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Tham khảo tài liệu 'a heat transfer textbook - third edition episode 2 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 6.2 Laminar incompressible boundary layer on a flat surface 289 Figure 6.11 Comparison of the third-degree polynomial fit with the exact b.l. velocity profile. Notice that the approximate result has been forced to u u 1 instead of 0.99 at y Ổ. We integrate this using the b.c. Ỗ2 0 at x 0 2 _ 280 vx 13 u or 0 x 4.64 VRẽX 6.31 This b.l. thickness is of the correct functional form and the constant is low by only 5.6 . The skin friction coefficient The fact that the function f n gives all information about flow in the b.l. must be stressed. For example the shear stress can be obtained from it 290 Laminar and turbulent boundary layers 6.2 by using Newton s law of viscous shear _ du Tw V- - dy yv y 0 df dn dn dy y 0 uy .uy d2f Jvx dn2 n 0 y 0 ỷy u But from Fig. 6.10 and Table 6.1 we see that d2f dn2 n 0 0.33206 so Tw 0.332 ụily VRẽ7 6.32 X The integral method that we just outlined would have given 0.323 for the constant in eqn. 6.32 instead of 0.332 Problem 6.6 . The local skin friction coefficient or local skin drag coefficient is defined as Cf - pU.i 2 0.664 VReX 6.33 The overall skin friction coefficient Cf is based on the average of the shear stress Tw over the length L of the plate Tw f L Tw dx 0 1 L puy fL 0.664 puy Ị V fy T --TT7TT dx 1.328 A J- 2L J07uyx v 2 l UyL so - _ 1.328 Cf eL 6.34 As a matter of interest we note that Cf x approaches infinity at the leading edge of the flat surface. This means that to stop the fluid that first touches the front of the plate dead in its tracks would require infinite shear stress right at that point. Nature of course will not allow such a thing to happen and it turns out that the boundary layer analysis is not really valid right at the leading edge. In fact the range X ồ 5Ổ is too close to the edge to use this analysis with accuracy because the b.l. is relatively thick and v is no longer u. With eqn. 6.2 this converts to X 600 v uy for a boundary layer to exist 6.2 Laminar incompressible boundary layer on a flat surface 291
