Integration of the Euler Equation

� For compressible flow, we need a working equation …

� Recall, the differential form is applicable,

�INTEGRATION CAN BE DONE

�WE NEED THERMODYNAMIC RELATIONS TO INTEGATE

� THE RESULT OF THE INTEGRATION IS:

dP VdV+ =ρ 0

2

P

P

Vc T + = constant

2

γwhere c = R (γ =1.4 for air)

γ-1

EQUATION SUMMARY

CONTINUITY EQUATION (INCOMPRESSIBLE)

BERNOULLI’S EQUATION

CONTINUITY EQUATION (COMPRESSIBLE)

ISENTROPIC RELATIONS

ENERGY

EQUATION OF STATE

… BE FAMILIAR WITH ALL EMBEDDED ASSUMPTIONS

AV A V1 1 2 2=

PV

PV

112

222

2 2+ = +ρ ρ

ρ ρ1 1 1 2 2 2AV A V=

P

P

T

T2

1

2

1

2

1

1=

=

−ρ

ρ

γγ

γ

P RT

P RT1 1 1

2 2 2

=

=

ρ

ρ

c TV

c TV

P P112

222

2 2+ = +

PHYSICAL PRINCIPLES

� CONSERVATION OF MASS

� LED TO THE CONTINUITY EQUATION

� DIFFERENT FOR COMPRESSIBLE OR INCOMPRESSIBLE

� CONSERVATION OF MOMENTUM

� COMES FROM NEWTON’S SECOND LAW: F = ma

� LED TO EULER’S EQUATION (COMPRESSIBLE OR INCOMPRESSIBLE)

� LED TO BERNOULLI’S EQUATION (INCOMPRESSIBLE ONLY)

NEXT : THE CONSERVATION OF ENERGY

TERMINOLOGY� WORK

� DEFINED : PRODUCT OF THE FORCE AND THE DISPLACEMENT

IN THE DIRECTION OF THE FORCE

� WORK = Fcos θ θ θ θ x s

�UNITS OF WORK: FOOT-LBS (ENGLISH),

NEWTON-METER = JOULE (MKS)

� ENERGY

� DEFINED AS THE ABILITY TO DO WORK

�UNITS ARE THE SAME AS WORK

� CLASSIFIED AS

�POTENTIAL ENERGY (ENERGY DUE TO POSITION)

�KINETIC ENERGY (ENERGY DUE TO MOTION)

WHY THE INTEREST IN WORK & ENERGY ?

� ENERGY CHANGES ARE REFLECTED IN

P , T and ρρρρ changes in the flow …

� HIGH “SPEED” FLOW LEADS TO COMPRESSIBILITY,

WHICH LEADS TO LARGE ENERGY CHANGES IN A GAS

� For us, COMPRESSIBILITY MEANS CHANGES IN TEMPERATURE,

WHICH AFFECTS OTHER PROPERTIES OF THE GAS.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

Mach no.

ρ / ρρ / ρρ / ρρ / ροοοο

.95

.3

“COMPRESSIBILITY” ERROR

Speed of sound (Mach One at STP):

1120 ft/sec

340 m/s

761 mph

FIRST LAW OF THERMODYNAMICS

� IN EQUATION FORM (one of many forms)

� IN WORDS:

PER UNIT MASS, THE CHANGE IN INTERNAL ENERGY ( e ) OF A SYSTEM IS

EQUAL TO THE SUM OF THE HEAT (q) ADDED TO (OR TAKEN AWAY FROM)

THE SYSTEM AND THE WORK (w) DONE ON (OR DONE BY) THE SYSTEM

q w deδ δ+ =

WORK DONE ON A SYSTEM

� FOR AN ELEMENTAL dA

� FROM OUR DEFINITION

OF WORK

� TO OBTAIN THE INCREMENT

OF WORK DONE ON

THE GAS BOUNDED BY “A”,

WE MUST INTEGRATE OVER THE AREA

� IF THE SURFACE IS IN THERMODYNAMIC EQUILIBRIUM,

“P” IS CONSTANT OVER “A”

W F s PdA s∆ = × = ×

A Aw PdA s Ps dAδ = =∫ ∫

A Aw P s dA; but, sdA d

thus, w Pd

where, is specific volume

δ ν

δ ν

ν

= ≡ −

= −

∫ ∫

� APPLYING PRESSURE TO DO WORK ON A VOLUME OF GAS

WORK DONE ON A SYSTEM

w Pdδ ν= −

FIRST LAW OF THERMODYNAMICS

� ALTERNATE FORM OF EQUATION FROM THE FIRST LAW

� HAVING ESTABLISHED THAT THE WORK DONE ON THE SYSTEM IS:

� SUBSTITUTING INTO THE ORIGINAL FORM OF THE FIRST LAW:

� DEFINING ENTHALPY AS:

