mach angle number

8/12/2019 Mach Angle Number

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Mach Angle, Mach Number -1

School of Aerospace Engineering

Copyright © 2001 by Jerry M. Seitzman. All rights reserved.

Mach Angle and Mach Number Looking for relationship between speed of sound and flow

speed (or speed of body moving through fluid)

• Consider small body (point) moving in stagnant fluid

– continuously “launches” weak pressure disturbances

(e.g., from “pushing” fluid)

• Disturbances travel outward spherically at sound speed (a)

• Look at disturbances generated at

equally spaced time intervals

• Start with body moving with v<<a

– e.g., nearly stationary or moving

through incompressible liquid

x=a×2∆t x=a×∆t

x=a×3∆t






Subsonic and Supersonic Motion• Now compare two bodies, one moving with v<a,

subsonic other body moving with v>a, supersonic

• Subsonic body always behind sound waves launchedfrom previous positions

• Supersonic body moves ahead of previous sound waves

v<a

0 -1 -2 -3

t=-3∆tt=-2∆tt=-∆t

t 0 -1 -2 -3t

v>a t=-3∆t

t=-2∆t

t=-∆t






Mach Wave and Mach Angle• For supersonic flow, can define region where

disturbance has had an effect (been “heard”)

• Conical region delineated by tangents to

sound wave spheres

• Waves coalesce at edge of cone,

produce largest disturbance

– Mach wave (Mach line)

• Angle between Mach lineand body motion, Mach angle

0 -1 -2

-3

at

Zone of

ActionZone of

Silence

=

=µ −−

v

asin

vt

atsin

11

vt






Mach Cone and Shock Waves• Change frame of reference, now let body be

stationary and flow is moving

• Weak disturbances from

presence of body

– can only be felt inside

Mach cone

– can not be felt upstream

• Strong pressure disturbances(nonisentropic) can occur,

they coalesce to form

shock waves

v∞

Zone of Action

Zone of

Silence

Zone of Action

Shock Waveµ≥β

v∞






Mach Number• So flow/body speed relative to sound speed is

fundamental ratio

• Mach number

• In terms of Mach number, the Mach angle is

• Loose demarkation of flow regimes (M∞=v∞ /a∞)

=µ −

M1sin 1

avM ≡ named for Ernst Mach,

Austrian ~1870

hypersonic

supersonic

transonic

subsonic“incompressible”

very strong shocks

stronger changes

shocks for M>1

changes with v<5% effect

3<M∞

1.2<M∞<3

0.8<M∞<1.2

0.3<M∞<0.8M∞<0.3

(VI.3)

(VI.4)






Adiabatic Flow Ellipse• Another way to look at M effects

• Energy equation

• Stagnation To also constant (thermally & calor. p.g.)

const2

vhh

2

o =+=

Maximum velocity possible

(no thermal energy left, T=0)

constR

v

2

1

TT

2

o =γ −γ

+=

2o

2max

22

2

a1

2vva1

2

constvRT1

2

−γ ==+−γ

=+γ −γ

Stagnation speed of sound

(no kinetic energy left, v=0)

adiabatic stream tube

(VI.5)






Adiabatic Flow Ellipse (con’t)• Transition from low speed (ao) to high speed (vmax)

2o

222max a1

2a1

2vv −γ =−γ +=

hypersonic

supersonic

transonic

subsonic

incomp.

v>>a, dv<<da, M change mostly due to a(T) changes

v>a, M changes through substantial changes in v and a(T)

|v-a| << v,a

v<a, M changes primarily to changes in v

v<<a, da<<dv, little change in a(T)

~

v

aa=vao

vmax

M=1Incomp.

subsonic

transonic

supersonichypersonic

mach angle number

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