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4liftgen

4liftgen - Lift Generation and Streamline Curvature Prof...

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Lift Generation and Streamline Curvature Prof. David L. Darmofal Department of Aeronautics and Astronautics Massachusetts Institute of Technology October 13, 2005 1 Objective The objective of these notes is to explain how an airfoil generates lift using a streamline curvature analysis. The analysis will show in a qualitative manner how the shape of an airfoil influences the pressure distribution, and therefore the lift. 2 Inviscid Momentum in Natural Coordinates The steady, two-dimensional momentum equations in ( x, y ) coordinates is, ρ ~ V · ∇ u = - ∂p ∂x , (1) ρ ~ V · ∇ v = - ∂p ∂y , (2) where ~ V = u ~ i + v ~ j . Instead of an ( x, y ) coordinate system, these equations can be written in a stream-aligned coordinated system ( r, s ). As shown in Figure 1, r is the direction normal to the local streamwise direction and s is the local streamwise direction. In this coordinate system, the inviscid momentum equations are, ρV ∂V ∂s = - ∂p ∂s , (3) ρ V 2 R = ∂p ∂r , (4) R r R s s r ~ V Figure 1: Natural coordinate system ( r, s ). r is the direction normal to the streamwise direction (along the radius of curvature), s is in the streamwise direction (tangent to the velocity vector, ~ V ). R is the radius of curvature. 1

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R p l p u Figure 2: Airfoil with a circular arc camber line with radius R and zero thickness. p u is the upper surface pressure, p l is the lower surface pressure. where V = | ~ V | is the magnitude of the velocity vector (i.e. the speed). For incompressible flow, we note that Equation (3) is equivalent to Bernoulli’s equation, ρV ∂V ∂s = - ∂p ∂s , ∂s 1 2 ρV 2 = - ∂p ∂s , ∂s p + 1 2 ρV 2 = 0 , p + 1 2 ρV 2 = constant . However, the key to understanding lift generation is not Bernoulli’s equation (contrary to popular believe) but rather is the normal momentum equation, Equation (4). 3 Impact of Camber The impact of camber on the pressure distributions can be demonstrated most simply by considering an airfoil with a circular arc camber line and zero thickness as shown in Figure 2. Far away from the airfoil, the pressure returns to the freestream pressure p .
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4liftgen - Lift Generation and Streamline Curvature Prof...

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