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
= ≡ −
= −
∫ ∫
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+ = +