MAE 5420 - Compressible Fluid Flow! 1!
Incompressible, Compressible, and Supersonic Flow Fields: Static, Dynamic, and
Total Pressure (1)!• In fluid mechanics static pressure is the pressure exerted by a fluid at rest. !Examples of static pressure are:!
1) Air pressure inside a latex balloon !2) Atmospheric (ambient) pressure (neglecting the effect of wind).!3) Hydrostatic pressure at the bottom of a dam is the static pressure !
Strictly Speaking, pressure inside a ventilation duct is not static pressure, !unless the air inside the duct is still.!
Strictly Speaking, On an aircraft, the pressure measured on a generic point of !the surface of the wing (or fuselage) is not, in general, the static pressure, unless !the aircraft is not moving with respect to the air.!
MAE 5420 - Compressible Fluid Flow! 2!
Incompressible, Compressible, and Supersonic Flow Fields: Static, Dynamic, and
Total Pressure (2)!• For fluids in motion the term static pressure is still applicable !(in particular with regard to external flows), and refers strictly to !the pressure in the fluid far upstream (freestream) of any object !immersed into it.!
•Freestream Pressure = Ambient pressure for atmospheric flight!
• When the fluid comes in proximity to a body, its pressure !deviates from the freestream value and strictly-speaking!should no longer be referred to as static pressure …. !quantity should be called simply “pressure”.!
MAE 5420 - Compressible Fluid Flow! 3!
Incompressible, Compressible, and Supersonic Flow Fields: Static, Dynamic, and
Total Pressure (3)!• However, to distinguish this local surface pressure from local!Total AND dynamic pressure, and the freestream pressure …!
… the local “pressure” by tradition is still called “static pressure”!
• The confusion between Total pressure, “pressure”, ! and static pressure arises from one of the basic laws of fluid !dynamics, the Bernoulli equation!
• Bernoulli is Strictly applicable for incompressible, inviscid flow,!With negligible gravitational effects:!
MAE 5420 - Compressible Fluid Flow! 4!
Incompressible Bernoulli Equation (1)!
• Consider Incompressible Flow Case!
MAE 5420 - Compressible Fluid Flow! 5!
Incompressible Bernoulli Equation (2)!
Pstagnation =12ρV 2 + Pstatic
"qbar "→
12ρV 2 =
12
γ Pstaticγ RgTstatic
V 2 =γ2Pstatic ⋅
V 2
γ RgTstatic=γ2PstaticM
2 ≡_
q
Incompressible!Bernoulli Equation!
Fictitious but useful quantity known as!Incompressible Dynamic Pressure, “qbar”!
MAE 5420 - Compressible Fluid Flow! 6!
Incompressible Bernoulli Equation (3)!
• More General Definition of Stagnation or Total Pressure!
• Applicable to Compressible flow fields!
• How does this reconcile with the Bernoulli Equation?!
p +12ρV 2 = const = Pstagnation!
Pstagnation = pstatic ⋅ 1+γ −12
M 2$%&
'()
γ
γ −1
MAE 5420 - Compressible Fluid Flow! 7!
Compressible Bernoulli Equation (subsonic flow) (1)!
• Adding and subtracting p to the equation!
Compressible form of !Bernoulli Equation!P0 = p + p 1+
γ −1( )2
M 2#
$%
&
'(
γ
γ −1−1
)
*+
,+
-
.+
/+≡ p + qc ⇒
qc ≡ p 1+γ −1( )2
M 2#
$%
&
'(
γ
γ −1−1
)
*+
,+
-
.+
/+
Compressible!Dynamic Pressure!
MAE 5420 - Compressible Fluid Flow! 8!
Compressible Bernoulli Equation (subsonic flow) (2)!
Fictitious quantity known as!Dynamic Pressure, “qbar”!
Compressible Dynamic !Pressure or “impact”!Pressure!
• Incompressible Flow!
• Compressible Flow (Subsonic) !
