Incompressible, Compressible, and Supersonic Flow Fields ...

1 MAE 5420 -Compressible Fluid Flow1 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 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 5420 -Compressible Fluid Flow2 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.

2 Freestream Pressure = Ambient pressure for atmospheric flight When the fluid comes in proximity to a body, its pressure deviates from the freestreamvalue and strictly-speakingshould no longer be referred to as static pressure .. quantity should becalled simply pressure .MAE 5420 -Compressible Fluid Flow3 Incompressible, Compressible, and Supersonic Flow Fields : Static, Dynamic, and Total Pressure (3) However, to distinguish this local surface pressure from localTotal 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 Flow4 Incompressible Bernoulli Equation (1) Consider Incompressible Flow CaseMAE 5420 -Compressible Fluid Flow5 Incompressible Bernoulli Equation (2)IncompressibleBernoulli EquationFictitious but useful quantity known asIncompressible Dynamic Pressure, qbar MAE 5420 -Compressible Fluid Flow6 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?

3 P+12 V2=const= PstagnationPstagnation=pstatic 1+ 12M2$%&'() 1 MAE 5420 -Compressible Fluid Flow7 Compressible Bernoulli Equation (subsonic flow) (1) Adding and subtracting pto the equationCompressible form of Bernoulli EquationP0=p+p1+ 1()2M2#$%&'( 1 1)*+,+-.+/+ p+qc qc p1+ 1()2M2#$%&'( 1 1)*+,+-.+/+CompressibleDynamic PressureMAE 5420 -Compressible Fluid Flow8 Compressible Bernoulli Equation (subsonic flow) (2)Fictitious quantity known asDynamic Pressure, qbar Compressible Dynamic Pressure or impact Pressure Incompressible Flow Compressible Flow (Subsonic) MAE 5420 -Compressible Fluid FlowThe Compressible Bernoulli Equation Consider (M < 1) Compressible Flow Through an orifice Entering Flow is IsentropicSubcritical Flow:pP0 >2 +1 1!m=Cd A2 1 P0 0 pP0 2 1 pP0 1 Define r=pP0 Kn=2 1 r2 1 r 1 !

4 M=Kn Cd A P0 0P0p=1+ 12 M2 1 P0p 1 =1+ 12 M2 =1+ 12 V2 RgT A2 MAE 5420 -Compressible Fluid FlowThe Compressible Bernoulli EquationP0p=1+ 12 M2 1 P0p 1 =1+ 12 M2 =1+ 12 V2 RgT P0p 1 =T0T substitute T0T=1+ 12 V2 RgT Solve 12V2= 1 RgT0 1 RgTSubstitute Gas Law p =RgTV22+ 1 p = 1P0 0 "Compressible Bernoulli Equation"Since Entering Flow is IsentropicMAE 5420 -Compressible Fluid FlowCompare to Incompressible Bernoulli"Compressible Bernoulli Equation"V22+ 1 p = 1P0 0 "Compressible Bernoulli Equation"P0=p+12 V2At Mach zero, Compressible Bernoulli Reduces to Incompressible Equation From Definition of Speed of SoundSonic Velocity c= Rg T= p s=0 Incompressible Flow =0 cincr= lim V22+ 1p = 1P0 =V22+p =P0 Solve for P0P0=p+12 V2!

5 MAE 5420 -Compressible Fluid FlowApplications of Bernoulli: The Venturi Massflow Equations Consider Flow Through a Venturi ShapeGiovanni Battista Ve n t u r i(1796)MAE 5420 -Compressible Fluid FlowIncompressible Venturi (1)13massflowA2 Bernoulli's Law for Incompressible Flow ( =constant) p1+12 V12=const=p2+12 V22 Continuity (Conservation of Mass) A1 V1=!m= A2 V2 V1=A2A1 V2 Solve for pressure differetial in terms of V2 p1 p2=12 V22 V12()=12 V22 A2A1 V2 2 First look at Incompressible Ve n t u r i w h e r e d e n s i t y i s constant throughout theFlow 5420 -Compressible Fluid FlowIncompressible Venturi (2)14massflowA2 Solve for V2 V2=2 p1 p2 1 A2A1 2 Calculate Massflow !m= A2 V2=A21 A2A1 22 p1 p2()Account for Friction Losses with discharge coefficient"fudge factor" !m=Cd A21 A2A1 2 2 p1 p2()MAE 5420 -Compressible Fluid FlowIncompressible Venturi (3)15massflowA2 Solve for V2 V2=2 p1 p2 1 A2A1 2 Calculate Massflow !

