Incompressible, Compressible, and Supersonic Flow Fields

Home , Compressible flow, Dynamic pressure, Incompressible flow, Mach number, Pressure coefficient, Stagnation pressure

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.

1 MAE 5420 - Compressible Fluid Flow 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”. 2 MAE 5420 - Compressible Fluid Flow 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: 3 MAE 5420 - Compressible Fluid Flow Incompressible Bernoulli Equation (1)

• Consider Incompressible Flow Case

4 MAE 5420 - Compressible Fluid Flow Incompressible Bernoulli Equation (2)

Incompressible Bernoulli Equation

Fictitious but useful quantity known as Incompressible Dynamic Pressure, “qbar” 5 MAE 5420 - Compressible Fluid Flow Incompressible Bernoulli Equation (3)

• More General Definition of Stagnation or Total Pressure

γ γ −1 $ γ − 1 2 ' P = p ⋅ 1+ M stagnation static %& 2 ()

• Applicable to Compressible flow fields

• How does this reconcile with the Bernoulli Equation? 1 p + ρV 2 = const = P 2 stagnation

6 MAE 5420 - Compressible Fluid Flow Compressible Bernoulli Equation (subsonic flow) (1)

• Adding and subtracting p to the equation ) γ - γ −1 +# (γ − 1) 2 & + Compressible form of P0 = p + p *%1+ M ( − 1. ≡ p + qc ⇒ Bernoulli Equation +$ 2 ' + , / ) γ - Compressible γ −1 +# (γ − 1) 2 & + Dynamic Pressure qc ≡ p *%1+ M ( − 1. +$ 2 ' + , /

7 MAE 5420 - Compressible Fluid Flow Compressible Bernoulli Equation (subsonic flow) (2) Fictitious quantity known as • Incompressible Flow Dynamic Pressure, “qbar”

• Compressible Flow (Subsonic)

Compressible Dynamic Pressure or “impact”

Pressure 8 MAE 5420 - Compressible Fluid Flow Alternate Form of Compressible Bernoulli

à Flow is Isentropic Equation

γ γ−1 ⎛ ⎛ ⎞ ⎞γ−1 ⎛ ⎞ γ ⎛ ⎛ ⎞ ⎞ ⎛ ⎛ ⎞ 2 ⎞ P0 γ −1 2 ⎟ P0 ⎟ γ −1 2 ⎟ ⎜ γ −1 V ⎟ =⎜1+⎜ ⎟ M ⎟ →⎜ ⎟ =⎜1+⎜ ⎟ M ⎟=⎜1+⎜ ⎟ ⎟ Compare to Incompressible Bernoulli Equation ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ p ⎝ ⎝ 2 ⎠ ⎠ ⎝ p ⎠ ⎝ ⎝ 2 ⎠ ⎠ ⎝⎜ ⎝ 2 ⎠γRgT ⎠⎟ 2 2 γ−1 V V p P p P 0 ⎛ ⎞ γ ⎛ ⎛ ⎞ 2 ⎞ +ρ ⋅ = 0 → + = P0 ⎟ T0 T0 ⎜ γ −1⎟ V ⎟ 2 2 ρ ρ ⎜ ⎟ = → substitute → =⎜1+⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎝ p ⎠ T T ⎝⎜ ⎝ 2 ⎠γRgT ⎠⎟ ⎛∂ p⎟⎞ Sonic Velocity → c = γ ⋅ R ⋅T = ⎜ ⎟ 1 2 γ γ g ⎜ ∂ρ ⎟ Solve → V = R T − R T ⎝ ⎠Δs=0 2 γ −1 g 0 γ −1 g 1 γ γ Incompressible Flow → ∂ρ = 0 → c = ∞ Solve → V 2 = R T − R T incr 2 γ −1 g p0 γ −1 g c = ∞ → = ∞ Substitute Gas Law → = R T incr γ ρ g p ⎡ 2 P ⎤ 2 P Substitute Gas Law → = R T V γ p γ 0 V p 0 2 g → lim ⎢ + = ⎥ = + = ! Compressible⎛ Bernoulli⎞ Eqn.ρ⎛ in⎞ “Standard Form” γ→∞ ⎢ ⎥ V γ ⎟ p γ ⎜ P0 ⎟ 2 γ −1 ρ γ −1 ρ 2 ρ ρ +⎜ ⎟ = ⎜ ⎟→ "Compressible Bernoulli Equatio⎣ n" ⎦ 2 ⎜⎛ ⎞⎟ ⎛ ⎜ ⎞ ⎟ V2 ⎝γ −γ 1⎠pρ γγ−1⎜P0ρ⎟ ⎟ +⎜ ⎟ = ⎜ ⎝ ⎟0→⎠ "Compressible Bernoulli Equation" ⎜ ⎟ ⎜ ⎟ 2 ⎝γ −1⎠ ρ γ −1⎝⎜ρ0 ⎠⎟ At Mach zero, Compressible Bernoulli Reduces to Incompressible Compare to Incompressible Bernoulli Equation

