The effect of back spin on a table tennis ball moving in a viscous fluid.

Transcription

1 How can planes fly? The phenomenon of lift can be produced in an ideal (non-viscous) fluid by the addition of a free vortex (circulation) around a cylinder in a rectilinear flow stream. This is known as the Magnus effect. It was mentioned by Newton in 1672 and was investigated experimentally by Magnus in In a real (viscous) fluid, this effect may be produced by a tennis ball, for example, by making it spin as it travels through the air. Because the relative velocity between the air and the ball is zero at the surface of the ball, the spin of the ball produces a circulation approximating a free vortex outside the boundary layer. A top spin produces a downward force, and a back spin upward force. Spin about a vertical axis produces a sideward force, known as a "hook" or "slice" in golf. In the case of both the ideal fluid and the real fluid, a circulation is necessary for the production of a lift force. Fig. 1 The effect of back spin on a table tennis ball moving in a viscous fluid. Lift is expressed as the product of a lift coefficient, the dynamic pressure of the free stream, and the chord area of the lifting vane. The lift coefficient depends on a number of parameters, including the shape and angle of attack of the vane, Reynolds number, Mach number, aspect ratio, etc. The lift force is generally defined by the equation F Lift = C L ½ ρ v s 2 A where C L is the lift coefficient; (ρ v s 2 /2) is the dynamic pressure of the free stream; and A is the chord area of the lifting vane. Fig. 2 shows the flow lines past an airfoil at a given angle of attack α, with and without circulation. The circulation required to swing the trailing stagnation streamline A tangent to the trailing edge of the airfoil is the desired quantity. The addition of circulation results in a higher velocity and lower pressure over the upper surface and a lower velocity and higher pressure over the lower surface airfoil. a03/p1/fluids/flight.doc 1

2 A simple model relates the lift coefficient to the angle of attack for an airfoil in an ideal fluid when the airfoil thickness and camber approach zero (flat plate) C L = 2 π sinα o for small angles of attack. The angle α o is the difference between the actual angle of attack and the angle of attack for which the lift is zero. For a flat plate and symmetrical airfoils without 'camber (curvature), α o is the same as α, since lift is zero for zero angle of attack. F lift A a b Fig. 2 Ideal fluid passing an airfoil. (a) Without circulation. No lift or drag. (b) With circulation lift but no drag. Circulation added to flow pattern in (a) to produce flow pattern in (b). Thus, ideal flow theory predicts actual lift performance amazingly well but gives zero drag in an infinite fluid for steady flow. An airfoil in a real fluid must create its own circulation, or vortex field, just as the spinning table tennis ball, in order to experience lift. The starting vortex is indicated in Fig. 3. As the motion begins, it is very slow and approaches that of irrotational flow in an ideal fluid. The fluid particles passing around the trailing edge must move very rapidly and must approach a stagnation condition at A. But because of the viscosity of the fluid, they have less velocity than if the fluid were ideal, and the fluid separates from the trailing edge of the airfoil in the form of a vortex Fig. 3b. As this vortex passes from the airfoil, an opposing reaction starts a counter circulation opposite in direction to that of the trailing vortex. It is this induced counter circulation that produces lift. Starting vortex (a) counter circulation (b) Fig. 3 Starting vortex (a) at the beginning of motion and (b) after vortex created. a03/p1/fluids/flight.doc 2