6 M= A2 V2=A21 A2A1 22 p1 p2()Account for Friction Losses with discharge coefficient"fudge factor" !m=Cd A21 A2A1 2 2 p1 p2()"Flow Coefficient"Cv Cd A21 A2A1 2 !m=Cv 2 p P=P1 P2 MAE 5420 -Compressible Fluid FlowCompressible Venturi (1)16massflowA2 Now look at subcritical Compressible Venturi Flow where density is not constant throughout the Flow 1 D Compressible Massflow Equation:!m=A1 P0 Rg T0 M11+ 12M12 12 +1 1 Venturi Inlet!m=A2 P0 Rg T0 M21+ 12M22 12 +1 1 Venturi ThroatIsentropic Flow P0p1 1 =1+ 12M12 P0p2 1 =1+ 12M22 M1=2 1P0p1 1 1 M2=2 1P0p2 1 1 P0 MAE 5420 -Compressible Fluid FlowCompressible Venturi (2)17massflowA2 Substitute and SimplifyP0 MAE 5420 -Compressible Fluid FlowCompressible Venturi (3)18massflowA2 But What is P0?

7 !m=A1 2 1 P02Rg T0p1P0 2 p1P0 +1 and!m=A2 2 1 P02Rg T0p2P0 2 p2P0 +1 P0 MAE 5420 -Compressible Fluid FlowCompressible Venturi (4)19massflowA2 Since !mP0 Rg T0 is constantA1 M11+ 12M12 12 +1 1 =A2 M21+ 12M22 12 +1 1 M1=2 1P0p1 1 1 M2=2 1P0p2 1 1 SubstituteA1A2=M2M1 P0p1 1 12 +1 1 P0p2 1 12 +1 1 =M2M1 p2p1 +12 =2 1P0p2 1 1 2 1P0p1 1 1 p2p1 +12 P0 MAE 5420 -Compressible Fluid FlowCompressible Venturi (5)20massflowA2 SOLVE for P0 A1A2 2 p1()2 P0() 1 p1() 1 =p2()2 P0() 1 p2() 1 P0() 1 A1A2 2 p1()2 p2()2 =A1A2 2 p1()2 p1() 1 p2()2 p2() 1 P0 P0=A12 p1() +1 A22 p2() +1 A12 p1()2 A22 p2()2 1=A1A2 2 p1() +1 p2() +1 A1A2 2 p1()2 p2()2 1 MAE 5420 -Compressible Fluid FlowCompressible Venturi (6)

8 21massflowA2 Allow for Frictional LossesP0P0=A1A2 2 p1() +1 p2() +1 A1A2 2 p1()2 p2()2 1!m=Cd A1 2 1 0 P0p1P0 2 p1P0 +1 with friction losses discharge coefficient(1) Venturi Inlet(2) Venturi Throat(3) Venturi Outlet Cd~1 Pinlet PoutletPinlet=Pinlet+PoutletPinletp1 Venturi Inlet Pressurep1 Venturi Inlet PressureP0 Venturi Stagnation PressureT0 Venturi Block TemperatureCd A1 Venturi InletDischarge AreaMAE 5420 -Compressible Fluid FlowCompressible Venturi (7)22massflowA2 Supercritical Flow(Choked Throat)P0>!m=Cd A2* P0T0 Rg 2 +1 +1 1P0=p2* +12 1 MAE 5420 -Compressible Fluid Flow23 How do we measure stagnationpressure? (1)transducerFlow Probe Looks into flow Captures or stagnates Incoming flow .. nearly transducer senses total pressure (included effects of flow kinetic energy)V ~ ProbeMAE 5420 -Compressible Fluid Flow24 How do we measure normal or staticpressure?

9 TransducerVVFreestreamDuct Port is perpendiculartoThe flow field ..Sensed pressure does not capture effect of externalflow kinetic SourcePort or pressure tapMAE 5420 -Compressible Fluid Flow25 How do we Measure Airspeed or Mach number? (1) Senses Both Total and StaticPressure from incoming flow field Incompressible .. density typically based on sea level number .. probe gives indicated airspeed .. Must be localized density effectsMAE 5420 -Compressible Fluid Flow26 How do we Measure Airspeed or Mach number? (2) Senses Both Total and StaticPressure from incoming flow field Compressible, Subsonic .. Typically Probe static pressure is influenced by theAircraft and indicated Machnumber .. must be localized vehicle effectsMAE 5420 -Compressible Fluid Flow27 Pitot / Static Probe Details (1) White --Total or Stagnation Pressure Yellow -- normal or Static PressurerrMAE 5420 -Compressible Fluid Flow28 Pitot / Static Probe Details (2)Flow Direction VanePitot ProbeStatic Source HolesLooking along flow directionAirliners have pitot/static probes protruding from the cockpit area for measuring airspeedFlow directionMAE 5420 -Compressible Fluid FlowTypical Small Aircraft Airspeed SystemMAE 5420 -Compressible Fluid FlowUS Military Airspeed Definitions Indicated Airspeed (IAS) As shown on the airspeed indicator.

10 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) Tr u e A i r s p e e d CAS corrected for temperature and pressure (actual speed through airstream)MAE 5420 -Compressible Fluid FlowIndicated Airspeed (IAS)Standard Formula:MAE 5420 -Compressible Fluid FlowIndicated Airspeed (Derivation) Airspeed as shown on the airspeed indicator calibrated to reflected standard atmosphere, adiabatic compressible flow with no static sensor position error calibrationsMAE 5420 -Compressible Fluid FlowIndicated Airspeed (Derivation) (2) Since Typical Airspeed system only measures impact pressure and not static pressure pinddirectly.