Introduction to Fluid Mechanics

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!
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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
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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

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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

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Lift

• Induced Drag

31
Lift

• Induced Drag (Continued)

Reduction in Effective Angle of Attack
Finite Wing Drag Coefficient
32
Lift

• Induced Drag (Continued)