� DIFFERENTIATING h AND SUBSTITUTING INTO

OUR ALTERNATIVE FORM OF THE FIRST LAW …

� WE GET A SECOND ALTERNATIVE EQUATION

Aw P sdA or w Pdδ δ ν= = −∫

q de Pdδ ν= +

h e P e RTν≡ + = +

dh de Pd dPν ν= + +

q dh dPδ ν= −

� SO FAR, WE HAVE THREE FORMS

OF THE FIRST LAW OF THERMO

� OTHER FORMS ARE USED FOR SPECIFIC TYPES OF

THERMODYNAMIC PROCESSES

FIRST LAW(S) OF THERMODYNAMICS

q dh dPδ ν= −

q de Pdδ ν= +q w deδ δ+ =

FIRST LAW OF THERMODYNAMICS

� SPECIFIC HEAT

� HEAT ADDED (OR TAKEN AWAY FROM) A SYSTEM PER UNIT CHANGE

IN TEMPERATURE

� THE VALUE OF SPECIFIC HEAT DEPENDS ON THE THERMO PROCESS

�SPECIFIC HEAT AT CONSTANT VOLUME:

�SPECIFIC HEAT AT CONSTANT PRESSURE:

� FOR A CONSTANT PRESSURE PROCESS

�AND, ASSUMING cP IS CONSTANT FOR AIR, AND SINCE δδδδq = cP dT ,

… ANOTHER FORM FOR THE FIRST LAW OF THERMO

dhqdPdhq =δ⇒ν−=δ

cq

dT≡

δ

cq

dTvat cons t volume

≡

δ

tan

cq

dTPat cons t pressure

≡

δ

tan

dh c dTP=dh c dTP=

FIRST LAW OF THERMODYNAMICS

� SIMILARLY, CONSIDER THE CONSTANT VOLUME PROCESS

� δδδδq = de because dνννν = 0

� assuming cv = constant and since dq = cvdT

� All forms of the First Law of Thermo represent the Conservation of Energy

q de Pd q deδ ν δ= + ⇒ =

de c dTν=de c dTν=

THERMODYNAMIC PROCESSES

� WE CONSIDERED TWO THERMO PROCESSES

� CONSTANT VOLUME

� CONSTANT PRESSURE

� OTHER IMPORTANT THERMO PROCESSES:

� ADIABATIC: A PROCESS IN WHICH NO HEAT (q) IS ADDED

OR TAKEN AWAY FROM THE SYSTEM.

� REVERSIBLE: A PROCESS IN WHICH NO FRICTION OR OTHER

DISSIPATIVE LOSSES OCCUR

� ISENTROPIC: A PROCESS WHICH IS BOTH ADIABATIC AND REVERSIBLE

(“constant entropy”)

ISENTROPIC FLOW RELATIONS

� ADIABATIC FLOW IMPLIES

� Divide de by dh … de/dh =

γγγγ is the

ratio of specific heats

( γγγγ = 1.4 for air)

� SUBSTITUTING AND INTEGRATING … along a streamline

q de Pd 0 Pd de c dTνδ ν ν= + = ⇒ − = =

Pq dh dP 0 dP dh c dTδ ν ν= − = ⇒ = =

P

P

c cPd dP d

dP c P c

ν

ν

ν ν

ν ν

−= ⇒ = −

γ ≡

c

cP

v

2 2

1 1

P

P

2 2 2 2

1 1 1 1

2 2

1 1

dP d dP d

P P

P Pln ln

P P

P1note, thus

P

ν

ν

γ

γ

ν νγ γ

ν ν

ν νγ

ν ν

ρν

ρ ρ

−

= − ⇒ = −

= − ⇒ =

= =

∫ ∫

� COMBINING THE EQUATION OF STATE WITH THE PRIOR RESULTS

� RETURN TO THE ENERGY BALANCE

EXPRESSED BY THE FIRST LAW OF THERMODYNAMICS

� THE PHYSICAL PRINCIPLE IS: ENERGY CAN NEITHER BE CREATED

NOR DESTROYED, BUT IT MAY CHANGE FORM.

�FOR ADIABATIC (OR ISENTROPIC) FLOW

�RECALLING EULER’S EQUATION:

where V = Velocity

�COMBINING THESE 2 EQUATIONS, noting νννν ==== 1111/ρ/ρ/ρ/ρ

ISENTROPIC FLOW RELATIONS

ρρ

ρ

γγ

γ= ⇒ =

=

−P

RT

P

P

T

T2

1

2

1

2

1

1

q dh dP 0 dh dPδ ν ν= − = ⇒ =

dP VdV= −ρ

dh VdV+ = 0

q dh dPδ ν= −

� INTEGRATING THIS ENERGY EQUATION ALONG A STREAMLINE

� RECALL, ADIABATIC FLOW IMPLIES

note h = 0 at T = 0

� ALONG A STREAMLINE, AND IF FLOW CONDITIONS FOR ALL

STREAMLINES PASS THROUGH A REGION OF UNIFORM FLOW,

ENERGY EQUATION

dh VdV dh VdV

h hV V

hV

hV

hV

CONSTANT

h

h

V

V+ = ⇒ + =

− + −

=

+ = + = + =

∫ ∫0 0

2 20

2 2 2

1

2

1

2

2 122

12

112

222 2

P

P

dh c dT

h c T

=

=

c TV

c TV

c TV

CONSTANTP P P112

222 2

2 2 2+ = + = + =

SOUND WAVES

� Consider a wave moving at Velocity “a” = the Speed of Sound

� OR, MAKE THE WAVE STATIONARY

�ONE DIMENSIONAL FLOW

�APPLY CONTINUITY …

( ) ( )

