MAE 1202: AEROSPACE PRACTICUM Lecture 5: Compressible and Isentropic Flow 1 February 11, 2013 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

READING AND HOMEWORK ASSIGNMENTS Reading: Introduction to Flight, by John D. Anderson, Jr. For this week’s lecture: Chapter 4, Sections 4.10 - 4.21, 4.27 For next week’s lecture: Chapter 5, Sections 5.1 - 5.13 Lecture-Based Homework Assignment: Problems: 4.7, 4.11, 4.18, 4.19, 4.20, 4.23, 4.27 DUE: Friday, February 22, 2013 by 5 PM Problems: 5.2, 5.3, 5.4, 5.6 DUE: Friday, March 1, 2013 by 5 PM Turn in hard copy of homework Also be sure to review and be familiar with textbook examples in Chapter 5

ANSWERS TO LECTURE HOMEWORK 5.2: L = 23.9 lb, D = 0.25 lb, Mc/4 = -2.68 lb ft Note 1: Two sets of lift and moment coefficient data are given for the NACA 1412 airfoil, with and without flap deflection. Make sure to read axis and legend properly, and use only flap retracted data. Note 2: The scale for cm,c/4 is different than that for cl, so be careful when reading the data 5.3: L = 308 N, D = 2.77 N, Mc/4 = - 0.925 N m 5.4: a = 2° 5.6: (L/D)max ~ 112

CREO DESIGN CONTEST Create most elaborate, complex, stunning Aerospace Related project in Creo Criteria: Assembly and/or exploded view First place Either increase your grade by an entire letter (C → B), or Buy your most expensive textbook next semester Second place: +10 points on final exam Third place: +10 points on final exam

CAD DESIGN CONTEST

CAD DESIGN CONTEST

If you do the PRO|E challenge… Do not let it consume you!

Constant along a streamline BERNOULLI’S EQUATION Constant along a streamline One of most fundamental and useful equations in aerospace engineering! Remember: Bernoulli’s equation holds only for inviscid (frictionless) and incompressible (r = constant) flows Bernoulli’s equation relates properties between different points along a streamline For a compressible flow Euler’s equation must be used (r is variable) Both Euler’s and Bernoulli’s equations are expressions of F = ma expressed in a useful form for fluid flows and aerodynamics

EXAMPLE: MEASUREMENT OF AIRSPEED (4.11) How do we measure an airplanes speed in flight? Pitot tubes are used on aircraft as speedometers (point measurement) 13

STATIC VS. TOTAL PRESSURE In aerodynamics, 2 types of pressure: Static and Total (Stagnation) Static Pressure, p Due to random motion of gas molecules Pressure we would feel if moving along with flow Strong function of altitude Total (or Stagnation) Pressure, p0 or pt Property associated with flow motion Total pressure at a given point in flow is the pressure that would exist if flow were slowed down isentropically to zero velocity p0 ≥ p 14

MEASUREMENT OF AIRSPEED: INCOMPRESSIBLE FLOW Static pressure Dynamic pressure Total pressure Incompressible Flow

Total and Static Ports

TOTAL PRESSURE MEASUREMENT (4.11) Measures total pressure Open at A, closed at B Gas stagnated (not moving) anywhere in tube Gas particle moving along streamline C will be isentropically brought to rest at point A, giving total pressure 17

EXAMPLE: MEASUREMENT OF AIRSPEED (4.11) Point A: Static Pressure, p Only random motion of gas is measured Point B: Total Pressure, p0 Flow is isentropically decelerated to zero velocity A combination of p0 and p allows us to measure V1 at a given point Instrument is called a Pitot-static probe p p0 18

MEASUREMENT OF AIRSPEED: INCOMPRESSIBLE FLOW Static pressure Dynamic pressure Total pressure Incompressible Flow 19

TRUE VS. EQUIVALENT AIRSPEED What is value of r? If r is measured in actual air around the airplane Measurement is difficult to do Practically easier to use value at standard seal-level conditions, rs This gives an expression called equivalent airspeed 20

TRAGIC EXAMPLE: Air France Crash Aircraft crashed following an aerodynamic stall caused by inconsistent airspeed sensor readings, disengagement of autopilot, and pilot making nose-up inputs despite stall warnings Reason for faulty readings is unknown, but it is assumed by accident investigators to have been caused by formation of ice inside pitot tubes, depriving airspeed sensors of forward-facing air pressure. Pitot tube blockage has contributed to airliner crashes in the past

HOW DOES AN AIRFOIL GENERATE LIFT? Lift due to imbalance of pressure distribution over top and bottom surfaces of airfoil (or wing) If pressure on top is lower than pressure on bottom surface, lift is generated Why is pressure lower on top surface? We can understand answer from basic physics: Continuity (Mass Conservation) Newton’s 2nd law (Euler or Bernoulli Equation) Lift Force = SPA

