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




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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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?




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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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 ?




Stagnation Properties: Example (con’t)

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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




Stagnation Properties: Example 2 (con’t)

Stagnation Properties and Mach...

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