V 2 V 2 p P p +ρ ⋅ = P → + = 0 2 0 2 ρ ρ

MAE 5420 - Compressible Fluid Flow How do we measure stagnation pressure? (1)

Flow

• Probe Looks into flow V ~ 0

• Captures or “stagnates” Incoming flow … nearly Isentropically transducer

… transducer senses total pressure (included effects of flow kinetic energy) ….Pitot Probe 10 MAE 5420 - Compressible Fluid Flow How do we measure “normal” or static pressure? ….Static Source V Port or pressure tap

Freestream Duct V • Port is perpendicular to The flow field ….

Sensed pressure does transducer not capture effect of external flow kinetic energy 11 MAE 5420 - Compressible Fluid Flow How do we Measure Airspeed or Mach number? (1)

• Senses Both Total and Static Pressure from incoming flow field

• Incompressible ….

• density typically based on sea level number … … probe gives “indicated” airspeed .. Must be calibrated for … localized density effects

12 MAE 5420 - Compressible Fluid Flow 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

13 MAE 5420 - Compressible Fluid Flow Pitot / Static Probe Details (1)

• White -- Total or Stagnation Pressure • Yellow -- “normal” or Static Pressure 14 MAE 5420 - Compressible Fluid Flow Pitot / Static Probe Details (2)

Flow Direction Vane Pitot Probe

Airliners have pitot/static Flow probes protruding from the cockpit area direction for measuring airspeed

Static Source Holes

Looking along flow direction 15 MAE 5420 - Compressible Fluid Flow Typical Small Aircraft Airspeed System

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:

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 à pind by psl

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 … dpstatic

MAE 5420 - Compressible Fluid Flow Calibrated Airspeed (CAS) (2) •Correct indicated airspeed for known static pressure measurement source errors … dpstatic

MAE 5420 - Compressible Fluid Flow Calibrated Airspeed (CAS) (3)

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

dpstatic

MAE 5420 - Compressible Fluid Flow Equivalent Airspeed (EAS)

Standard Formula:

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)

s 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)

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 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 by normal shockwave (position error must be calibrated for) 33 MAE 5420 - Compressible Fluid Flow Supersonic Pitot Probe Calculation (2)

• Assume direct measurement

• Calculate Indicated Mach Number from Rayleigh-Pitot Equation

Lets derive this ….

34 MAE 5420 - Compressible Fluid Flow Rayleigh-Pitot Equation Derivation

• From Isentropic Flow P P P γ 02 02 0∞ = ⋅ γ−1 P0 ⎛ γ −1 ⎞ ∞ ⎜ 2 ⎟ p∞ P0 p∞ =⎜1+ M ⎟ ∞ ⎜ ∞ ⎟ p∞ ⎝ 2 ⎠ • From Normal Shockwave Eqs.

⎪⎧ γ ⎪⎫ ⎪ ⎡ 2 ⎤ γ−1 ⎪ ⎪ ⎢ ⎛γ +1 ⎟⎞ ⎥ ⎪ ⎪ ⎜ M ⎟ ⎪ P ⎪ ⎢ ⎜ 2 ∞ ⎟ ⎥ ⎪ 02 ⎪ 2 ⎢ ⎝ ⎠ ⎥ ⎪ = ⎨⎪ ⎬⎪ ⎪ 1 ⎢ ⎥ ⎪ P0 ⎢ γ −1 2 ⎥ ∞ ⎪ ⎛ ⎞γ−1 1 M ⎪ ⎪ 2 γ −1⎟ + ∞ ⎪ ⎪(γ +1)⋅⎜γ M − ⎟ ⎢ 2 ⎥ ⎪ ⎪ ⎝⎜ ∞ 2 ⎠⎟ ⎣⎢ ⎦⎥ ⎪ ⎩⎪ ⎭⎪

35 MAE 5420 - Compressible Fluid Flow Rayleigh-Pitot Equation Derivation

Rayleigh-Pitot Equation 36 MAE 5420 - Compressible Fluid Flow Supersonic Mach Number Calibration

• Apply Calibration Correction for errors In p∞ (indicated)

