4.12 - .._.._._W_.e.¢ s. - ,.,V_~..Afi W, ,...

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Unformatted text preview: .._.._._W_.e.¢ s. - ,.,V_~..Afi W, , fl/ph—Sfl/I‘V'flr'hj 'V(r\/ /(7W KI’an/I/I/Vd FIGURE 3.4] Flow over a cirwlnr cylinder. Re- LS4. (Phalogmph by Sadatoxhi Taneda, from Van Dyke, Milton, An Album of Fluid Motion. The Parabolic Pun, Stanford, Cal'f I982.) (0) 5/7/10!) I'Mj 0’) FIGURE 3.29 Th5: flow-field pictures were obtained in water, where aluminum filings were scattered on the surface In show the direction of the streamlines. (From Prandll and Tietjens, Raj? 8.) (a) Shown above is the use for the nonspinning cylinder. (b) Spinning cylinder: peripheral velocity of the surface;= 3V.” .fi...» -q.--—- .~.'—--w-.-...- -———-—vW« .W..W.. A ...-.. ..._.-..-... fa: FUNDAMENTALS OF AERODYNAMICS simply does not model the proper physics for drag calculations. On the other hand, the prediction of lift via Eq. (3.140) is quite realistic. Let us return to the wind-tunnel experiments mentioned at the beginning of this chapter. If a station- ary, nonspinning cylinder is placed in a low-speed wind tunnel, the flow field will appear as shown in Fig. 3.2%. The streamlines over the front of the cylinder are similar to theoretical predictions, as sketched at the right of Fig. 3.21. However, because of viscous effects, the flow separates over the rear of the cylinder, creating a recirculating flow in the wake downstream of the body. This separated flow greatly contributes to the finite drag measured for the cylinder. On the other hand, Fig. 3.29a shows a reasonably symmetric flow about the horizontal axis, and the measurement of lift is essentially zero. Now let us spin the cylinder in a clockwise direction about its axis. The resulting flow fields are shown in Fig. 3.2% and c. For a moderate amount of spin (Fig. 3.2%), the stagnation points move to the lower part of the cylinder, similar to the theoretical flow. sketched in Fig. 3.28a. Ifthe spin is sufficiently increased (Fig. 3.29c), the stagnation point lifts ofi the surface, similar to the theoretical flow sketched in Fig. 3.28c. And what is most important, a finite lift is measured for the spinning cylinder in the wind tunnel. What is happening here? Why does spinning the cylinder produce lift? In actuality, the friction between the fluid and the surface of the cylinder tends to drag the fluid near the surface in the same direction as the rotational thion. Superimposed on top of the usual nonspinning flow, this “extra” velocity contribution creates a higher-than-usual velocity at the top of the cylinder and a lower-than-usual velocity at the bottom, as sketched in Fig. 3.30. These velocities are assumed to be just outside the viscous boundary layer on the surface. Recall (c') FICURE 3.29 (c) Spinning cylinder: peripheral velocity of the surface=6Vw .‘.; (3.114) Note that since «[1: s that the streamlines nes are circ es. Thus, . Also, note from Eq. :., straight radial lines 5 and streamlines are - our four elementary blet to synthesize the lition, we proved that the streamline pattern theox ally possible lSiStCITF/With zero lift. a cirCular cylinder— ~ cylinder. Such lifting ed by the question as nder. Is not the body result in a symmetric discussed? You might e a stationary cylinder sfaction, you measure section is ridiculous-— wind tunnel, and this itively high revolutions his time you might be s it to curve, and spin here are nonsymmetric bodies. So, maybe the llfIndeed, as you will ider will start us on a nerated by airfoils, as t nonlifting flow over a 7. The stream function V w 4 __..._,,-. fi+= l Nonlifting flow Vortex of ' I _ over a c linder stre ‘f" Lifting flow over y d « "ngm a cylinder J FIGURE 3.27 - - v ‘ The synthesis of lifting flow over a circular cylinder. for nonlifting flow over a circular cylinder of radius