Pstagnation =_
q + Pstatic
qc = 1+ γ −12
M 2#$%
&'(
γγ −1
−1)*+
,+
-.+
/+⋅Pstatic
_
q =γ2PstaticM
2
Pstagnation = qc + Pstatic
MAE 6530 - Propulsion Systems
Solve→ 12V 2 =
γγ−1
RgT0−γγ−1
RgT
Substitute Gas Law→ pρ= RgT
V 2
2+γγ−1⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟pρ=γγ−1
P0
ρ0
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟→ "Compressible Bernoulli Equation"
Compressible Bernoulli Eqn. in “Standard Form”
Alternate Form of Compressible Bernoulli Equationà Flow is Isentropic
P0p= 1+ γ−1
2⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟M
2⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
γγ−1
→P0p⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
γ−1γ
= 1+ γ−12
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟M
2⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟= 1+ γ−1
2⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟V 2
γRgT
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
P0p⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
γ−1γ
=T0T→ substitute→
T0T= 1+ γ−1
2⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟V 2
γRgT
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
Compare to Incompressible Bernoulli Equation
p+ρ ⋅V2
2= P0→
V 2
2+
pρ=
P0ρ
Compare to Incompressible Bernoulli Equation
p+ρ ⋅V2
2= P0→
V 2
2+
pρ=
P0
ρ
Sonic Velocity→ c= γ ⋅Rg ⋅T =∂ p∂ρ
⎛
⎝⎜⎜⎜⎞
⎠⎟⎟⎟⎟Δs=0
Incompressible Flow→∂ρ= 0→ cincr =∞
cincr =∞→ γ=∞
→ limγ→∞
V 2
2+γγ−1
pρ=γγ−1
P0
ρ
⎡
⎣⎢⎢
⎤
⎦⎥⎥=
V 2
2+
pρ=
P0
ρ!
Solve→ 12V 2 =
γγ−1
RgT0−γγ−1
RgT
Substitute Gas Law→ pρ= RgT
V 2
2+γγ−1⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟pρ=γγ−1
P0
ρ0
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟→ "Compressible Bernoulli Equation"
At Mach zero, Compressible Bernoulli Reduces to Incompressible
MAE 5420 - Compressible Fluid Flow! 9!
How do we measure stagnation"pressure? (1)!
transducer!
Flow!
• Probe Looks into flow!
• Captures or “stagnates” !Incoming flow … nearly Isentropically!
… transducer senses total pressure !(included effects of flow kinetic energy)!
V ~ 0!
….Pitot Probe!
MAE 5420 - Compressible Fluid Flow! 10!
How do we measure “normal” or static"pressure?!
transducer!
V!
V!Freestream! Duct!
• Port is perpendicular to!The flow field ….!
Sensed pressure does !not capture effect of external!flow kinetic energy!
….Static Source!Port or pressure tap!
MAE 5420 - Compressible Fluid Flow! 11!
How do we Measure Airspeed or Mach number? (1)!
• Senses Both Total and Static!Pressure from incoming flow field!
• Incompressible …. ! Pstagnation =12ρV 2 + Pstatic → V = 2
Pstagnation − Pstatic( )ρ
• density typically based on sea level number …! … probe gives “indicated” airspeed .. Must be calibrated for!… localized density effects!
MAE 5420 - Compressible Fluid Flow! 12!
How do we Measure Airspeed or Mach number? (2)!
• Senses Both Total and Static!Pressure from incoming flow field!
• Compressible, Subsonic …. !
•.. Typically Probe static pressure is influenced by the!Aircraft and indicated Mach number … must be calibrated for!… localized vehicle effects!
M =2
γ −1PstagnationPstatic
#
$%&
'(
γ −1γ
−1)
*
+++
,
-
.
.
.
MAE 5420 - Compressible Fluid Flow! 13!
Pitot / Static Probe Details (1)!
• White -- Total or Stagnation Pressure!• Yellow -- “normal” or Static Pressure!
ρ
ρ
qc = Ptotal − Pstatic
MAE 5420 - Compressible Fluid Flow! 14!
Pitot / Static Probe Details (2)!
Flow Direction Vane! Pitot Probe!
Static Source Holes!
Looking along !flow direction!
Airliners have pitot/static !probes protruding !from the cockpit area !for measuring airspeed!
Flow !direction!
MAE 5420 - Compressible Fluid Flow!
Typical Small Aircraft Airspeed System
MAE 6530 - Propulsion Systems II
Section 4.2. The Standard Atmosphere
MAE 6530 - Propulsion Systems II
Atmospheric Model Background• Earth's atmosphere is an extremely thin sheet of air extending from surface of Earth to edge of space.
