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Introduction to Fluid Mechanics

Introduction to Fluid Mechanics. Chapter 9 External Incompressible Viscous Flow. Main Topics. The Boundary-Layer Concept Boundary-Layer Thicknesses Laminar Flat-Plate Boundary Layer: Exact Solution Momentum Integral Equation Use of the Momentum Equation for Flow with Zero Pressure Gradient

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Introduction to Fluid Mechanics

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  1. Introduction to Fluid Mechanics Chapter 9 External Incompressible Viscous Flow

  2. Main Topics • The Boundary-Layer Concept • Boundary-Layer Thicknesses • Laminar Flat-Plate Boundary Layer: Exact Solution • Momentum Integral Equation • Use of the Momentum Equation for Flow with Zero Pressure Gradient • Pressure Gradients in Boundary-Layer Flow • Drag • Lift

  3. The Boundary-Layer Concept

  4. The Boundary-Layer Concept

  5. Boundary Layer Thicknesses

  6. Boundary Layer Thicknesses • Disturbance Thickness, d • Displacement Thickness, d* • Momentum Thickness, q

  7. Laminar Flat-PlateBoundary Layer: Exact Solution • Governing Equations

  8. Laminar Flat-PlateBoundary Layer: Exact Solution • Boundary Conditions

  9. Laminar Flat-PlateBoundary Layer: Exact Solution • Equations are Coupled, Nonlinear, Partial Differential Equations • Blasius Solution: • Transform to single, higher-order, nonlinear, ordinary differential equation

  10. Laminar Flat-PlateBoundary Layer: Exact Solution • Results of Numerical Analysis

  11. Momentum Integral Equation • Provides Approximate Alternative to Exact (Blasius) Solution

  12. Momentum Integral Equation • Equation is used to estimate the boundary-layer thickness as a function of x: • Obtain a first approximation to the freestream velocity distribution, U(x). The pressure in the boundary layer is related to the freestream velocity, U(x), using the Bernoulli equation • Assume a reasonable velocity-profile shape inside the boundary layer • Derive an expression for tw using the results obtained from item 2

  13. Use of the Momentum Equation for Flow with Zero Pressure Gradient • Simplify Momentum Integral Equation(Item 1) • The Momentum Integral Equation becomes

  14. Use of the Momentum Equation for Flow with Zero Pressure Gradient • Laminar Flow • Example: Assume a Polynomial Velocity Profile (Item 2) • The wall shear stress tw is then (Item 3)

  15. Use of the Momentum Equation for Flow with Zero Pressure Gradient • Laminar Flow Results(Polynomial Velocity Profile) • Compare to Exact (Blasius) results!

  16. Use of the Momentum Equation for Flow with Zero Pressure Gradient • Turbulent Flow • Example: 1/7-Power Law Profile (Item 2)

  17. Use of the Momentum Equation for Flow with Zero Pressure Gradient • Turbulent Flow Results(1/7-Power Law Profile)

  18. Pressure Gradients in Boundary-Layer Flow

  19. Drag • Drag Coefficient with or

  20. Drag • Pure Friction Drag: Flat Plate Parallel to the Flow • Pure Pressure Drag: Flat Plate Perpendicular to the Flow • Friction and Pressure Drag: Flow over a Sphere and Cylinder • Streamlining

  21. Drag • Flow over a Flat Plate Parallel to the Flow: Friction Drag Boundary Layer can be 100% laminar, partly laminar and partly turbulent, or essentially 100% turbulent; hence several different drag coefficients are available

  22. Drag • Flow over a Flat Plate Parallel to the Flow: Friction Drag (Continued) Laminar BL: Turbulent BL: … plus others for transitional flow

  23. Drag • Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag Drag coefficients are usually obtained empirically

  24. Drag • Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag (Continued)

  25. Drag • Flow over a Sphere and Cylinder: Friction and Pressure Drag

  26. Drag • Flow over a Sphere and Cylinder: Friction and Pressure Drag (Continued)

  27. Streamlining • Used to Reduce Wake and hence Pressure Drag

  28. Lift • Mostly applies to Airfoils Note: Based on planform area Ap

  29. Lift • Examples: NACA 23015; NACA 662-215

  30. Lift • Induced Drag

  31. Lift • Induced Drag (Continued) Reduction in Effective Angle of Attack: Finite Wing Drag Coefficient:

  32. Lift • Induced Drag (Continued)

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