Rayleigh-Pitot Equation

37 MAE 5420 - Compressible Fluid Flow Alternate Form For Rayleigh Pitot Equation

• Use …. p2 (indicated) ⎪⎧ γ ⎪⎫ ⎪ ⎡ 2 ⎤ γ−1 ⎪ ⎪ ⎢ ⎛γ +1 ⎞ ⎥ ⎪ ⎪⎧ ⎪⎫ ⎪ ⎜ M ⎟ ⎪ ⎪⎧ γ ⎪⎫ ⎪ ⎪ ⎢ ⎜ ∞ ⎟ ⎥ γ−1 ⎪ ⎪ P0 P0 P0 p ⎪ 2 ⎜ 2 ⎟ ⎪ ⎪⎛ 1 ⎞ ⎪ 1 2 2 ∞ ∞ ⎪ ⎢ ⎝ ⎠ ⎥ ⎪ ⎪ γ − 2 ⎟ ⎪ ⎪ ⎪ = ⋅ ⋅ = ⎨ ⎬×⎨⎜1+ M ⎟ ⎬×⎨ ⎬ ⎪ 1 ⎢ 1 ⎥ ⎪ ⎪⎜ ∞ ⎟ ⎪ ⎪ 2 ⎪ p2 P0 p∞ p2 ⎢ γ − 2 ⎥ ⎝ 2 ⎠ γ 2 ∞ ⎪ ⎛ γ −1⎞γ−1 1+ M ⎪ ⎪ ⎪ ⎪1+ M −1 ⎪ ⎪ 1 ⎜ M 2 ⎟ ⎢ ∞ ⎥ ⎪ ⎩⎪ ⎭⎪ ⎪ ( ∞ )⎪ (γ + )⋅⎜γ ∞ − ⎟ 2 ⎪ γ +1 ⎪ ⎪ ⎝⎜ 2 ⎠⎟ ⎣⎢ ⎦⎥ ⎪ ⎩ ⎭ ⎩⎪ ⎭⎪

• Simplify

γ ⎡γ +1 ⎤ γ−1 ⎧ ⎫ ⎢ M 2 ⎥ ⎪ ⎪ ∞ ⎪ ⎪ P0 P0 p ⎢ 2 ⎥ 1 2 2 ∞ ⎣ ⎦ ⎪ ⎪ = ⋅ = 1 ×⎨ ⎬ p p p ⎪ 2γ 2 ⎪ 2 ∞ 2 ⎛ 2γ γ −1⎞γ−1 ⎪1+ M −1 ⎪ ⎜ M 2 ⎟ ⎪ ( ∞ )⎪ ⎜ ∞ − ⎟ ⎪ γ +1 ⎪ ⎝⎜γ +1 γ +1⎠⎟ ⎩ ⎭

38 MAE 5420 - Compressible Fluid Flow Maximum Pressure Coefficient • Across a Normal Shock wave (bow shock at nose of aircraft) Maximum pressure coefficient (Cpmax) on a flying body is

P0 − p Cp = 2 ∞ max 1 ρ V 2 2 ∞ ∞

¥--> free stream condition

39 MAE 5420 - Compressible Fluid Flow Maximum Pressure Coefficient (cont’d) • Rewriting in terms of free stream mach number P0 2 − 1 P02 − p∞ P02 − p∞ P02 − p∞ p∞ Cpmax = = = = 1 2 1 γ p∞ 2 γ 2 γ 2 ρ∞V∞ V∞ p∞ M ∞ M ∞ 2 2 γ RgT∞ 2 2 • For supersonic Conditions: 1 + γ . 5 - γ −10 $ ' 3 + γ + 1 2 . , / 3 3 - M ∞ 0 3 &(P02 − p∞ )) (P02 − p∞ ) 1 3 , 2 / 3 Cp 1 max = & ) = = 2 + 1 . − 6 1 2 γ 2 γ 2 3 ,- γ −1/0 3 & ρ∞V∞ ) p∞ M ∞ M ∞ + 2γ γ − 1 . % 2 ( 2 2 3 M 2 ( ) 3 - ∞ − 0 43, (γ + 1) (γ + 1)/ 73 • Proof left as exercise

40 MAE 5420 - Compressible Fluid Flow Compressible Bernoulli Equation (Supersonic Flow) (1) • Adding and subtracting p to the Rayleigh Pitot equation

