MAE 1202: AEROSPACE PRACTICUMLecture 5: Compressible and Isentropic Flow 1

February 11, 2013

Mechanical and Aerospace Engineering DepartmentFlorida Institute of Technology

D. R. Kirk

READING AND HOMEWORK ASSIGNMENTSReading: Introduction to Flight, by John D. Anderson, Jr.For this weeks lecture: Chapter 4, Sections 4.10 - 4.21, 4.27For next weeks 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, 4.27DUE: Friday, February 22, 2013 by 5 PMProblems: 5.2, 5.3, 5.4, 5.6DUE: Friday, March 1, 2013 by 5 PMTurn in hard copy of homeworkAlso be sure to review and be familiar with textbook examples in Chapter 5

ANSWERS TO LECTURE HOMEWORK5.2: L = 23.9 lb, D = 0.25 lb, Mc/4 = -2.68 lb ftNote 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 CONTESTCreate most elaborate, complex, stunning Aerospace Related project in CreoCriteria: Assembly and/or exploded view

First placeEither increase your grade by an entire letter (C B), orBuy your most expensive textbook next semesterSecond place: +10 points on final examThird 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!

BERNOULLIS EQUATIONOne of most fundamental and useful equations in aerospace engineering!Remember:Bernoullis equation holds only for inviscid (frictionless) and incompressible (r = constant) flowsBernoullis equation relates properties between different points along a streamlineFor a compressible flow Eulers equation must be used (r is variable)Both Eulers and Bernoullis equations are expressions of F = ma expressed in a useful form for fluid flows and aerodynamicsConstant along a streamline

*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)

*STATIC VS. TOTAL PRESSUREIn aerodynamics, 2 types of pressure: Static and Total (Stagnation)

Static Pressure, pDue to random motion of gas moleculesPressure we would feel if moving along with flowStrong function of altitude

Total (or Stagnation) Pressure, p0 or ptProperty associated with flow motionTotal pressure at a given point in flow is the pressure that would exist if flow were slowed down isentropically to zero velocity

p0 p

MEASUREMENT OF AIRSPEED: INCOMPRESSIBLE FLOWStaticpressureDynamicpressureTotalpressureIncompressible 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

*EXAMPLE: MEASUREMENT OF AIRSPEED (4.11)Point A: Static Pressure, pOnly random motion of gas is measured

Point B: Total Pressure, p0Flow 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 probep0p

*MEASUREMENT OF AIRSPEED: INCOMPRESSIBLE FLOWStaticpressureDynamicpressureTotalpressureIncompressible Flow

*TRUE VS. EQUIVALENT AIRSPEEDWhat is value of r?

If r is measured in actual air around the airplaneMeasurement is difficult to do

Practically easier to use value at standard seal-level conditions, rs

This gives an expression called equivalent airspeed

TRAGIC EXAMPLE: Air France CrashAircraft crashed following an aerodynamic stall caused by inconsistent airspeed sensor readings, disengagement of autopilot, and pilot making nose-up inputs despite stall warningsReason 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 generatedWhy is pressure lower on top surface?

We can understand answer from basic physics:Continuity (Mass Conservation)Newtons 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 surfaceStreamtube A senses upper portion of airfoil as an obstructionStreamtube A is squashed to smaller cross-sectional areaMass continuity rAV=constant: IF A THEN VStreamtube A is squashedmost in nose region(ahead of maximum thickness) AB

HOW DOES AN AIRFOIL GENERATE LIFT?As V pIncompressible: Bernoullis EquationCompressible: Eulers EquationCalled Bernoulli EffectWith lower pressure over upper surface and higher pressure over bottom surface, airfoil feels a net force in upward direction LiftMost of lift is producedin first 20-30% of wing(just downstream of leading edge)Can you express these ideas in your own words?

Incorrect Lift Theoryhttp://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 areaMost important variables: p and VT and r are constants throughout flowcontinuityBernoulliWhat if flow is high speed, M > 0.3?

What if there are temperature effects?

How does density change?

1st LAW OF THERMODYNAMICS (4.5)System

e (J/kg)BoundarySurroundingsSystem (gas) composed of molecules moving in random motionEnergy 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) systemWork, dw, is done on (or by) system

THOUGHT EXPERIMENT #1Do not allow size of balloon to change (hold volume constant)Turn on a heat lampHeat (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 sizeWhen volume varies work is doneWho 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)System (gas) composed of molecules moving in random motionEnergy 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) systemWork, dw, is done on (or by) systemSYSTEM(unit mass of gas)BoundarySURROUNDINGSdq

1st LAW IN MORE USEFUL FORM (4.5)1st Law: de = dq + dwFind more useful expression for dw, in terms of p and r (or v = 1/r)

When volume varies work is doneWork done on balloon, volume Work done by balloon, volume Change inVolume (-)

ENTHALPY: A USEFUL QUANTITY (4.5)Define a new quantitycalled enthalpy, h:(recall ideal gas law: pv = RT)

Differentiate

Substitute into 1st law(from previous slide)

