Integration with variables Notes_Part_21

Integration with variables Notes_Part_21 - 15 30 Figure...

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Unformatted text preview: 15 30 Figure 7.36. Kutta Flow Past a Tilted Airfoil. which remains asymptotically 1 at large distances. By Cauchys Theorem 7.48 coupled with formula (7.123), if C is a curve going once around the disk in a counter-clockwise direction, then contintegraldisplay C f ( z ) dz = contintegraldisplay C parenleftbigg 1 1 z 2 + i z parenrightbigg dz = 2 . Therefore, when negationslash = 0, the circulation integral is non-zero, and the cylinder experiences a net lift, which is upward provided the circulation is negative: < 0. In Figure 7.35, the streamlines for the flow corresponding to a few representative values of are plotted. Note the asymmetry of the streamlines that accounts for the lift experienced by the disk. In particular, assuming | | < 1, the stagnation points have moved from 1 to radicalbig 1 2 i . When we compose the modified lift potentials (7.132) with the Joukowski transforma- tion (7.101), we obtain a complex potential for flow around the corresponding airfoil the image of the unit disk. The conformal mapping does not affect the value of the complex integrals, and hence, for any negationslash = 0, there is a nonzero circulation around the airfoil under the modified fluid flow. A negative circulation will cause a net upward lift on the airfoil, and at last our airplane will fly! However, there is now a slight embarrassment of riches, since we have designed flows around the airfoil with an arbitrary value 2 for the circulation integral, and hence having an arbitrary amount of lift! Which of these possible flows most closely realizes the true physical version with the correct amount of lift? In his 1902 thesis, the German mathematician Martin Kutta hypothesized that Nature chooses the constant...
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Integration with variables Notes_Part_21 - 15 30 Figure...

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