MAE 1202: AEROSPACE PRACTICUM Lecture 3: Introduction to Basic Aerodynamics 2 January 28, 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 4.1 - 4.9 For next week’s lecture: Chapter 4, Sections 4.10 - 4.21, 4.27 Lecture-Based Homework Assignment: Problems: 4.1, 4.2, 4.4, 4.5, 4.6, 4.8, 4.11, 4.15, 4.16 DUE: Friday, February 8, 2013 by 11:00 am Turn in hard copy of homework Also be sure to review and be familiar with textbook examples in Chapter 4 Lab this week: Machine shop (remember to dress appropriately, no ‘open-toe’ shoes) Team Challenge #1

ANSWERS TO LECTURE HOMEWORK 4.1: V2 = 1.25 ft/s 4.2: p2-p1 = 22.7 lb/ft2 4.4: V1 = 67 ft/s (or 46 MPH) 4.5: V2 = 102.22 m/s Note: it takes a pressure difference of only 0.02 atm to produce such a high velocity 4.6: V2 = 216.8 ft/s 4.8: Te = 155 K and re = 2.26 kg/m3 Note: you can also verify using equation of state 4.11: Ae = 0.0061 ft2 (or 0.88 in2) 4.15: M∞ = 0.847 4.16: V∞ = 2,283 MPH Notes: Outline problem/strategy clearly – rewrite question and discuss approach Include a brief comment on your answer, especially if different than above Write as neatly as you possibly can If you have any questions come to office hours or consult GSA’s

3 FUNDAMENTAL PRINCIPLES Mass is neither created nor destroyed (mass is conserved) Conservation of Mass Often called Continuity Force = Mass x Acceleration (F = ma) Newton’s Second Law Momentum Equation Bernoulli’s Equation, Euler Equation, Navier-Stokes Equation Energy Is Conserved Energy neither created nor destroyed; can only change physical form Energy Equation (1st Law of Thermodynamics) How do we express these statements mathematically?

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

CONSERVATION OF MASS (4.1) Physical Principle: Mass can be neither created nor destroyed Stream tube Funnel wall A2 A1 V1 V2 As long as flow is steady, mass that flows through cross section at point 1 (at entrance) must be same as mass that flows through point 2 (at exit) Flow cannot enter or leave any other way (definition of a stream tube) Also applies to solid surfaces, pipe, funnel, wind tunnels, airplane engine “What goes in one side must come out the other side”

CONSERVATION OF MASS (4.1) Stream tube A1: cross-sectional area of stream tube at 1 V1: flow velocity normal (perpendicular) to A1 Consider all fluid elements in plane A1 During time dt, elements have moved V1dt and swept out volume A1V1dt Mass of fluid swept through A1 during dt: dm=r1(A1V1dt)

SIMPLE EXAMPLE Given air flow through converging nozzle, what is exit velocity, V2? p1 = 1.2x105 N/m2 T1 = 330 K V1 = 10 m/s A1 = 5 m2 p2 = ? T2 = ? V2 = ? m/s A 2= 1.67 m2 IF flow speed < 100 m/s assume flow is incompressible (r1=r2) Conservation of mass could also give velocity, A2, if V2 was known Conservation of mass tells us nothing about p2, T2, etc.

INVISCID MOMENTUM EQUATION (4.3) Physical Principle: Newton’s Second Law How to apply F = ma for air flows? Lots of derivation coming up… Derivation looks nasty… final result is very easy is use… What we will end up with is a relation between pressure and velocity Differences in pressure from one point to another in a flow create forces

APPLYING NEWTON’S SECOND LAW FOR FLOWS x y z Consider a small fluid element moving along a streamline Element is moving in x-direction V dy dz dx What forces act on this element? Pressure (force x area) acting in normal direction on all six faces Frictional shear acting tangentially on all six faces (neglect for now) Gravity acting on all mass inside element (neglect for now) Note on pressure: Always acts inward and varies from point to point in a flow

APPLYING NEWTON’S SECOND LAW FOR FLOWS x y z p (N/m2) dy dz dx Area of left face: dydz Force on left face: p(dydz) Note that P(dydz) = N/m2(m2)=N Forces is in positive x-direction

APPLYING NEWTON’S SECOND LAW FOR FLOWS x y z Pressure varies from point to point in a flow There is a change in pressure per unit length, dp/dx p+(dp/dx)dx (N/m2) p (N/m2) dy dz dx Area of left face: dydz Force on left face: p(dydz) Forces is in positive x-direction Change in pressure per length: dp/dx Change in pressure along dx is (dp/dx)dx Force on right face: [p+(dp/dx)dx](dydz) Forces acts in negative x-direction

APPLYING NEWTON’S SECOND LAW FOR FLOWS x y z p (N/m2) p+(dp/dx)dx (N/m2) dy dz dx Net Force is sum of left and right sides Net Force on element due to pressure

APPLYING NEWTON’S SECOND LAW FOR FLOWS Now put this into F=ma First, identify mass of element Next, write acceleration, a, as (to get rid of time variable)

SUMMARY: EULER’S EQUATION Euler’s Equation (Differential Equation) Relates changes in momentum to changes in force (momentum equation) Relates a change in pressure (dp) to a chance in velocity (dV) Assumptions we made: Neglected friction (inviscid flow) Neglected gravity Assumed that flow is steady

WHAT DOES EULER’S EQUATION TELL US? Notice that dp and dV are of opposite sign: dp = -rVdV IF dp ↑ Increased pressure on right side of element relative to left side dV ↓

WHAT DOES EULER’S EQUATION TELL US? Notice that dp and dV are of opposite sign: dp = -rVdV IF dp ↑ Increased pressure on right side of element relative to left side dV ↓ (flow slows down) IF dp ↓ Decreased pressure on right side of element relative to left side dV ↑ (flow speeds up) Euler’s Equation is true for Incompressible and Compressible flows

INVISCID FLOW ALONG STREAMLINES 2 1 Points 1 and 2 are on same streamline! Relate p1 and V1 at point 1 to p2 and V2 at point 2 Integrate Euler’s equation from point 1 to point 2 taking r = constant

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

WHEN AND WHEN NOT TO APPLY BERNOULLI YES NO

SIMPLE EXAMPLE Given air flow through converging nozzle, what is exit pressure, p2? p1 = 1.2x105 N/m2 T1 = 330 K V1 = 10 m/s A1 = 5 m2 p2 = ? T2 = 330 K V2 = 30 m/s A2 = 1.67 m2 Since flow speed < 100 m/s assume flow is incompressible (r1=r2) Since velocity is increasing along flow, it is an accelerating flow Notice that even with a 3-fold increase in velocity pressure decreases by only about 0.8 %, which is characteristic of low velocity flow

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 = PA

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?

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 What if flow is high speed, M > 0.3? What if there are temperature effects? How does density change?

ONLINE REFERENCES http://www.aircraftenginedesign.com/enginepics.html http://www.pratt-whitney.com/ http://www.geae.com/ http://www.geae.com/education/engines101/ http://www.ueet.nasa.gov/StudentSite/engines.html http://www.aeromuseum.org/Education/Lessons/HowPlaneFly/HowPlaneFly.html http://www.nasm.si.edu/exhibitions/gal109/NEWHTF/HTF532.HTM http://www.aircav.com/histturb.html http://inventors.about.com/library/inventors/bljjetenginehistory.htm http://inventors.about.com/library/inventors/blenginegasturbine.htm http://www.gas-turbines.com/primer/primer.htm