• If Earth were size of a basketball, a tightly held pillowcase would represent thickness of atmosphere.
• Gravity holds atmosphere to Earth's surface. Within atmosphere, very complex chemical, therrmodynamic, and fluid dynamics effects occur.
• Atmosphere is not uniform; fluid properties are constantly changing with time and place. We call this change “weather”.
• Variations in air properties extend upward from surface of Earth. Sun heats surface of Earth, and some of this heat goes into warming air near surface, heated air is also diffused or convected up through atmosphere.
MAE 6530 - Propulsion Systems II
Atmospheric Model Background (2)• Thus the air temperature (and the local speed of sound) is highest near the surface and decreases as altitude increases.
• Air pressure is proportional to the weight of the air over a given location. Tus air pressure decreases as we increase altitude.
• Air density depends on both the temperature and the pressure through the equation of state and also decreases with increasing altitude.
• Due to localized sun angles, terrain, gravitational changes with latitude, and coriolis forces due to earth’s rotation, properties within the atmosphere change significantly with time of year, and global latitude/longitude.
• To help with establishing “mean designs” for air-vehicles, it is extremely useful to define “standard” conditions, and develop models of how the atmospheric properties vary as a function of altitude .. Thus the so-called Standard Day, and Standard Atmosphere.”
MAE 6530 - Propulsion Systems II
HydrostaticEquation
IntegratingoveradepthofatmosphereAssumingconstantgravity
Hydrostatic Model of the Atmosphere• For a fluid in rest without shearing stresses, any elementary fluid element will be subjected to two types of forces, namely, surface forces due to the pressure and a body force equal to the weight of the element.
∂ p∂z=−ρ ⋅g=− p
Rg ⋅T⋅g
NeedtemperatureprofileTocompleteintegral
dpp
p1
p2
∫ = lnp1p1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟=−
gRg
dξT (ξ)
z1
z2
∫
Temperature“LapseRate”ModelT (z)=T1−αL ⋅ z→αL ≡ Lapse Rate
MAE 6530 - Propulsion Systems II
International Standard Atmospheric Model• There are a myriad of atmospheres available .. Each serving a certain technical purpose …
see link on section 4 wep page .. http://www.spacewx.com/Docs/AIAA_G-003B-2004.pdf
• Basically, working models break into distinct altitude layers
1) Troposphere • à h < 11 km (36,000 ft)
2) Lower Stratosphere • à 11 km< h < 25 km (82,000 ft)
3) Upper Stratosphere • à 25 km < h < 50 km
4) Mesosphere • à h > 50 km
By Cmglee - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=23450463
MAE 6530 - Propulsion Systems II
International Standard Atmospheric Model (2)
MAE 6530 - Propulsion Systems II
Hydrostatic Model of the Atmosphere (2)
lnp2p1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟=−
gRg
dξT (ξ)
z1
z2
∫ =−gRg
dξT1−αL ⋅ξz1
z2
∫
T (z)=T1−αL ⋅ z→α
lnp2p1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟=−
gRg
dξT1−αL ⋅ξz1
z2
∫ =
gαLRg
ln T1−αL ⋅ z2( )− ln T1−αL ⋅ z1( )⎡⎣⎢
⎤⎦⎥=
gαLRg
lnT1−αL ⋅ z2T1−αL ⋅ z1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟
MAE 6530 - Propulsion Systems II
Hydrostatic Model of the Atmosphere (3)
Let→ z1= 0,T =Tsl
→ lnp2pSL
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟=
gαLRg
lnT1−αL ⋅ z2
T1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟=g ⋅TslαLRg
ln 1−αLTsl⋅ z2
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟
p(z)= psl ⋅ 1−αLTsl⋅ z
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟
gαLRg
DecayofPressurewithAltitudeinTroposphere
MAE 6530 - Propulsion Systems II
Hydrostatic Model of the Atmosphere (4)
• For the stratosphere atmospheric layer (between 11.0 and 20.1 km), the temperature has a constant value (isothermal conditions) which is 56.5 C (or 69.7 F, 389.97 R, 216.65K).
p(z)= ps ⋅e−
gRg⋅Ts
z−zs( )
DecayofPressurewithAltitudeinStratosphere
MAE 5420 - Compressible Fluid Flow!
