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

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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

8

9

10

11 Laminar Flat-Plate Boundary Layer: Exact Solution Governing Equations

12 Laminar Flat-Plate Boundary Layer: Exact Solution Boundary Conditions

13 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

14 Laminar Flat-Plate Boundary Layer: Exact Solution Results of Numerical Analysis

15 Momentum Integral Equation Provides Approximate Alternative to Exact (Blasius) Solution

16 Momentum Integral Equation Equation is used to estimate the boundarylayer thickness as a function of x: 1. 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 2. Assume a reasonable velocity-profile shape inside the boundary layer 3. Derive an expression for t w using the results obtained from item 2

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

18 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 t w is then (Item 3)

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

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

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

22

23

24 Pressure Gradients in Boundary-Layer Flow

25

26

27 Drag Drag Coefficient with or

28 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

29 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

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

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

32 Drag Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag (Continued)

33 Drag Flow over a Sphere and Cylinder: Friction and Pressure Drag

34 Drag Flow over a Sphere and Cylinder: Friction and Pressure Drag (Continued)

35 Streamlining Used to Reduce Wake and hence Pressure Drag

36

37

38

39 Lift Mostly applies to Airfoils Note: Based on planform area A p

40 Lift Examples: NACA 23015; NACA

41

42 Lift Magnus Effect

43

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