7
School of Aerospace Engineering
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
Stagnation Properties: Example 2• Projectile flying through still air at 170 m/s• Static conditions: T∞=288 K, p∞=1 atm• Nose of projectile = point B• Find:
1. poA (relative to observeron projectile)
2. pB
3. TB
• Hint, use RTa γ=
Bstagnationstreamline 170 m/s
A
AE3450Stag. Props. and Mach Number -8
School of Aerospace Engineering
Copyright © 2001-2002 by Jerry M. Seitzman. All rights reserved.
Stagnation Properties: Example 2 (con’t)