US Military Airspeed Definitions
• Indicated Airspeed (IAS)—As shown on the airspeed indicator.
• Calibrated Airspeed (CAS) —Indicated corrected for instrument and installation error.
• Equivalent Airspeed (EAS) – CAS corrected for adiabatic compressible flow at a the measurement altitude. EAS and CAS are equal for a standard atmosphere at sea level … typically expressed in units of nautical miles/hour (kts)
• True Airspeed—CAS corrected for temperature and pressure (actual speed through airstream)
MAE 5420 - Compressible Fluid Flow!
Indicated Airspeed (IAS)
Standard Formula:!
IAS = 7 pslρsl
⋅Ptotal − p∞
psl+1
%
&'
(
)*
13.5−1
+
,
---
.
/
000→
psl =101.325kPa = 2116.2 psf =14.696 psiaρsl =1.225kg/m3 = 7.6314×10
−2lbm/ ft 3
Ptotal − p∞ = qcindicated
MAE 5420 - Compressible Fluid Flow!
Indicated Airspeed (Derivation) • Airspeed as shown on the airspeed indicator – calibrated to reflected standard atmosphere, adiabatic compressible flow with no static sensor position error calibrations
MAE 5420 - Compressible Fluid Flow!
Indicated Airspeed (Derivation) (2)
• Since Typical Airspeed system only measures impact pressure and not static pressure pind directly … replace pind by psl
IAS = 2 ⋅γγ −1
⋅pslρsl
%
&'
(
)*
qcindpsl
+1%
&'
(
)*
γ−1γ
−1+
,
---
.
/
000= 7 psl
ρsl⋅Ptotal − p∞
psl+1
%
&'
(
)*
13.5−1
+
,
---
.
/
000
MAE 5420 - Compressible Fluid Flow!
Indicated Airspeed (Derivation) (3) • Evaluate using standard values for parameters …. Express results in Kts (minute of longitude at the equator is equal to 1 nautical mile. )
MAE 5420 - Compressible Fluid Flow!
Indicated Airspeed (Derivation) (4) • Alternate version of IAS .. Sometimes still in use
Example …. = 250 psf
MAE 5420 - Compressible Fluid Flow!
Calibrated Airspeed (CAS)
• Primary Instrumentation is for the “static source position error” of just “position error” …. corrects the indicated airspeed for known static pressure measurement source errors … δpstatic
MAE 5420 - Compressible Fluid Flow!
Calibrated Airspeed (CAS) (2)
• Correct indicated airspeed for known static pressure measurement source errors … δpstatic
MAE 5420 - Compressible Fluid Flow!
Calibrated Airspeed (CAS) (3)
• Typically … δpstatic is a calibrated function of indicated impact pressure
δpstatic!
MAE 5420 - Compressible Fluid Flow!
Equivalent Airspeed (EAS)
Standard Formula:!
EAS = 7 p∞ρsl
⋅Ptotal − p∞
p∞+1
%
&'
(
)*
13.5−1
+
,
---
.
/
000
pind ≈ p∞
MAE 5420 - Compressible Fluid Flow!
Equivalent Airspeed (Derivation) (2)
• CAS corrected for adiabatic compressible flow at the measurement altitude.
MAE 5420 - Compressible Fluid Flow!
True Airspeed (TAS)
• CAS corrected for temperature and pressure (actual speed through airstream)
1σ=
ρ∞ρsealevel
ρ∞ρsealevel
Air density of free-stream flow divided by standard sea level air density!
MAE 5420 - Compressible Fluid Flow!
True Airspeed (2)
• CAS corrected for temperature and pressure (actual speed through airstream)
Mtrue =TASc∞
c∞
MAE 5420 - Compressible Fluid Flow!
Airspeed Definition Comparisons
MAE 5420 - Compressible Fluid Flow!
Airspeed Definition Comparisons (2)
MAE 5420 - Compressible Fluid Flow!