R is given by Eq. (3.92): . R2 [/1] = (er sin 6)(1--r—2> (3.92) The stream function for a vortex of strength F is given by Eq. (3.114). Recall that the stream function is determined within an arbitrary constant; hence, Eq. (3.114) can be written as F ¢2=Eln r+c0nst (3.115) Since the value of the constant is arbitrary, let P _ Const= —-—ln R (3.116) 277 Combining Eqs. (3.115) and (3.116), we obtain I‘ r =——l — . ((12 2” n R (3117) Equation (3.117) is the stream function for a vortex of strength F and is just as valid as Eq. (3.114) obtained earlier; the only difierence between these two equations is a constant of the value given by Eq. (3.116). The resulting stream function for the flow shown at the right of Fig. 3.27 is 11’ = Wi'i' ‘1’2 , R2 F r 7 or ¢=(Vwrsm 6)<1——2—)+—ln— (3.118) r 27r R From Eq. (3.118), if r= R, then 1/1 = 0 for all values of 9. Since {/1 =: constant is the equation of a streamline, r= R is therefore a streamline of the flow, but 3.118) is a valid stream ular cylinder of radius .115 result given by Eq. is sketched at the right symmetrical about the :ct (correctly) that the owever, the streamlines .s a result the drag will ortex of strength I‘ has is now finite and equal Eq. (3.118). An equally elocity field of a vortex because of the linearity d elementary flows add .ng flow over a cylinder we have, for the lifting (3.119) (3.120) = Ve=0 in Eqs. (3.119) ): (3.121) 0 (3.122) 1. (3.122) and solving for (3.123) e in the third and fourth m the bottom half of the 3.28a. These points are Never, this result is valid .123) has no meaning. If e surface of the cylinder, 3.28b. For the case of FUNDAMENTALS OF lNVlSCID. INCOMPRESSIBLE FLOW Twila/Mal? c (a) I‘ < 41erR 71< (b)I‘=41erR FIGURE 3.28 (c) 1‘ > 41IV00R Stagnation points for the lifting flow over a circular cylinder. 7" y V > "I .0 ll F/47erR > 1, return to Eq. (3.121). We saw earlier that it is satisfied by r= R; however, it is also satisfied by 0 = 7r/2 or —77/2. Substituting 0 = —1r/2 into Eq. (3.122), and solving for r, we have T l" 2 2 = i __ I r 4w... (4.1V...) R (3 124) Hence, for F/47rVaoR> 1, there are two stagnation points, one inside and the other outside the cylinder, and both on the vertical axis, as shown by points 4 and 5 in Fig. 3.28c. [How does one stagnation point fall inside the cylinder? Recall that r=R, or 41:0, is just one of the allowed streamlines of the flow. There is a theoretical flow inside the cylinder—flow that is issuing from the doublet at the origin superimposed with the vortex flow for r< R The circular streamline r = R is the dividing streamline between this flow and the flow from the freestream. Therefore, as before, we can replace the dividing streamline by a solid body—our circular cylinder—and the external flow will not know the difference. Hence, although one stagnation point falls inside the body (point 5), we are not realistically concerned about it. Instead, from the point of view of flow over a solid cylinder of radius R, point 4 is the only meaningful stagnation point for the case F/41rVooR > 1.] The results shown in Fig. 3.28 can be visualized as follows. Consider the inviscid incompressible flow of given freestream velocity V00 over a cylinder of given radius R. If there is no circulation, i.e., if r=o, the flow is given by the sketch at the right of Fig. 3.21, with horizontally opposed stagnation points A and B. Now assume that a circulation is imposed on the flow, such that F (477mG The flow sketched in Fig. 3.28a will result; the two stagnation points will move to the lower surface of the cylinder as shown by points 1 and 2. Assume that F is further increased until F = 477VwR. The flow sketched in Fig. 3.28b will result, ...
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4.12 - .._.._._W_.e.¢ s. - ,.,V_~..Afi W, ,...

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