3241 Lecture 4

# 3241 Lecture 4 - MAE 3241 AERODYNAMICS AND FLIGHT MECHANICS...

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MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS Review: Bernoulli Equation and Examples Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

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LECTURE OUTLINE Review of Euler’s Equation Euler’s equation for incompressible flow → Bernoulli’s Equation Review of Basic Aerodynamics How does an airfoil or wing generate lift? What are effects of viscosity? Why does an airfoil stall? Why are golf balls dimpled?
WHAT DOES EULER’S EQUATION TELL US? 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: Steady flow Neglected friction (inviscid flow), body forces, and external forces dp and dV are of opposite sign IF dp increases dV decreases → flow slows down IF dp decreases dV increases → flow speeds up Valid for Incompressible and Compressible flows Valid for Irrotational and Rotational flows VdV dp ρ - =

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INVISCID FLOW ALONG STREAMLINES 0 2 2 0 0 2 1 2 2 1 2 2 1 2 1 = - + - = + = + V V p p VdV dp VdV dp V V p p ρ ρ ρ Relate p 1 and V 1 at point 1 to p 2 and V 2 at point 2 Integrate Euler’s equation from point 1 to point 2 taking ρ =constant
BERNOULLI’S EQUATION = + + = + 2 2 2 2 2 1 1 2 2 2 V p V p V p ρ ρ ρ If flow is irrotational p+½ ρ V 2 = constant everywhere Remember: Bernoulli’s equation holds only for inviscid (frictionless) and incompressible ( ρ =constant) flows Relates properties between different points along a streamline or entire flow field if irrotational For a compressible flow Euler’s equation must be used ( ρ is a variable) Both Euler’s and Bernoulli’s equations are expressions of F =m a expressed in a useful form for fluid flows and aerodynamics Constant along a streamline

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HOW DOES AN AIRFOIL GENERATE LIFT? Lift is mainly due to imbalance of pressure distribution over the top and bottom surfaces of airfoil If pressure 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 Newton’s 2 nd law
HOW DOES AN AIRFOIL GENERATE LIFT? 1. Flow velocity over the 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 ρ AV=constant, velocity must increase Streamtube A is squashed most in nose region (ahead of maximum thickness) A B

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HOW DOES AN AIRFOIL GENERATE LIFT? 2. As velocity increases pressure decreases Incompressible: Bernoulli’s Equation Compressible: Euler’s Equation Called Bernoulli Effect 2. With lower pressure over upper surface and higher pressure over bottom surface, airfoil feels a net force in upward direction → Lift VdV dp V p ρ ρ - = = + constant 2 1 2 Most of lift is produced in first 20-30% of wing (just downstream of leading edge)
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