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-Plate Boundary Layer: Exact Solution
Governing Equations

8. Laminar Flat-Plate Boundary Layer: Exact Solution
Boundary Conditions

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

10. Laminar Flat-Plate Boundary 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

30. Lift
Induced Drag

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

32. Lift
Induced Drag (Continued)