incompressible, compressible, and supersonic flow fields...

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.!



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”.!



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:!


Incompressible Bernoulli Equation (1)!

• Consider Incompressible Flow Case!



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”!

Text



• 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


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!


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

5420 Compressible Fluids


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!


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!


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!


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)

*

+++

,

-

.

.

.


Pitot / Static Probe Details (1)!

• White -- Total or Stagnation Pressure!• Yellow -- “normal” or Static Pressure!

ρ

ρ

qc = Ptotal − Pstatic


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 5420 Compressible Fluids


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.



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.”



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

Stephen Whitmore


2


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


International Standard Atmospheric Model (2)


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

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟




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




• 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


http://www.neng.usu.edu/classes/mae/5420/Compressible_fluids/section8/StandardAtmosphere.txt

Link to US 1976 Standard Atmosphere Table


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)


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


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


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


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. )


Indicated Airspeed (Derivation) (4) • Alternate version of IAS .. Sometimes still in use

Example …. = 250 psf


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


Calibrated Airspeed (CAS) (2)

• Correct indicated airspeed for known static pressure measurement source errors … δpstatic


Calibrated Airspeed (CAS) (3)

• Typically … δpstatic is a calibrated function of indicated impact pressure

δpstatic!


Equivalent Airspeed (EAS)

Standard Formula:!

EAS = 7 p∞ρsl

⋅Ptotal − p∞

p∞+1

%

&'

(

)*

13.5−1

+

,

---

.

/

000

pind ≈ p∞


Equivalent Airspeed (Derivation) (2)

• CAS corrected for adiabatic compressible flow at the measurement altitude.


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!


True Airspeed (2)

• CAS corrected for temperature and pressure (actual speed through airstream)

Mtrue =TASc∞

c∞


Airspeed Definition Comparisons


Airspeed Definition Comparisons (2)


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)


Supersonic Pitot Probe Calculation (2)

• Calculate Indicated MachNumber from Rayleigh-Pitot Equation

• Assume direct measurement

Lets derive this ….


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


Rayleigh-Pitot Equation Derivation

Rayleigh-Pitot Equation


Supersonic Mach Number Calibration

• Apply CalibrationCorrection for errors In p∞ (indicated)

Rayleigh-Pitot Equation


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( )

⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎫

⎬

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪


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!

Impact Pressure and


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!


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


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


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


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

%

&

''''''

(

)

******


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$

%&'

()

$

%

&&&&&&&&&

'

(

)))))))))


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$

%&

'

()


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!


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%

&'

(

)*


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


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


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

.

.

.

Due Friday November 2, 2018

6


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!

Stephen Whitmore

Subsonic

6


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

6


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

*

+

,,

-

.

//

6


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#

$%&

'(

6

incompressible, compressible, and supersonic flow fields...

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