� Fluids – Lecture 19 Notes 1. Airfoils – Overview Reading: Anderson 4.1–4.3 Airfoils – Overview 3-D wing context The cross-sectional shape of a wing or other streamlined surface is called an airfoil. The importance of this shape arises when we attempt to model or approximate the flow about the 3-D surface as a collection of 2-D flows in the cross-sectional planes. zy V 3−D Wing 2−D Airfoil section flows L L’ Γ x In each such 2-D plane, the airfoil is the a erodynamic body shape of interest. 2-D section properties become f unctions of the spanwise coordinate y. Examples are L′ (y), Γ(y), etc. Quantities of interest for the whole wing cn then be obtained by integrating over all the sectional flows. For example, b/2 ′ L = L (y) dy −b/2 where b is the wing span. The airf oil shape is therefore an important item of interest, since it is key in defining the individual section flows. It must be stressed that the 2-D section flows are not completely independent, but rather they influence each other’s effective a ngle of attack, or the apparent ~V∞ direction in each 2-D plane. Fortunately this complication does not prevent us from treating each 2-D plane as though it was truly independent, since the angle of attack corrections can be added separately later. Nomenclature The figure below shows the key terms used when dealing with airfoil geometry. The Mean Camber Line is defined to lie halfway between the upper and lower surfaces. Chord c Chord Line Maximum Thickness Maximum Camber Mean Camber Line Trailing Edge Leading Edge 1Aerodynamic Characterization V∞ is defined by its magnitude V∞ = |~The freestream velocity vector ~V∞|, and the angle of Rattack α it makes with the airfoil’s chord line. The overall aerodynamic loads on the airfoil ′are the resultant force/span vector ~and the moment/span M′ , by convention taken about Dthe quarter-chord location. The resultant force is resolved into a lift force L′ and drag force ′ perpendicular and parallel to ~V∞. c/4α M’ D’ L’ R’ V The forces and moment are more conveniently nondimensionalized using the freestream dy-1namic pressure q∞ ≡ 2 ρ∞V2 and the chord c, giving the lift, drag, and moment coefficients. ∞ cL′ D′ M′ ℓ ≡ , cd ≡ , cm ≡ q∞ c q∞ c q∞ c2 Dimensional analysis reveals that these will depend o nly on the angle of attack α, the Reynolds number Re ≡ ρ∞V∞c/µ∞, the Mach number M∞ ≡ V∞/a∞, and on the airfoil shape. cℓ , cd , cm = f ( α , Re , M∞ , airfoil shape ) For low speed flows, M∞ has virtually no effect. And for a given airfoil shape, we therefore have cℓ , cd , cm = f ( α , Re ) (low speed flow, g iven airfoil) Typical cℓ(α) and cm(α curves for any given Re have a number of important features, as shown in the figure. For moderate angles of attack, the cℓ(α) curve is nearly linear, and very closely matches the one predicted by potential-flow theory (e.g. a panel method). At some larger angle of attack, cℓ curve reaches a maximum value of cℓmax and then decreases. For α’s beyond cℓmax the airfoil is said to be stalled, and exhibits varying amounts of separated flow. An analogous situation occurs for large negative α ’s. Within the linear region, the cℓ(α) curve can be closely approximated with a linear fit. cℓ(α) = a0 (α − αL=0) (away from stall) Here, a0 is the lift-curve slope, and α is the zero-lift angle. These can be measured or computed reasonably accurately with a potential-flow method. L=0 2cl cm a = d c d αl 0 L=0α cl max potential−flow prediction αα The moment coefficient cm(α), when defined about the quarter-chord point, is very nearly constant away from stall. Again this is predicted well by potential-flow methods. Past stall, the cm(α) curve deviates sharply from its constant value. The drag coefficient cd can be plotted versus α, as shown in the figure on the left. However, a more useful and more standard way is to plot cℓ vs cd, with α simply a dummy parameter along the curve. This plot is called a drag polar, and is shown in the figure on the right. cd cl c mind cl max low drag range α c c l d max α c mind cd One reason for using t he drag polar format is that when evaluating the aerodynamic per-formance of an airfoil, the α values are not really relevant. All that matters is the drag and how it compares to lift. The drag polar format compares these directly, and hence summarizes the most important f eatures of the airfoil’s drag characteristics in one plot. One such feature is the maximum lift-to-drag ratio, or (cℓ/cd)max, which is where a line from the origin lies tangent to the polar curve. The cℓmax and cdmin values are a lso directly visible. An aerodynamicist might also note the low-drag range of lift coefficients where the airfoil naturally wants t o operate. It must be stressed that cd values are roughly 100 times smaller than typical maximum cℓ values. Hence, the cd axis on a polar plot is greatly enlarged. 3A sample polar plot and cℓ(α; Re) and cm(α; Re) curves for an actual sailplane airfoil are shown below, fo r two different Reynolds numbers.
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