HOW DOES AN AIRFOIL GENERATE LIFT? Flow velocity over top of airfoil is faster than over bottom surface Streamtube A senses upper portion of airfoil as an obstruction Streamtube A is squashed to smaller cross-sectional area Mass continuity rAV=constant: IF A↓ THEN V↑ Streamtube A is squashed most in nose region (ahead of maximum thickness) A B

HOW DOES AN AIRFOIL GENERATE LIFT? As V ↑ p↓ Incompressible: Bernoulli’s Equation Compressible: Euler’s Equation Called Bernoulli Effect With lower pressure over upper surface and higher pressure over bottom surface, airfoil feels a net force in upward direction → Lift Most of lift is produced in first 20-30% of wing (just downstream of leading edge) Can you express these ideas in your own words?

Incorrect Lift Theory http://www.grc.nasa.gov/WWW/k-12/airplane/wrong1.html

SUMMARY OF GOVERNING EQUATIONS (4.8) Steady, incompressible flow of an inviscid (frictionless) fluid along a streamline or in a stream tube of varying area Most important variables: p and V T and r are constants throughout flow continuity Bernoulli What if flow is high speed, M > 0.3? What if there are temperature effects? How does density change?

1st LAW OF THERMODYNAMICS (4.5) Boundary System e (J/kg) Surroundings System (gas) composed of molecules moving in random motion Energy of molecular motion is internal energy per unit mass, e, of system Only two ways e can be increased (or decreased): Heat, dq, added to (or removed from) system Work, dw, is done on (or by) system

THOUGHT EXPERIMENT #1 Do not allow size of balloon to change (hold volume constant) Turn on a heat lamp Heat (or q) is added to the system How does e (internal energy per unit mass) inside the balloon change?

THOUGHT EXPERIMENT #2 *You* take balloon and squeeze it down to a small size When volume varies work is done Who did the work on the balloon? How does e (internal energy per unit mass) inside the balloon change? Where did this increased energy come from?

1st LAW OF THERMODYNAMICS (4.5) Boundary e (J/kg) SYSTEM (unit mass of gas) SURROUNDINGS dq System (gas) composed of molecules moving in random motion Energy of all molecular motion is called internal energy per unit mass, e, of system Only two ways e can be increased (or decreased): Heat, dq, added to (or removed from) system Work, dw, is done on (or by) system

1st LAW IN MORE USEFUL FORM (4.5) 1st Law: de = dq + dw Find more useful expression for dw, in terms of p and r (or v = 1/r) When volume varies → work is done Work done on balloon, volume ↓ Work done by balloon, volume ↑ Change in Volume (-)

ENTHALPY: A USEFUL QUANTITY (4.5) Define a new quantity called enthalpy, h: (recall ideal gas law: pv = RT) Differentiate Substitute into 1st law (from previous slide) Another version of 1st law that uses enthalpy, h:

HEAT ADDITION AND SPECIFIC HEAT (4.5) Addition of dq will cause a small change in temperature dT of system dq dT Specific heat is heat added per unit change in temperature of system Different materials have different specific heats Balloon filled with He, N2, Ar, water, lead, uranium, etc… ALSO, for a fixed dq, resulting dT depends on type of process…

SPECIFIC HEAT: CONSTANT PRESSURE Addition of dq will cause a small change in temperature dT of system System pressure remains constant dq dT Extra Credit #1: Show this step

SPECIFIC HEAT: CONSTANT VOLUME Addition of dq will cause a small change in temperature dT of system System volume remains constant dq dT Extra Credit #2: Show this step

HEAT ADDITION AND SPECIFIC HEAT (4.5) Addition of dq will cause a small change in temperature dT of system Specific heat is heat added per unit change in temperature of system However, for a fixed dq, resulting dT depends on type of process: Constant Pressure Constant Volume Specific heat ratio For air, g = 1.4

ISENTROPIC FLOW (4.6) g = ratio of specific heats g = cp/cv gair=1.4 Goal: Relate Thermodynamics to Compressible Flow Adiabatic Process: No heat is added or removed from system dq = 0 Note: Temperature can still change because of changing density Reversible Process: No friction (or other dissipative effects) Isentropic Process: (1) Adiabatic + (2) Reversible (1) No heat exchange + (2) no frictional losses Relevant for compressible flows only Provides important relationships among thermodynamic variables at two different points along a streamline g = ratio of specific heats g = cp/cv gair=1.4

