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= Velocity component along the cylinder wall (m=s) n = Any integer value Velocity Pro…le Adjacent to Cylinder It can be instructive to examine the velocity distribution along the y axis (x ´ 0):¯¯¯¯ ¯u = · @ª @y ¸ x=0 = U1 ¢ · R2 y2 +1 ¸¯¯¯¯ ¯ (3.98) u decreases to U1 inversely with £ y R ¤2 . It is equal to 2 U1 for y = R just as was found above; this value is worth remembering by the way! 3-26 CHAPTER 3. CONSTANT POTENTIAL FLOW PHENOMENA In‡uence of Circulation If a circulation is added to the ‡ow, then vµ along the cylinder wall comes directly from the superposition of equations 3.95 and 3.69: vµ = 2U1 sin µ + ¡ 2¼ R (3.99) again at the cylinder surface. Notice that vµ is no longer symmetric with respect to the x axis, as shown in …gure 3.14. In this case, a clockwise circulation has been added to a ‡ow from left to right. This, too, will be important for the hydrodynamic force discussion in the next main section of this chapter. A discussion of how this circulation is generated comes up later in this chapter. Figure 3.14: Cylinder in Uniform Flow with added Circulation 3.6.2 Pressures Now that the potential ‡ow around the cylinder is known, it is appropriate to discuss pressures and the forces which result from them. This treatment starts, again, with a single isolated cylinder in an in…nite ‡ow; there is no circulation. Stagnation Points It has already been pointed out via equation 3.96 that the tangential velocity component at the two locations where the cylinder wall and the x axis cross is identically equal to zero. Since the cylinder wall is impervious, the radial velocity component, vr, at this location is also zero. There is therefore no velocity at these two stagnation points. These points were labeled in …gure 3.12. Stagnation pressures are generally de…ned in terms of the ambient ‡ow velocity so that: ps = 1 2 ½U 21 (3.100) 3.6. SINGLE CYLINDER IN A UNIFORM FLOW 3-27 Pressure Distribution on Cylinder Wall The next step is to predict the complete pressure distribution along the entire circumference of the cylinder. This is done using the Bernoulli equation 3.44 and the known pressure and velocity at the stagnation point: 1 2 ½ U21+ 0 = p+ 1 2 ½ v2µ = const: (3.101) The elevation is left out of this balance because it is constant here. Equation 3.95 can be substituted for vµ here and equation 3.101 can be solved for the pressure, p, along the cylinder wall. This yields: p = 1 2 ½U 21 ¢ £ 1¡ 4 sin2 µ¤ (3.102) This (dimensionless) pressure distribution (the term in brackets above) is shown in …gure 3.15 in polar coordinates. It is also plotted in a rectangular coordinate form in chapter 4, by the way. Figure 3.15: Pressure Distribution on a Cylinder in a Potential Flow 3.6.3 Resulting Forces The next step is to determine the forces from the pressure distribution. Force Computation Principle The procedure used to compute the x and y components of the resulting force on the cylinder is illustrated for the x direction using …gure 3.16. This …gure shows a ’slice’ which extends from front to back across the cylinder; it isolates an arc length ds = Rdµ of the cylinder wall. The pressure on each end of this slice is directed normal to the surface (radially) so that its x-component is then dFx = p ¢ ds cos µ. Now, one only has to integrate this over the angle µ to get the total force Fx. 3-28 CHAPTER 3. CONSTANT POTENTIAL FLOW PHENOMENA Figure 3.16: Typical Slice used for Computing Resulting Force in X Direction D’Alembert’s Paradox The French mathematician Jean LeRond D’Alembert who died in 1783, is credited with having carried out the above computation and with discovering that Fx = 0 and - from an analogous computation in the y direction - Fy = 0 for a circular cylinder placed in a constant, uniform, potential ‡ow! This is D’Alembert’s paradox - a trap into which most every beginner falls. It could have been avoided by observing - beforehand! - that the pressure distribution in …gure 3.15 is completely symmetric about both the x and y axes so that any forces must cancel out in each of these directions. The conclusion should be very clear: There are no hydrodynamic forces on a circular cylinder in a constant uniform, potential ‡ow. Circulation and Lift If a circulation is added to the ‡ow used in these computations, a force will be generated. This comes about because the circulation will increase the tangential ‡ow velocity on one side of the cylinder - say the +y side - while it decreases the ‡ow velocity on the opposite - in this case ¡y side; see …gure 3.14. This is often referred to as the Magnus e¤ect. This extra ‡ow will obviously unbalance the symmetry of the ‡ow with respect to the x-axis; one could expect the pressure to be disturbed, too, so that a resultant force in the y direction will result. The tangential velocity will now be as given in equation 3.99. By letting C = ¡ ¡ 4¼RU1 = constant (3.103) then 3.99 becomes: vµ = 2U1 ¢ (sin µ¡ C) (3.104) and the equation for the pressure distribution, 3.102, now becomes: p = 1 2 ½U21 © 1 ¡ 4(sin µ¡ C)2ª (3.105) 3.6. SINGLE CYLINDER IN A UNIFORM FLOW 3-29 The drag force can be evaluated just as was done above; this still yields a zero resultant, independent of the value of C: The resulting force crosswise to the ‡ow can be found by integrating the y-component of the pressure force as follows: Fy = R Z 2¼ 0 p sin µ dµ (3.106) After a bit of substitution and algebra one should …nd that: Fy = 4¼ R ½ U 2 1 C (3.107) This resultant force is called lift; it is directed perpendicular to the direction of the undis- turbed approaching ‡ow. Notice that the lift force is only present when C 6= 0: Lift forces are responsible for keeping aircraft airborne. The necessary tangential velocity di¤erential is achieved by making the ‡ow path over the top of a wing or helicopter rotor longer than the ‡ow path on the underside so that the ‡ow over the top must travel faster to pass the object in the same time interval. This same phenomenon is important with ship’s propellers, too; see chapter 4. Lift is largely responsible for driving a sailboat forward when it is sailing ’close-hauled’. The necessary low pressure ’behind’ the main sail is now created by the jet caused by the jib or foresail. In tennis or with a baseball pitch, a spinning ball will curve one way or another. Now the ambient air remains stationary while the ball speeds through it. The necessary circulation is created by friction between the spinning ball and the air. Since friction is involved in this case, discussion will have to be delayed until the next chapter. 3-30 CHAPTER 3. CONSTANT POTENTIAL FLOW PHENOMENA . Chapter 4 CONSTANT REAL FLOW PHENOMENA 4.1 Introduction Now that the basics of constant idealized potential ‡ows have been discussed in the previous chapter, the next step is to consider more realistic - but still constant - ‡ows. The facts that there is no drag force in a potential ‡ow and that a circulation is required to generate a lift force is true only for a potential (non-viscous) ‡ow. The results from this chapter with real ‡ows which include viscous e¤ects will be more realistic. The initial discussion of some basic concepts motivates attention for two common dimen- sionless numbers, Rn and Fn, which are handled more generally in Appendix B. These are used to characterize constant ‡ows abound cylinders and ships - and the resultant forces - discussed in the latter part of the chapter. Propulsion systems are handled toward the end of the chapter. 4.2 Basic Viscous Flow Concepts This section discusses a few basic principles of constant viscous ‡ows. These supplement the discussion on potential ‡ows given in the …rst parts of chapter 3. 4.2.1 Boundary