Airspeed Definition Comparisons (3)
MAE 5420 - Compressible Fluid Flow 32
What About Supersonic Flow? (1)• Cannot Directly Measure Supersonic Total Pressure• Normal Shock wave forms in front of probe …
• Probe senses P02 (behind shock wave)Not! .. P01
• Static orifices far enough aft to not be strongly influenced bynormal shockwave (position error must be calibrated for)
MAE 5420 - Compressible Fluid Flow 33
Supersonic Pitot Probe Calculation (2)
• Calculate Indicated MachNumber from Rayleigh-Pitot Equation
• Assume direct measurement
Lets derive this ….
MAE 5420 - Compressible Fluid Flow 34
Rayleigh-Pitot Equation Derivation
• From Normal Shockwave Eqs.
P02p∞=P02P0∞⋅P0∞p∞
P02P0∞=
2
γ+1( )⋅ γM∞2 −γ−12
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
1γ−1
γ+12M∞
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
2
1+ γ−12M∞
2
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
γγ−1
⎧
⎨
⎪⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪⎪⎪⎪
P0∞p∞= 1+ γ−1
2M∞
2⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
γγ−1
• From Isentropic Flow
MAE 5420 - Compressible Fluid Flow 35
Rayleigh-Pitot Equation Derivation
Rayleigh-Pitot Equation
MAE 5420 - Compressible Fluid Flow 36
Supersonic Mach Number Calibration
• Apply CalibrationCorrection for errors In p∞ (indicated)
Rayleigh-Pitot Equation
MAE 5420 - Compressible Fluid Flow 37
Alternate Form For Rayleigh Pitot Equation• Use …. p2 (indicated)
• Simplify
P02p2=P02P0∞⋅P0∞p∞⋅p∞p2=
2
γ+1( )⋅ γM∞2 −γ−12
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
1γ−1
γ+12M∞
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
2
1+ γ−12M∞
2
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
γγ−1
⎧
⎨
⎪⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪⎪⎪⎪
× 1+ γ−12M∞
2⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
γγ−1
⎧
⎨⎪⎪⎪⎪
⎩⎪⎪⎪⎪
⎫
⎬⎪⎪⎪⎪
⎭⎪⎪⎪⎪
×1
1+ 2γγ+1
M∞2 −1( )
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
P02p2=P0 2p∞⋅p∞p2=
γ+12M∞
2⎡
⎣⎢⎢
⎤
⎦⎥⎥
γγ−1
2γγ+1
M∞2−γ−1γ+1
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
1γ−1
×1
1+ 2γγ+1
M∞2 −1( )
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
MAE 5420 - Compressible Fluid Flow! 39!
Maximum Pressure Coefficient!• Across a Normal Shock wave (bow shock at nose of aircraft)!Maximum pressure coefficient (Cpmax) on a flying body is!
Schematic of the Bow Shock near the Stagnation Point
P ∞, T ∞
Q∞, Pt ∞,
M∞ >> 1
CpMax= P2 − P ∞ = f(M ∞, T ∞, Equlibrium Gas Energy, State Equations )
Real Gas Effects Significantin Stagnation region for High Mach Numbers
Q∞
Cpmax =P02 − p∞12ρ∞V∞
2
∞--> free stream condition!
MAE 5420 - Compressible Fluid Flow! 40!
Maximum Pressure Coefficient (cont’d)!
• Rewriting in terms of free stream mach number!
Cpmax =P02 − p∞12ρ∞V∞
2=
P02 − p∞12γ p∞γ RgT∞
V∞2=P02 − p∞γ2p∞M∞
2=
P02p∞
−1
γ2M∞
2
• For supersonic Conditions:!
Cpmax =P02 − p∞( )12ρ∞V∞
2
$
%
&&&
'
(
)))=
P02 − p∞( )γ2p∞M∞
2=
1γ2M∞
2
γ +12
M∞2+
,-./0
γ
γ −1+
,-.
/0
2γγ +1( )
M∞2 −
γ −1( )γ +1( )
+
,-.
/0
1γ −1+
,-.
/0
−1
1
2
333
4
333
5
6
333
7
333
• Proof left as exercise!
MAE 5420 - Compressible Fluid Flow 40
Compressible Bernoulli Equation (Supersonic Flow) (1)
• Adding and subtracting p to the Rayleigh Pitot equation
Supersonic form of Bernoulli Equation
P02 = p∞ + p∞
γ +12
M∞2#
$%&'(
γ
γ −1#
$%&
'(
2γγ +1( )
M∞2 −
γ −1( )γ +1( )
#
$%&
'(
1γ −1#
$%&
'(
−1
*
+
,,,
-
,,,
.