DERIVATION: ENERGY EQUATION (4.7) Energy can neither be created nor destroyed Start with 1st law Adiabatic, dq=0 1st law in terms of enthalpy Recall Euler’s equation Combine Integrate Result: frictionless + adiabatic flow

ENERGY EQUATION SUMMARY (4.7) Energy can neither be created nor destroyed; can only change physical form Same idea as 1st law of thermodynamics Energy equation for frictionless, adiabatic flow (isentropic) h = enthalpy = e+p/r = e+RT h = cpT for an ideal gas Also energy equation for frictionless, adiabatic flow Relates T and V at two different points along a streamline

SUMMARY OF GOVERNING EQUATIONS (4.8) STEADY AND INVISCID FLOW Incompressible flow of fluid along a streamline or in a stream tube of varying area Most important variables: p and V T and r are constants throughout flow continuity Bernoulli continuity Compressible, isentropic (adiabatic and frictionless) flow along a streamline or in a stream tube of varying area T, p, r, and V are all variables isentropic energy equation of state at any point

EXAMPLE: SPEED OF SOUND (4.9) Sound waves travel through air at a finite speed Sound speed (information speed) has an important role in aerodynamics Combine conservation of mass, Euler’s equation and isentropic relations: Speed of sound, a, in a perfect gas depends only on temperature of gas Mach number = flow velocity normalizes by speed of sound If M < 1 flow is subsonic If M = 1 flow is sonic If M > flow is supersonic If M < 0.3 flow may be considered incompressible

KEY TERMS: CAN YOU DEFINE THEM? Streamline Stream tube Steady flow Unsteady flow Viscid flow Inviscid flow Compressible flow Incompressible flow Laminar flow Turbulent flow Constant pressure process Constant volume process Adiabatic Reversible Isentropic Enthalpy

MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW If M > 0.3, flow is compressible (density changes are important) Need to introduce energy equation and isentropic relations cp: specific heat at constant pressure M1=V1/a1 gair=1.4

MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW So, how do we use these results to measure airspeed p0 and p1 give Flight Mach number Mach meter M1=V1/a1 Actual Flight Speed Actual Flight Speed using pressure difference What is T1 and a1? Again use sea-level conditions Ts, as, ps (a1=340.3 m/s)

EXAMPLE: TOTAL TEMPERATURE Static temperature Vehicle flight Mach number A rocket is flying at Mach 6 through a portion of the atmosphere where the static temperature is 200 K What temperature does the nose of the rocket ‘feel’? T0 = 200(1+ 0.2(36)) = 1,640 K!

MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW What can happen in supersonic flows? Supersonic flows (M > 1) are qualitatively and quantitatively different from subsonic flows (M < 1)

HOW AND WHY DOES A SHOCK WAVE FORM? Think of a as ‘information speed’ and M=V/a as ratio of flow speed to information speed If M < 1 information available throughout flow field If M > 1 information confined to some region of flow field

MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW Notice how different this expression is from previous expressions You will learn a lot more about shock wave in compressible flow course

SUMMARY OF AIR SPEED MEASUREMENT Subsonic, incompressible Subsonic, compressible Supersonic

HOW ARE ROCKET NOZZLES SHAPPED?

MORE ON SUPERSONIC FLOWS (4.13) Isentropic flow in a streamtube Differentiate Euler’s Equation Since flow is isentropic a2=dp/dr Area-Velocity Relation

CONSEQUENCES OF AREA-VELOCITY RELATION IF Flow is Subsonic (M < 1) For V to increase (dV positive) area must decrease (dA negative) Note that this is consistent with Euler’s equation for dV and dp IF Flow is Supersonic (M > 1) For V to increase (dV positive) area must increase (dA positive) IF Flow is Sonic (M = 1) M = 1 occurs at a minimum area of cross-section Minimum area is called a throat (dA/A = 0)

2: OUTLET 1: INLET TRENDS: CONTRACTION M1 < 1 M1 > 1 V2 > V1

2: OUTLET 1: INLET TRENDS: EXPANSION M1 < 1 M1 > 1 V2 < V1

PUT IT TOGETHER: C-D NOZZLE 1: INLET 2: OUTLET

MORE ON SUPERSONIC FLOWS (4.13) A converging-diverging, with a minimum area throat, is necessary to produce a supersonic flow from rest Supersonic wind tunnel section Rocket nozzle

SUMMARY OF GOVERNING EQUATIONS (4.8) STEADY AND INVISCID FLOW Incompressible flow of fluid along a streamline or in a stream tube of varying area Most important variables: p and V T and r are constants throughout flow continuity Bernoulli continuity Compressible, isentropic (adiabatic and frictionless) flow along a streamline or in a stream tube of varying area T, p, r, and V are all variables isentropic energy equation of state at any point