Another version of 1st lawthat uses enthalpy, h:

HEAT ADDITION AND SPECIFIC HEAT (4.5)Addition of dq will cause a small change in temperature dT of systemSpecific heat is heat added per unit change in temperature of systemDifferent materials have different specific heatsBalloon filled with He, N2, Ar, water, lead, uranium, etcALSO, for a fixed dq, resulting dT depends on type of processdqdT

SPECIFIC HEAT: CONSTANT PRESSUREAddition of dq will cause a small change in temperature dT of systemSystem pressure remains constantdqdTExtra Credit #1:Show this step

SPECIFIC HEAT: CONSTANT VOLUMEAddition of dq will cause a small change in temperature dT of systemSystem volume remains constantdqdTExtra Credit #2:Show this step

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

ISENTROPIC FLOW (4.6)Goal: Relate Thermodynamics to Compressible FlowAdiabatic Process: No heat is added or removed from system dq = 0Note: Temperature can still change because of changing densityReversible Process: No friction (or other dissipative effects)

Isentropic Process: (1) Adiabatic + (2) Reversible(1) No heat exchange + (2) no frictional lossesRelevant for compressible flows onlyProvides important relationships among thermodynamic variables at two different points along a streamlineg = ratio of specific heatsg = cp/cvgair=1.4

DERIVATION: ENERGY EQUATION (4.7)Energy can neither be created nor destroyedStart with 1st law

Adiabatic, dq=01st law in terms of enthalpy

Recall Eulers equation

Combine

Integrate

Result: frictionless + adiabatic flow

ENERGY EQUATION SUMMARY (4.7)Energy can neither be created nor destroyed; can only change physical formSame idea as 1st law of thermodynamicsEnergy equation for frictionless,adiabatic flow (isentropic)

h = enthalpy = e+p/r = e+RTh = cpT for an ideal gas

Also energy equation forfrictionless, adiabatic flow

Relates T and V at two different points along a streamline

SUMMARY OF GOVERNING EQUATIONS (4.8)STEADY AND INVISCID FLOWIncompressible flow of fluid along a streamline or in a stream tube of varying areaMost important variables: p and VT and r are constants throughout flowCompressible, isentropic (adiabatic and frictionless) flow along a streamline or in a stream tube of varying area

T, p, r, and V are all variablescontinuityBernoullicontinuityisentropicenergyequation of stateat any point

EXAMPLE: SPEED OF SOUND (4.9)Sound waves travel through air at a finite speedSound speed (information speed) has an important role in aerodynamicsCombine conservation of mass, Eulers 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 subsonicIf M = 1 flow is sonicIf M > flow is supersonicIf M < 0.3 flow may be considered incompressible

KEY TERMS: CAN YOU DEFINE THEM?StreamlineStream tube

Steady flowUnsteady flow

Viscid flowInviscid flow

Compressible flowIncompressible flow

Laminar flowTurbulent flow

Constant pressure processConstant volume process

Adiabatic

Reversible

Isentropic

Enthalpy

MEASUREMENT OF AIRSPEED:SUBSONIC COMRESSIBLE FLOWIf M > 0.3, flow is compressible (density changes are important)Need to introduce energy equation and isentropic relationscp: specific heat at constant pressureM1=V1/a1gair=1.4

MEASUREMENT OF AIRSPEED:SUBSONIC COMRESSIBLE FLOWSo, how do we use these results to measure airspeedp0 and p1 giveFlight Mach numberMach meterM1=V1/a1Actual Flight SpeedActual Flight Speedusing pressure differenceWhat is T1 and a1?Again use sea-level conditions Ts, as, ps (a1=340.3 m/s)

EXAMPLE: TOTAL TEMPERATUREA 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!Total temperatureStatic temperatureVehicle flightMach number

MEASUREMENT OF AIRSPEED:SUPERSONIC FLOWWhat 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 FLOWNotice how different this expression is from previous expressionsYou 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

Eulers Equation

Since flow is isentropica2=dp/dr

Area-Velocity Relation

CONSEQUENCES OF AREA-VELOCITY RELATIONIF Flow is Subsonic (M < 1)For V to increase (dV positive) area must decrease (dA negative)Note that this is consistent with Eulers 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-sectionMinimum area is called a throat (dA/A = 0)

TRENDS: CONTRACTIONM1 < 1

M1 > 1V2 > V1

V2 < V11: INLET2: OUTLET

TRENDS: EXPANSIONM1 < 1

M1 > 1V2 < V1

V2 > V11: INLET2: OUTLET

PUT IT TOGETHER: C-D NOZZLE1: INLET2: 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 sectionRocket nozzle

SUMMARY OF GOVERNING EQUATIONS (4.8)STEADY AND INVISCID FLOWIncompressible flow of fluid along a streamline or in a stream tube of varying areaMost important variables: p and VT and r are constants throughout flowCompressible, isentropic (adiabatic and frictionless) flow along a streamline or in a stream tube of varying area

T, p, r, and V are all variablescontinuityBernoullicontinuityisentropicenergyequation of stateat any point