/
,,,
0
,,,
= p∞ + qc 2
qc 2 = p∞
γ +12
M∞2#
$%&'(
γ
γ −1#
$%&
'(
2γγ +1( )
M∞2 −
γ −1( )γ +1( )
#
$%&
'(
1γ −1#
$%&
'(
−1
*
+
,,,
-
,,,
.
/
,,,
0
,,,qc2 =Cpmax ⋅
γ2⋅ p∞ ⋅M∞
2
MAE 5420 - Compressible Fluid Flow 41
Compressible Bernoulli Equation (Supersonic Flow)(continued)• From earlier
• Comparing Equations
Compressible Form of Bernoulli Equation .. Valid for both Subsonic and Supersonic Flows
P0 − p
q_ = Cpmax
Cpmax =1
γ2M 2
γ +12
M 2"#$
%&'
γ
γ −1"
#$%
&'
2γγ +1( )
M 2 −γ −1( )γ +1( )
"
#$%
&'
1γ −1"
#$%
&'
−1
)
*
+++
,
+++
-
.
+++
/
+++
P0 = p + Cpmax q_
For Subsonic Conditions
.. thus
p02p∞
=p0∞p∞
= 1+γ −1( )2
M∞2$
%&
'
()
γ
γ −1
Cpmax =
1+γ −1( )2
M∞2$
%&
'
()
γ
γ −1−1
*
+,
-,
.
/,
0,
γ2M∞
2
supersonic
sunsonic
For Supersonic Conditions
MAE 5420 - Compressible Fluid Flow 42
Maximum Pressure Coefficient (cont’d)
Cpmax =
1+γ −1( )2
M∞2$
%&
'
()
γ
γ −1−1
*
+,
-,
.
/,
0,
γ2M∞
2
Cpmax =1
γ2M 2
γ +12
M 2"#$
%&'
γ
γ −1"
#$%
&'
2γγ +1( )
M 2 −γ −1( )γ +1( )
"
#$%
&'
1γ −1"
#$%
&'
−1
)
*
+++
,
+++
-
.
+++
/
+++
• M¥ --> 0 (incompressible flow)Cpmax -->1 P0 = p∞ + q
_"#
$% |M =0
MAE 5420 - Compressible Fluid Flow! 45!
Compressible Bernoulli Equation (collected)!P0 = p∞ + Cpmax
⋅q→ q =12ρ∞V∞
2 =γ2P∞M∞
2 → M∞ = 0(incompressible flow) → Cpmax= 1
0 < M∞ < 1→ Cpmax=
1+ γ −12
M∞2'
()*+,
γ
γ −1'
()*
+,
−1
γ2M∞
2
M∞ = 1→ Cpmax=1+ γ −1
2%&'
()*
γ
γ −1%
&'(
)*
−1
γ2
=2γ
γ +12
%&'
()*
γ
γ −1%
&'(
)*
−1%
&
''
(
)
**
1 < M∞ < ∞→ Cpmax=
1γ2M∞
2
γ +12
M∞2%
&'()*
γ
γ −1%
&'(
)*
2γγ +1%
&'(
)*M∞
2 −γ −1γ +1%
&'(
)*%
&'(
)*
1γ −1%
&'(
)*
−1
%
&
''''''
(
)
******
MAE 5420 - Compressible Fluid Flow! 46!
Compressible Bernoulli Equation (Hypersonic Limit)!What happens at “infinite” mach number?!
