In level flight, lift equals weight W and thrust equals drag, so The forces involved are obtained from the coefficients by multiplication with (ρ/2).S V 2, where ρ is the density of the atmosphere at the flight altitude, S is the wing area and V is the speed. One example of the way the polar is used in the design process is the calculation of the power required ( P R) curve, which plots the power needed for steady, level flight over the operating speed range. The effect of C L0 is to lift the polar curve upwards physically this is caused by some vertical asymmetry, such as a cambered wing or a finite angle of incidence, which ensures the minimum drag attitude produces lift and increases the maximum lift to drag ratio. ![]() If so, then C D = C D0 + K.( C L - C L0) 2. The other drag mechanisms, parasitic and wave drag, have both constant components, totalling C D0 say, and C L dependent contributions that are often assumed to increase as C L 2. Prandtl's lifting line theoretical work shows that this increases as C L 2. One component here is the induced drag of the wing, an unavoidable companion of the wing's lift, though one that can be reduced by increasing the aspect ratio. Such diagrams identify a minimum C D point at the left-most point on the plot, where the drag is locally independent of lift to the right, the drag is lift related. The accompanying diagram shows a drag polar for a typical light aircraft. A particular aircraft may have different polar plots even at the same R e and M values, depending for example on whether undercarriage and flaps are deployed. During the evolution of the design the drag polar will be refined. The design of a fighter will involve a set at different Mach numbers, whereas gliders, which spend their time either flying slowly in thermals or rapidly between them may require polars at different Reynolds numbers but are unaffected by compressibility effects. īecause of the Reynolds and Mach number dependence of the coefficients, families of drag polars may be plotted together. Eiffel was the first to use the name drag polar. Lift and drag data was gathered in this way in the 1880s by Otto Lilienthal and around 1910 by Gustav Eiffel, though not presented in terms of the more recent coefficients. When this measurement is repeated at different angles of attack the drag polar is obtained. If, in a wind tunnel or whirling arm system an aerodynamic surface is held at a fixed angle of attack and both the magnitude and direction of the resulting force measured, they can be plotted using polar coordinates. As L and D are at right angles, the latter parallel to the free stream velocity or relative velocity of the surrounding, distant, air, the resultant force R lies at the same angle to that direction as the line from the origin of the polar plot to the corresponding C L, C D point does to the C D axis. Since the lift and the drag forces, L and D are scaled by the same factor to get C L and C D, L/ D = C L/ C D. The drag polar of an aircraft contains almost all the information required to analyse its performance and hence to begin a design. ![]() Similar plots can be made for other components or for whole aircraft in all cases they are referred to as drag polars. C L and C D are often presented individually, plotted against α, but an alternative graph plots C L as a function of C D, using α parametrically. Like other such aerodynamic quantities, they are functions only of the angle of attack α, the Reynolds number R e and the Mach number M. The significant aerodynamic properties of aircraft wings are summarised by two dimensionless quantities, the lift and drag coefficients C L and C D. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.Drag polar for the Nairfoil, colour-coded as opposite plot. In fluid dynamics, Stokes' law is an empirical law for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Not to be confused with Stokes' theorem in vector calculus, or Stokes shift in luminescence and Raman spectroscopy.
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