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Touchard 3 o laminar transition turbulent x zone zone zone U∞ C. Boundary layer We are now going to examine the effect of the airflow around a body to understand better the need of aerodynamic actuators. A vehicle in motion is submitted to the friction of the air around its body. This is due to the attachment of the air molecules at the interface air/body which induces a null relative velocity of the air at this interface, then, the viscosity of the air produces the so called "boundary layer" all around the body in motion. In this layer the relative velocity of air around the body evolves from zero at the body wall, to a constant velocity far from the body. This relative external velocity is equal and opposite from the body velocity. We first are going to examine the boundary layer development on a flat plate. D. Boundary layer on a flat plate We consider a flat plate and a uniform flow velocity U∞ before the plate (figure 8). Then from the beginning of the plate (generally called the leading edge) a laminar boundary layer develops. The width of this layer is increasing with the distance to the leading edge. At the limit of the boundary layer the velocity is U∞ . (in practice, the limit is defined when the velocity is 0.99 U∞ ). The pressure is more important upstream that downstream, we say that the pressure gradient is negative. The velocity profile depends on the flow regime. Indeed two flow regimes exist: the laminar flow and the turbulent flow with a transition between laminar to turbulent. The profile given in figure 8 corresponds to a laminar flow. Fig. 9. Transition from laminar to turbulent on a flat plate. E. Separation In the case of an airfoil, due to the curvature of the wall an adverse pressure gradient appears. When the pressure gradient becomes positive, then separation takes place. In other words, the velocity gradient perpendicular to the plate decreases and when it becomes null the boundary layer detach the plate. This is the point of separation. Then instabilities appear and a back flow exists, this is the separation zone (figure 10). The existence of such zone on an airfoil induces a lift reduction and an increasing of the drag. This can be easily observed on the experiments made by Prandtl (reported by Schlichting [2]) concerning a flow around a typical airfoil for two different inclinations (figure 10). At small incidence angles (up to about 10°) the flow does not separate on either side (left picture in figure 11). Increasing the incidence the adverse pressure gradient on the upper side of the airfoil becomes larger. For a given incidence angle (about 15°) separation occurs (right picture in figure 11). The separation point is located close behind the leading edge. As a consequence the lift is widely decreased and the drag increased (figure 12). In this figure CL and CD represent the lift and the drag coefficients, these two parameters are defined by the following equations: (5) (6) U∞ U∞ U∞ U∞ boundary layer leading edge Fig. 8. Laminar boundary layer development on a flat plate. At a given distance from the leading edge the transition from laminar to turbulent appears. This transition to turbulence is clearly discernible by a sudden and large increase of the boundary layer thickness (figure 9). Considering the Reynolds number , (where U∞ is the external velocity, x the distance from the leading edge and ν the air kinematic viscosity), then the transition on a flat plate appears for the so called critical value of the Reynolds number R c ≈ 3.2 10 5 . In the case of an airfoil the same phenomenon occurs with roughly the same characteristics. Where L is the lift, D the drag, ρ the air density, U the air stream velocity and A a characteristic area, e.g. the frontal area exposed by the body to the flow direction. In extreme conditions these consequences may be responsible for airplane stalling and must obviously be avoided, but, even in "normal" conditions, the separation can be important, for instance during takeoff, and induces a very bad efficiency of the airplane. Thus, delaying separation on a wing plane is one of the most important challenges to aeronautics for many years. CL= L 1/2ρU2 A ∞ CD= D 1/2ρU2 A ∞ ∞ R = U∞x νPDF Image | Plasma actuators for aeronautics applications
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