Cpmax
limM∞→∞
=1
γ2
M∞
limM∞→∞
$
%&&
'
())
2
γ +12
M∞
limM∞→∞
$
%&&
'
())
2$
%
&&
'
(
))
γ
γ −1$
%&'
()
2γγ +1$
%&'
()M∞
limM∞→∞
$
%&&
'
())
2
−γ −1γ +1$
%&'
()
$
%
&&
'
(
))
1γ −1$
%&'
()
−1
$
%
&&&&&&&&&
'
(
)))))))))
=
1
γ2
M∞
limM∞→∞
$
%&&
'
())
2
γ +12
M∞
limM∞→∞
$
%&&
'
())
2$
%
&&
'
(
))
γ
γ −1$
%&'
()
2γγ +1$
%&'
()M∞
limM∞→∞
$
%&&
'
())
2$
%
&&
'
(
))
1γ −1$
%&'
()
$
%
&&&&&&&&&
'
(
)))))))))
=
2γ
γ +12
$%&
'()
γ
γ −1$
%&'
()
2γγ +1$
%&'
()
1γ −1$
%&'
()
M∞
limM∞→∞
$
%&&
'
())
2γγ −1$
%&'
()
M∞
limM∞→∞
$
%&&
'
())
2 γ −1( )
M∞
limM∞→∞
$
%&&
'
())
2$
%
&&&
'
(
)))
1γ −1$
%&'
()
$
%
&&&&&&&&&
'
(
)))))))))
MAE 5420 - Compressible Fluid Flow! 47!
Compressible Bernoulli Equation (Hypersonic Limit (2) )!Collecting exponents on ! M∞
limM∞→∞
#
$%%
&
'((
Cpmax
limM∞→∞
=
2γ
γ +12
$%&
'()
γ
γ −1$
%&'
()
2γγ +1$
%&'
()
1γ −1$
%&'
()
M∞
limM∞→∞
$
%&&
'
())
2γγ −1$
%&'
()− 2 γ −1( )+2( ) 1
γ −1$
%&'
()$
%
&&&
'
(
)))=
2γ
γ +12
$%&
'()
γ
γ −1$
%&'
()
2γγ +1$
%&'
()
1γ −1$
%&'
()
M∞
limM∞→∞
$
%&&
'
())
0$
%
&&
'
(
))=
2γ
γ +12
$%&
'()
γ
γ −1$
%&'
()
2γγ +1$
%&'
()
1γ −1$
%&'
()
=
2γ
γ +12
$%&
'()
γ
γ −1$
%&'
()
2γγ +1$
%&'
()
1γ −1$
%&'
()
=1γ⋅
2
2( )γ
γ −1$
%&'
() ⋅ 2( )1
γ −1$
%&'
()
γ +1( )γ
γ −1$
%&'
()
γγ +1$
%&'
()
1γ −1$
%&'
()
=1γ⋅
1
2( )2
γ −1$
%&'
()
γ +1( )γ
γ −1$
%&'
()
γγ +1$
%&'
()
1γ −1$
%&'
()
=
1
γ1+ 1
γ −1
⋅1
2( )2
γ −1$
%&'
()
γ +1( )γ
γ −1+γ −1
$
%&'
() =γ
γ
1−γ$
%&'
()
2( )2γ −1$
%&
'
()
γ +1( )γ +1γ −1$
%&
'
()
MAE 5420 - Compressible Fluid Flow! 48!
Compressible Bernoulli Equation (Hypersonic Limit (3) )!Check .. Substitute γ = 1.40 !
Cpmax
limM∞→∞
=γ
γ
1−γ%
&'(
)*
2( )2γ −1%
&'
(
)*
γ +1( )γ +1γ −1%
&'
(
)*
= 1.41.4
1 1.4−" #$ %
22
1.4 1−" #$ %
1.4 1+( )
1.4 1+1.4 1−" #$ %
" #& '& '$ %
= 1.8394!
Fundamental property of!Hypersonic flow … Mechanical!Properties on surface (i.e. pressure) become!independent of free stream!Mach number!
MAE 5420 - Compressible Fluid Flow! 49!
Compressible Bernoulli Equation (Hypersonic Limit (4) )!
Fundamental property of!Hypersonic flow … Mechanical!Properties on surface (i.e. pressure) become!independent of free stream!Mach number!
Cpmax
limM∞→∞
=γ
γ
1−γ%
&'(
)*
2( )2γ −1%
&'
(
)*
γ +1( )γ +1γ −1%
&'
(
)*
MAE 5420 - Compressible Fluid Flow! 50!
Example Problem!• Supersonic Pitot Tube!
• p∞ (indicated) ~ 250 kPa! P02 ~ 1200 kPa!
• Estimate Indicated M∞" (assume γ = 1.4)!P02!
p∞ (indicated)!