3 For a finite lifting vane (the preceding discussion applies vane of infinite span), additional explanations are necessary, since a vortex cannot terminate within the fluid. The lift of a finite wing is zero at its tips, and thus the circulation would appear to be there also, so that the vortex system cannot extend out to infinity. A closed vortex loop is necessary, and this loop consists of tip vortices that trail behind the airfoil back to the starting vortex (Fig. 4). The starting vortex degenerates to zero with time because of viscous dissipation, and the vortex pattern is thus more horseshoe shaped than rectangular or toroidal. The tip vortices have low-pressure cores, and for ship propellers they are seen as threadlike cavities in the shape of helixes peeling off the blade tips. Under certain conditions they may be observed as vapour trails behind aircraft flying at high altitudes. Air within the vortex core expands and cools, and vapours condense to become visible. For an airfoil of finite span, the tip vortices produce a downward wash. Circulation around an airfoil Trailing tip vortices Starting vortices Trailing tip vortices Fig. 4 Closed loop vortex pattern for a finite wing. Lift (C L ) and drag (C D ) data can be plotted against the angle of attack α (Fig. 5). A desirable characteristic of an airfoil is a high lift to drag ratio. In Fig. 5b, the maximum value of this ratio is found by finding the tangent to the curve through the origin that has the largest slope. This is the point A in Fig. 5b. Fig. 5 shows that the attack angle can t be to large, otherwise a stall condition is reached and lift is no longer provided by the circulation around the airfoil. a03/p1/fluids/flight.doc 3

4 (a) (b) stall C L Decreasing Reynolds number C L Aspect ratio: span / chord Angle of attack α C D (c) (d) Fig. 5 Airfoil characteristics. (a) and (b)typical lift and drag coefficients. (c) Effect of Reynolds number on lift coefficient and stall angle. (d) Effect of airfoil aspect ratio. a03/p1/fluids/flight.doc 4

5 Lift of airfoils may be calculated or they may be measured directly in a wind tunnel by the integration of the pressures over the airfoil section. The algebraic difference between the predominantly negative pressure on the top surface of the foil and the predominantly positive pressure on the bottom surface will result in a net force normal to the chord of the airfoil. As a first approximation, the component of this force normal to the approaching free stream is the net lift on the airfoil. If p x is the surface pressure at a distance x from the leading edge of an airfoil of chord length C, p S is the free stream pressure, and is the free stream velocity of approach, then the surface pressures may be expressed as a dimensionless pressure coefficient C p = (p x p S ) / (1/2 ρ 2 ) The variation of C p over an airfoil (NACA 0015: symmetrical foil with maximum thickness 15% of chord length) in a small wind tunnel is shown in Fig. 6. The average height between the two curves in Fig. 6 is called the normal force coefficient C N and is related to the angle of attack by C L ~ C N cosα For the data in given in Fig. 6, C L = 0.85 and from the equation C L = 2π sinα = Thus, calculations from an ideal fluid theory agree well with measured values of lift. According to the Bernoulli equation, the maximum pressure from the free stream value p s is the dynamics pressure. Thus, at a stagnation point on an airfoil, the max value of p x p S = ½ 2 and the maximum pressure coefficient is C p = 1. C p Most of the lift is generated by the drop in pressure Over the upper front surface of the airfoil Pressure on upper surface Pressure on lower surface x / C Fig. 6 Dimensionless plot of measured pressure coefficient over an NACA 0015 airfoil at an attack angle of 8 with Reynolds number ~ a03/p1/fluids/flight.doc 5

6 The low pressures indicated in Fig. 6 for the front upper surface provide most of the lift for this airfoil. Associated with these low pressures are high velocities, so that at high subsonic free stream velocities, local velocities along the foil surface may become supersonic and shocks may affect the flow. DRAG The total drag on an aircraft is due to pressure differences and viscous shear and the drag force acts in a direction parallel and opposite to the direction of motion of the aircraft. Viscous shear forces may play an important role in the development of the boundary layer and influencing the point of separation of the boundary layer from the surface of the body. This affects the size of the pressure differences and pressure drag. This skin friction is the force that the air exerts on a surface in the direction of flow and is a direct consequence of momentum transfer through the boundary layer. The pressure differences around the aircraft, arise from a force known as the form drag and is a consequence of the acceleration of the fluid through which the aircraft is moving. It is very depend upon the shape and orientation of an object. F drag = C ½ ρ 2 A C is the drag coefficient (for skin friction or other forms), ½ ρ 2 is the dynamic pressure of the free stream and A is the projected frontal area, or area being sheared for pure skin friction, or the chord area for lifting vanes. Reference: Essentials of Engineering Fluid Mechanics (3 rd Ed) Reuben M. Olson a03/p1/fluids/flight.doc 6