* # γ & . % γ −1( , # γ + 1 2 & $ ' , % M ∞ ( , $ 2 ' , P = p + p − 1 = p + q 02 ∞ ∞ + # 1 & / ∞ c 2 ,# 2γ γ − 1 & $% γ −1'( , , M 2 ( ) , Supersonic form of % ∞ − ( -,$ (γ + 1) (γ + 1)' 0, Bernoulli Equation

* # γ & . % γ −1( , # γ + 1 2 & $ ' , % M ∞ ( , $ 2 ' , q = p − 1 c 2 ∞ + # 1 & / , $% γ −1'( , γ 2 # 2γ 2 (γ − 1)& , M − , qc = Cp ⋅ ⋅ p∞ ⋅ M∞ % ∞ ( 2 max 2 -,$ (γ + 1) (γ + 1)' 0, 41 MAE 5420 - Compressible Fluid Flow Compressible Bernoulli Equation (Supersonic Flow)(continued) • From earlier

P0 − p ) " γ % - Cp $ γ −1' _ = max For Supersonic Conditions+ " γ + 1 2 % # & + $ M ' q 1 + # 2 & + Cp 1 • Comparing Equations max = * " 1 % − . _ supersonic γ 2 M +" 1 % #$ γ −1&' + 2 2γ 2 (γ − ) P0 = p + Cpmax q +$ M − ' + ,+# (γ + 1) (γ + 1)& /+ For Subsonic Conditions γ γ −1 p02 p0∞ $ (γ − 1) 2 ' = = &1+ M ∞ ) p∞ p∞ % 2 ( Compressible Form of Bernoulli .. thus Equation .. Valid for both * γ . γ −1 ,$ (γ − 1) 2 ' , Subsonic and Supersonic Flows +&1+ M ∞ ) − 1/ ,% 2 ( , - 0 Cp = max γ sunsonic M 2 2 ∞ 42 MAE 5420 - Compressible Fluid Flow Maximum Pressure Coefficient (cont’d)

) " γ % - $ γ −1' + " γ + 1 2 % # & + $ M ' 1 + # 2 & + Cp 1 max = * " 1 % − . γ 2 M +" 1 % #$ γ −1&' + 2 2γ 2 (γ − ) +$ M − ' + ,+# (γ + 1) (γ + 1)& /+

* γ . γ −1 ,$ (γ − 1) 2 ' , +&1+ M ∞ ) − 1/ ,% 2 ( , - 0 Cp = max γ M 2 2 ∞

• M¥ --> 0 (incompressible flow) " _ $ Cpmax -->1 P = p + q # 0 ∞ % |M =0 43 MAE 5420 - Compressible Fluid Flow Compressible Bernoulli Equation (collected)

44 MAE 5420 - Compressible Fluid Flow Compressible Bernoulli Equation (Hypersonic Limit) What happens at “infinite” mach number?

45 MAE 5420 - Compressible Fluid Flow Compressible Bernoulli Equation (Hypersonic Limit (2) ) Collecting exponents on

46 MAE 5420 - Compressible Fluid Flow Compressible Bernoulli Equation (Hypersonic Limit (3) ) Check .. Substitute g = 1.40

$ 1.4 % " # 1.4 + 1 1 − 1.4 $ $ % % 1.4 & "1.4 − 1# ' = 1.8394 & (1.4 + 1) ' $ 2 % " # " # 2 1.4 − 1

Fundamental property of Hypersonic flow … Mechanical Properties on surface (i.e. pressure) become independent of free stream Mach number

47 MAE 5420 - Compressible Fluid Flow 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

48 MAE 5420 - Compressible Fluid Flow Supersonic Pitot Probe Summary • Max Pressure Coefficient

) " γ % - $ γ −1' + " γ + 1 2 % # & + $ M ' 1 + # 2 ∞ & + Cp 1 max = * " 1 % − . γ 2 M + #$ γ −1&' + " 2γ 2 (γ − 1)% 2 ∞ + M − + $ ∞ ' ,+# (γ + 1) (γ + 1)& /+

• Rayleigh Pitot Equation γ γ −1 ⎛ γ +1 2 ⎞ • Stagnation Pressure Ratio Behind Shock ⋅M P ⎜ 2 ∞ ⎟ 02 γ 2 ⎝ ⎠ 1 M C γ 2 = + ⋅ ∞ ⋅ p = 1 p 2 max P P 1+ ⋅M ⋅C ∞ γ −1 0 0 p ∞ pmax ⎛ 2γ γ −1⎞ 2 2 2 2 ⋅M2 − = = ⎝⎜ γ +1 ∞ γ +1⎠⎟ p p p ⎛ 2⋅γ ⎞ 2 ∞ ∞ 1+ ⋅ M2 −1 ⎝⎜ γ +1⎠⎟ ( ∞ ) • Static Pressure Ratio Across Shock p ⎛ 2⋅γ ⎞ 2 = 1+ ⋅ M2 −1 p ⎜ 1⎟ ( ∞ ) ∞ ⎝ γ + ⎠