( )( )

ρ ρ

ρ ρ ρ

ρ ρ ρ

ρ ρ ρ ρ

ρρ

1 1 1 2 2 2

1 2

A V A V

A a d A a da

a d a da

a a ad da

ada

d

=

= + +

= + +

= + +

= −

� APPLYING EULER’S EQUATION (“MOMENTUM”) AT V = a

� SUBSTITUTING INTO THE EXPRESSION FOR a

� FOR ISENTROPIC FLOW

� SUBSTITUTING INTO a AND USING THE EQUATION OF STATE

SPEED OF SOUND

dp ada dadp

a= − ⇒ = −ρ

ρ

ada

d d

dp

aa

dp

d= − = ⇒ =ρ

ρ

ρ

ρ ρ ρ2

isentropic

dpa

dρ

=

p

p

p pcons t2

1

2

1

2

2

1

1

=

⇒ = =

ρ

ρ ρ ρ

γ

γ γtan

dp

d

d

dc c

p p

isentropicρ ρ

ρ γρρ

γργ

ργ γ

γγ

= = = =− −1 1

isentropic

dp pa

d

γ

ρ ρ

= =

a RT= γ

SPEED OF SOUND

PHYSICAL RELATIONSHIP OF SPEED SOUND AND TEMPERATURE

� TEMPERATURE IS A MEASURE OF RANDOM MOLECULAR MOTION

� SPEED OF SOUND IS A MEASURE OF DIRECTED MOTION

TRANSMITTED BY MOLECULAR COLLISIONS

a RT= γ

� ≈ 50 �

� ≈ 20 �

“a” for air

English

Metric

MACH NUMBER, M

THE MACH NUMBER …

� IS THE RATIO OF VELOCITY TO SPEED OF SOUND

� RELATES DIRECTED

KINETIC ENERGY TO INTERNAL ENERGY

� IS A NON-DIMENSIONAL MEASURE OF FLOW COMPRESSIBILITY.

� FLOW REGIMES

� SUBSONIC AND SUPERSONIC FLOW ARE EASY TO ANALYZE

� TRANSONIC FLOW IS THE MOST DIFFICULT REGIME

( because shocks (non-isentropic) form )

VM

a≡

TRANSONIC FLOW

� FREESTREAM FLOW IS STILL SUBSONIC

� TRANSONIC FLOW IS A MIXED FLOW REGIME; IT IS PARTLY SUBSONICAND PARTLY SUPERSONIC … and a (non-isentropic) shock may form.

THE LOCAL FLOW SPEEDS UP

OVER A CURVED SURFACE

M > 1

M∞∞∞∞ < 1

STAGNATION EQUATIONS

� RECALL, THE ENTHALPY DEFINITION

AND

� DIVIDING BY cp

� ALSO, RECALL THE ENERGY EQUATION

where To is the stagnation T

(a.k.a. “total”)

� DIVIDING BY T1 AND SUBSTITUTING FOR cp

h e pv≡ +

c TV

c Tp p112

02+ =

h c T AND e c Tp v= =

c T c T RT OR c c Rp v p v= + − =

v

p

p

p

v

p c

c where

1

Rc

11

c

R

c

c

11 =γ

−γ

γ=⇒

γ−==−

2

1

1

0

1

2

1

1

2

1

1p

2

1

1

0 M2

11

T

T

RT

V

2

11

T1

R2

V1

Tc2

V1

T

T −γ+=⇒

γ

−γ+=

−γ

γ+=+=

EQUATION SUMMARY

CONTINUITY EQUATION (INCOMPRESSIBLE)

BERNOULLI’S EQUATION

STEADY ? INCOMPRESSIBLE ? INVISCID ? .

CONTINUITY EQUATION (COMPRESSIBLE)

ISENTROPIC RELATIONS

ENERGY

EQUATION OF STATE

… BE FAMILIAR WITH ALL EMBEDDED ASSUMPTIONS

AV A V1 1 2 2=

PV

PV

112

222

2 2+ = +ρ ρ

ρ ρ1 1 1 2 2 2AV A V=

P

P

T

T2

1

2

1

2

1

1=

=

−ρ

ρ

γγ

γ

P RT

P RT1 1 1

2 2 2

=

=

ρ

ρ

c TV

c TV

P P112

222

2 2+ = +