P0 2P∞(indicated )
=
γ +12
Mind2#
$%&'(
γγ −1
2γγ +1
Mind2 −
γ −1γ +1
*
+,-
./
1γ −1
MAE 5420 - Compressible Fluid Flow! 51!
Example Problem!• Supersonic Pitot Tube .. in terms of P02/P∞!
P02p∞
=1200250
= 4.8 • trial and error solution!
1.4 1+2
1.82822! "# $
1.41.4 1−( )
2 1.4⋅
1.4 1+( )1.82822 1.4 1−( )
1.4 1+−! "
# $
11.4 1−( )
=!
=4.800011-->M∞ ~ 1.8282!
P0 2P∞(indicated )
=
γ +12
Mind2#
$%&'(
γγ −1
2γγ +1
Mind2 −
γ −1γ +1
*
+,-
./
1γ −1
MAE 5420 - Compressible Fluid Flow! 52!
Homework 5: (1)!
• Show That for supersonic free stream conditions !
• Prove that when …. M0 …. CpMax 1.0!
Careful here … subsonic flow when M=0!!
CP max=
P02 − p∞12⋅ ρ∞ ⋅V∞
2%&'
()*
=1
γ2M∞
2
γ +12
M∞2%
&'()*
γ
γ −1
2γγ +1
M∞2 −
γ −1γ +1
%
&'(
)*
1γ −1
−1
,
-
.
.
/
.
.
.
0
1
.
.
2
.
.
.
MAE 5420 - Compressible Fluid Flow! 53!
Homework 5:(2)!
• Write iterative Solver for “Rayleigh Pitot Equation” .. !Given pressure ratio …. Calculate Mach number !
P02p∞
=
γ +12
M∞2#
$%&'(
γ
γ −1#
$%&
'(
2γγ +1( )
M∞2 −
γ −1( )γ +1( )
#
$%&
'(
1γ −1#
$%&
'(
*
+
,,,
-
,,,
.
/
,,,
0
,,,
Carefully comment your code .. Hand in code with assignment …. !Make program “smart enough” to solve for both supersonic !and subsonic free stream conditions!
Supersonic Flow!
P0p∞
= 1+ γ −12
M∞2$
%&'()
γ
γ −1 Supersonic Flow!
MAE 5420 - Compressible Fluid Flow! 54!
Homework 5: (3)
• Consider a Pitot / Static Probe on X-1 experimental rocket plane!• Probe measures free stream static pressure and local impact pressure .. May or MAY NOT!!Be shockwave in front of probe!
• Use Rayleigh-Pitot Code to compute free stream Mach number for following conditions !a)P0 y
= 1.22 ×105 Nt / m2 → px = 1.01×105 Nt / m2
b)P0 y= 7222 lbf / ft 2 → px = 2116 lbf / ft 2
c)P0 y= 13107 ×105 lbf / ft 2 → px = 1020 lbf / ft 2
assume...γ = 1.4
MAE 5420 - Compressible Fluid Flow! 55!
Homework 5: (4)
• Verify that CpMax is a continuous curve at Mach 1. !
i.e. .. Show that When … !M∞ = 11!
CP max( )supersonic
=1
γ2M∞
2
γ +12
M∞2#
$%&'(
γ
γ −1
2γγ +1
M∞2 −
γ −1γ +1
#
$%&
'(
1γ −1
−1
*
+
,,
-
,,
.
/
,,
0
,,
M∞ → 1=
CP max( )subsonic
=1+ γ −1
2M∞
2$%&
'()
γ
γ −1−1
γ2M∞
2
*
+
,,,,,
-
.
/////
M∞ → 1=2γ
γ +12
$%&
'()
γ
γ −1−1
*
+
,,
-
.
//
MAE 5420 - Compressible Fluid Flow! 56!
Homework 5: (5)!
• Key Piece of Knowledge!
∂
∂M
γ +12
M 2#$%
&'(
γ
γ −1#
$%&
'(
2γγ +1( )
M 2 −γ −1( )γ +1( )
#
$%&
'(
1γ −1#
$%&
'(
*
+
,,,,,,
-
.
//////
=
γM 2M 2 −1( ) M 2 γ +1( )2
*
+,
-
./
1γ −1#
$%&
'(
2γγ +1( )
M 2 −γ −1γ +1( )
#
$%&
'(
γ
γ −1#
$%&
'(