49 MAE 5420 - Compressible Fluid Flow Supersonic Pitot Probe Summary (2)

50 MAE 5420 - Compressible Fluid Flow Example Problem • Supersonic Pitot Tube

• p¥ (indicated) ~ 250 kPa P02 ~ 1200 kPa

• Estimate Indicated M¥ Assume g = 1.4 P02

p¥ (indicated)

51 MAE 5420 - Compressible Fluid Flow Example Problem

• Supersonic Pitot Tube .. in terms of P02/P¥ P 1200 02 = = 4.8 • trial and error solution p 250 ∞ 1.4 # 1.4 + 1 2$ (1.4 − 1) ! 1.8282 " = 2 1 2⋅1.4 (1.4 − 1) (1.4 − 1) # 1.82822 − $ ! (1.4 + 1) 1.4 + 1 "

=4.800011-->M¥ ~ 1.8282

52 MAE 5420 - Compressible Fluid Flow Homework 5: (1) • Show That for supersonic free stream conditions

, γ 0 γ −1 . % γ + 1 2 ( . ' M ∞ * P0 − p∞ 1 . & 2 ) . C 2 1 P max = = - 1 − 1 % 1 2 ( γ 2 ⋅ ρ ⋅V M .% 2γ γ − 1( γ −1 . ' ∞ ∞ * 2 ∞ 2 & 2 ) .' M ∞ − * . /.& γ + 1 γ + 1) 2.

• Prove that when …. Mà0 …. CpMax à 1.0 Careful here … subsonic flow when M=0!

53 MAE 5420 - Compressible Fluid Flow Homework 5:(2) • Write iterative Solver for “Rayleigh Pitot Equation” .. Given pressure ratio …. Calculate Mach number

* # γ & . % γ −1( , # γ + 1 2 & $ ' , , % M ∞ ( , Supersonic Flow P02 , $ 2 ' , = + / p # 1 & ∞ ,# 2γ γ − 1 & $% γ −1'( , , M 2 ( ) , % ∞ − ( -,$ (γ + 1) (γ + 1)' 0, γ γ −1 P0 $ γ − 1 2 ' Supersonic Flow = &1+ M ∞ ) p∞ % 2 ( Carefully comment your code .. Hand in code with assignment …. Make program “smart enough” to solve for both supersonic and subsonic free stream conditions 54 MAE 5420 - Compressible Fluid Flow 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

5 2 5 2 a)P 1.22 10 Nt / m → p = 1.01 10 Nt / m 0 y = × x × b)P 7222 lbf / ft 2 → p = 2116 lbf / ft 2 0 y = x

5 2 2 c)P 13107 10 lbf / ft → p = 1020 lbf / ft 0 y = × x

assume...γ = 1.4 55 MAE 5420 - Compressible Fluid Flow Homework 5: (4)

• Verify that CpMax is a continuous curve at Mach 1. i.e. .. Show that When … à1 M ∞ = 1 * γ . γ −1 # γ + 1 2 & , M , 1 , $% 2 ∞ '( , C 1 P max = + 1 − / ( )supersonic γ 2 γ −1 M ∞ ,# 2γ 2 γ − 1& , 2 , M ∞ − , $% γ + 1 γ + 1'( - 0 = M ∞ → 1 γ * γ −1 - ,$ γ − 1 2 ' / &1+ M ∞ ) − 1 ,% 2 ( / C P max = ( )subsonic , γ 2 / , M ∞ / 2 * γ - +, ./ 2 $ γ + 1' γ −1 = ,& ) − 1/ M → 1 γ ,% 2 ( / ∞ + . 56 MAE 5420 - Compressible Fluid Flow Homework 5: (5)

• Key Piece of Knowledge

* # γ & - # 1 & % ( % γ −1( $ γ −1' , # γ + 1 2 & $ ' / 2 * 2 (γ + 1)- M γ M (2M − 1),M / ∂ , $% 2 '( / 2 , / = + . ∂M # 1 & # γ & , $% γ −1'( / $% γ −1'( # 2γ 2 (γ − 1)& # 2γ 2 γ − 1 & , M − / M − ,% ( / % ( +$ (γ + 1) (γ + 1)' . $ (γ + 1) (γ + 1)'

57 MAE 5420 - Compressible Fluid Flow