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482 Chapter 9 ■ Flow over Immersed Bodies 1.0 Linear Cubic Sine wave y __ δ 0.5 Blasius Parabolic 0 0 0.5 u__ U 1.0 F I G U R E 9.12 Typical approximate boundary layer profiles used in the momentum integral equation. to express the wall shear stress. From Eq. 9.30 we obtain the approximate value while the Blasius solution result is given by c f 12C 1 C 2 A m rUx 12C 1C 2 1Re x c f 0.664 1Re x (9.32) These results are also indicated in Table 9.2. For a flat plate of length / and width b, the net friction drag, d f , can be expressed in terms of the friction drag coefficient, C Df , as The friction drag coefficient is an integral of the local friction coefficient. or / C Df d b f t w dx 1 2 rU 2 b/ 0 1 2 rU 2 b/ C Df 1 / / c f dx 0 (9.33) TABLE 9.2 Flat Plate Momentum Integral Results for Various Assumed Laminar Flow Velocity Profiles Profile Character DRe x 12 x c f Re x 12 C Df Re 12 a. Blasius solution 5.00 0.664 1.328 b. Linear uU yd 3.46 0.578 1.156 c. Parabolic uU 2yd 1yd2 2 5.48 0.730 1.460 d. Cubic uU 31 yd22 1yd2 3 2 4.64 0.646 1.292 e. Sine wave uU sin3p1 yd224 4.79 0.655 1.310
C Df 0.04 0.03 0.02 0.01 0.00 Laminar boundary layer Re We use the above approximate value of c f 12C 1 C 2 mrUx2 1 2 to obtain 9.2 Boundary Layer Characteristics 483 C Df 18C 1C 2 1Re / where Re / U/n is the Reynolds number based on the plate length. The corresponding value obtained from the Blasius solution 1Eq. 9.322 and shown by the figure in the margin gives C Df 1.328 1Re / These results are also indicated in Table 9.2. The momentum integral boundary layer method provides a relatively simple technique to obtain useful boundary layer results. As is discussed in Sections 9.2.5 and 9.2.6, this technique can be extended to boundary layer flows on curved surfaces 1where the pressure and fluid velocity at the edge of the boundary layer are not constant2 and to turbulent flows. The boundary layer on a flat plate will become turbulent if the plate is long enough. 9.2.4 Transition from Laminar to Turbulent Flow The analytical results given in Table 9.2 are restricted to laminar boundary layer flows along a flat plate with zero pressure gradient. They agree quite well with experimental results up to the point where the boundary layer flow becomes turbulent, which will occur for any free-stream velocity and any fluid provided the plate is long enough. This is true because the parameter that governs the transition to turbulent flow is the Reynolds number—in this case the Reynolds number based on the distance from the leading edge of the plate, Re x Uxn. The value of the Reynolds number at the transition location is a rather complex function of various parameters involved, including the roughness of the surface, the curvature of the surface 1for example, a flat plate or a sphere2, and some measure of the disturbances in the flow outside the boundary layer. On a flat plate with a sharp leading edge in a typical airstream, the transition takes place at a distance x from the leading edge given by Re to 3 10 6 xcr 2 10 5 . Unless otherwise stated, we will use Re xcr 5 10 5 in our calculations. The actual transition from laminar to turbulent boundary layer flow may occur over a region of the plate, not at a specific single location. This occurs, in part, because of the spottiness of the transition. Typically, the transition begins at random locations on the plate in the vicinity of Re x Re xcr . These spots grow rapidly as they are convected downstream until the entire width of the plate is covered with turbulent flow. The photo shown in Fig. 9.13 illustrates this transition process. The complex process of transition from laminar to turbulent flow involves the instability of the flow field. Small disturbances imposed on the boundary layer flow 1i.e., from a vibration of the plate, a roughness of the surface, or a “wiggle” in the flow past the plate2 will either grow 1instability2 or decay 1stability2, depending on where the disturbance is introduced into the flow. If these disturbances occur at a location with Re x 6 Re xcr they will die out, and the boundary layer will return to laminar flow at that location. Disturbances imposed at a location with Re x 7 Re xcr will grow and transform the boundary layer flow downstream of this location into turbulence. The study of the initiation, growth, and structure of these turbulent bursts or spots is an active area of fluid mechanics research. V9.5 Transition on flat plate F I G U R E 9.13 Turbulent spots and the transition from laminar to turbulent boundary layer flow on a flat plate. Flow from left to right. (Photograph courtesy of B. Cantwell, Stanford University.)
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Achieve Positive Learning Outcomes
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WileyPLUS combines robust course ma
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Sixth Edition F undamentals of Flui
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About the Authors vii Bruce R. Muns
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P reface This book is intended for
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Preface xi Life Long Learning Probl
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Preface xiii WileyPLUS offers today
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Featured in this Book xv LEARNING O
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C ontents 1 INTRODUCTION 1 Learning
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Contents xix 6.4.3 Irrotational Flo
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12.6 Axial-Flow and Mixed-Flow Pump
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10 8 Jupiter red spot diameter 2 Ch
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4 Chapter 1 ■ Introduction and do
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10 Chapter 1 ■ Introduction E XAM
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12 Chapter 1 ■ Introduction TABLE
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14 Chapter 1 ■ Introduction TABLE
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16 Chapter 1 ■ Introduction Crude
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18 Chapter 1 ■ Introduction Visco
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20 Chapter 1 ■ Introduction Kinem
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22 Chapter 1 ■ Introduction As th
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24 Chapter 1 ■ Introduction Boili
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26 Chapter 1 ■ Introduction E XAM
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28 Chapter 1 ■ Introduction The r
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30 Chapter 1 ■ Introduction fluid
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32 Chapter 1 ■ Introduction Secti
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34 Chapter 1 ■ Introduction avail
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36 Chapter 1 ■ Introduction compr
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2 Fluid Statics CHAPTER OPENING PHO
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40 Chapter 2 ■ Fluid Statics in a
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42 Chapter 2 ■ Fluid Statics p Fo
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60 Chapter 2 ■ Fluid Statics The
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66 Chapter 2 ■ Fluid Statics SOLU
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68 Chapter 2 ■ Fluid Statics COMM
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70 Chapter 2 ■ Fluid Statics V2.8
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88 Chapter 2 ■ Fluid Statics 2.81
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92 Chapter 2 ■ Fluid Statics 2.12
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94 Chapter 3 ■ Elementary Fluid D
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100 Chapter 3 ■ Elementary Fluid
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148 Chapter 4 ■ Fluid Kinematics
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264 Chapter 6 ■ Differential Anal
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532 Chapter 9 ■ Flow over Immerse
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10 Open-Channel Flow CHAPTER OPENIN
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646 Chapter 12 ■ Turbomachines Ex
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724 Appendix D ■ Compressible Flo
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Answers to Selected Even-Numbered H
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I ndex Absolute pressure, 13, 48 Ab
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Index I-3 guidelines for selection,
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Index I-5 hydrostatic: on curved su
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Index I-7 piezometer tube, 50, 105,
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Index I-11 Stress field, 277 Strouh
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Index I-13 Weber, M., 29, 350 Weber
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V4.8 Jupiter red spot V4.9 Streamli
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V10.3 Water strider V10.13 Triangul
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■ TABLE 1.4 Conversion Factors fr
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■ TABLE 1.7 Approximate Physical
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