m341 12 lecture25 compressible flow intro

8/13/2019 m341 12 Lecture25 Compressible Flow Intro

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Compressible flows

- Introduction -Darko Matovic, 2012

Queen's niversit!Dept" of Mec#anical and Materials $n%ineerin%

MECH 341

Fluid Mechanics II


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Class objectives:

Identify when are the flows compressible Classify flows according to Mach number

Derive speed of sound from first principles

Make distinction between adiabatic andisentropic flow

Derive Mach relations for adiabatic flow

Derive Isentropic pressure and densityrelations


3/17

Compressible flows around us:

Punctured tire

pray painting

!ullets

"#plosions

IC engine intake$e#haust% knocking

Compressed air jet & cleaning


4/17

'lows according to Mach number:

Incompressible Ma ( )*+

ubsonic )*+ ( Ma ( )*,

-ransonic )*, ( Ma ( .*/

upersonic .*/ ( Ma ( +*)

0ipersonic +*) ( MaC

om

pres

sible

flow

s


5/17

Compressible flows re1uire

Continuity e1uation

Momentum e1uation "nergy e1uation

"1uation of state


6/17

2ew relations:

Mach 2umber: Ma=Ua

a speed of soundU velocity

pecific&0eat

3atio:k=

cp

cv

cpsp* heat at const* pressure

cv sp* heat at const* volume

"1uation of state

4perfect gas5:

p=RT

R=cpcv

p pressure

R gas constant

T6 temperature density

cv= R

k. cp=k cv=

k R

k.a=k RT


7/17

Characteristic values for air:

R=c pcv=/,78 /(kg 9)

=.*/kg /m+

;T=/+*.;9;a=+t standard conditions:

2ote: ?nits forRand cin the te#tbook are m/$4s/95

which is e1uivalent to 8$4kg 95% but less intuitive*


8/17

peed of ound 4propagation speed

for small pressure disturbances5&ave interior

p

T

V=C

pp

T T

V=C V

Continuity across the wave:

-his is a .D problem

AC= AC V

V=C

VC

Momentum conservation alongx:

Fx=m VoutVi n pAp pA=ACCVC

p=C V. Combining: C/= p

.


9/17

Combining p= C V and p=C V

C/

=p

.

-he result indicates that the wave propagation is faster for strong

disturbances 4large 5% i.e.e#plosion waves* 'or sound waves

)* then a/

=p

Combining

In order to determine the derivative 4slope5% we need to know

what kind of process is this* @iven that there is no net heat

e#change% it ought to be an adiabaticprocess* ince the density

change is also infinitesimal it is also an isentropicflow* -hus:

a=p s./ /

= kp T.//

'or perfect gas: a=k p=k R T


10/17


11/17

Internal energy change: u/ u.=cv T/T.

"nthalpy change: h/h.=cp T/T.

!ut if we take into account

that specific heats depend

on temperature% then

h/h.=.

/

cp dTand

u/ u.=.

/

cv dT


12/17

Isentropic Process

"ntropy change is obtained from T ds=dhdp /

Introducing dh=cp dT and p=RT

.

/

ds=

.

/

cpdT

T R

.

/dp

p

'or constant specific heat% these can be integrated to yield

s/s.=cp lnT/

T.

R lnp/

p.

=cv lnT/

T.

R ln/

.'or isentropic 4constant entropy5 process% s.=s/ * -hen

p/

p.=

T/

T.

k

k.=

/

.

k


13/17

>diabatic and Isentropic teady 'low

>pplying energy e1uation for .D flow:

h.V.

/

/ g .=h/

V//

/ g /!"visc

Ae can neglect viscous work% heat e#change and elevation4potential5 energy* -hen the .D energy e1uation reduces to:

hV

/

/

=h)=const where h) is the stagnation entha#p$

where T)is the stagnation temperature.cp T

V/

/ =cp T)

hV

/

/

=h)=const where h) is the stagnation entha#p$.


14/17

Mach number relations for adiabaticflow 4isentropic or notB5

dividing cp T V/

/=cp T) by cp T

. V

/

/ cp T=

T)

T % and using cp T=

k R

k.T=

a/

k.we arrive at

T)

T =.

k.

/

V/

a/

or

T)

T =.

k.

/Ma

/ and

a)

a=T)T

./ /

=[. k./ Ma/].//

since aT.//


15/17

Mach number relations for isentropicflow 4must be adiabatic% tooB5

T)

T =.

k.

/Ma

/ 4from previous slide5

?sing isentropic relations p/

p.=/.

k

=T/

T. kk.

and

p)p=T)

T k

k.=[. k./ Ma/]

k

k.

)=T)T.

k.=[ . k./ Ma/]

.

k.

Must also be isentropic

(diabatic


16/17

Critical values 4at Ma.*)5alues at critical point% where Mach number is e1ual to one

4sonic conditions5 are of special importance for compressibleflow calculations* 'or that reason% we mark these values by anasterisk:

pE

p)=

/

k.

k

k. E

)=

/

k.

.

k.

TE

T)= /k.

aE

a)= /k.

.//

'or air 4k .*


17/17

Isentropic flow tables Do not confuse A (nozzlearea) with a(sound speed)!

m341 12 lecture25 compressible flow intro

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