Fluids – Lecture 20 Notes 1. Airfoils – Detailed Look Reading: http://www.av8n.com/how/htm/ Sections 3.1–3.3 (optional) Airfoils – Detailed Look Flow curvature and pressure gradients The pressures acting on an airf oil are determined by the airfoil’s overall shape and the angle of attack. However, it’s useful t o examine how local pressures are approximately affected by local geometry, and the surface curvature in particular. Consider a location near the airfoil surface, ignoring the thin boundary layer. The local flow speed is V , the local streamline radius of curvature is R. Another equivalent way to define the curvature is κ = 1/R = dθ/ds, where θ is the inclination angle of the surface or streamline, and s is the arc length. Positive κ is defined to be concave up as shown. s n θ R =κ−1 u v x y V V θ V local cartesian xy axes 6 6n p To determine how the pressure varies normal to the surface, we align local xy axes t angent and normal to the surface, and employ the y-momentum equation, with the viscous forces neglected. ∂p ∂v ∂v = −ρu − ρv ∂y ∂x ∂y The Cartesian velocity components are related to the speed and the surface angle as follows. u = V cos θ , v = V sin θ At the or ig in where θ = 0, we then have ∂v ∂V ∂θ v = 0 , u = V , = sin θ + V cos θ = V κ ∂x ∂x ∂x Therefore, the normal pressure g r adient along y = n is ∂p = −ρ V2κ ∂n This is the normal-momentum equation, sometimes also called the centrifugal formula. It describes the physical requirement that there must be a transverse pressure gradient to force fluid to flow alo ng a curved streamline. It is valid for inviscid flows, at any Mach number. 1Implications for surface pressures Because of the influence on normal pressure gradients, changes in surface curvature are expected to cause changes in surface pressure. If a common reference pressure exists away from the wall, a concave corvature will produce a higher pressure towards the wall, while a convex curvature will produce a lower pressure towards the wall. The figure below illustrates the situation for a simple bump. The “+” and “-” symbols indicate exp ected changes in pressure. nnn p 66np p p −+ + >0 66np66np< 0 < 0 The curvature/pressure-gradient relation also indicates the pressures which can be expected on the surface of a body such as an airfoil. Examination of the streamline curvatures indicates that for a symmetric airfoil at zero angle of attack, higher pressure is expected at the leading and trailing edges, while lower pressure is expected a lo ng the sides. − + + − − − Fo r the same symmetrical airfoil at an angle of attack, the streamline pattern and the pressures near the leading edge are now considerably different. The stagnation point moves under the leading edge, and a strongly reduced pressure, called a “leading edge spike”, forms at the leading edge point itself. − + − ++ − 2The actual surface pressure force vectors −Cpˆn ar e shown for the NACA 0015 airfoil, atα = 0◦(cℓ= 0 ) , and α = 10◦(cℓ= 1.23). The Cp(x) distributions are also shown plotted.These results were computed using a panel method, and therefore correspond to inviscidirrotational incompressible flow. The drag is predicted to be zero (d’Alembert’s Paradox),and the possibility of boundary layer separation is ignored. Despite these limitations, thecalculations are useful in that they simply reveal the intense pressure spike, which is knownto promote separation of the upper surface boundary layer, and thus degrades the airfoil’sstall resistance. A corrective redesign of the airfoil would normally be undertaken if theleading edge spike is deemed to be too strong.Use of camberAn effective way to reduce the intensity of the leading edge spike is to add camber to theairfoil. The NACA 4415 airfoil has the same 15% maximum thickness (relative to chord) asthe 0015, but it has a nonzero 4 % maximum camber. The figures below show the camberedNACA 4415 airfoil at the same same cℓ= 0 and cℓ= 1.23 as in the NACA 0015 case(comparing at the same lift or cℓis more meaningful than comparing at the same α).3The leading edge spike at the high angle of att ack is indeed reduced considerably. The lowangle of attack case now has a “negative” spike on the bottom surface, but this is muchweaker and appears tolerable.Although camber is seen to be attractive in the case above, too much camber is usuallydetrimental. The figure below shows the NACA 84 15 at the same cℓ= 0 and cℓ= 1 .2 3 con-ditions. The intense spike on the bottom surface shows the drawback of using the excessive8% camber – the low cℓ(high speed) condition is likely t o have excessive drag. Selection ofthe ideal amount of camber is a major desig n choice for the airplane
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