Introduction to Fluid Mechanics
Chapter 9
External Incompressible
Viscous Flow
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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
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The Boundary-Layer Concept
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The Boundary-Layer Concept
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Boundary Layer Thicknesses
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Boundary Layer Thicknesses
Disturbance Thickness, d
Displacement Thickness, d*
Momentum Thickness, q
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Laminar Flat-Plate
Boundary Layer: Exact Solution
 Governing Equations
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Laminar Flat-Plate
Boundary Layer: Exact Solution
 Boundary Conditions
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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
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Laminar Flat-Plate
Boundary Layer: Exact Solution
Results of Numerical Analysis
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Momentum Integral Equation
Provides Approximate Alternative
to Exact (Blasius) Solution
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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 tw using the results
obtained from item 2
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Use of the Momentum Equation for
Flow with Zero Pressure Gradient
 Simplify Momentum Integral Equation
(Item 1)
 The Momentum Integral Equation becomes
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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)
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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)
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Use of the Momentum Equation for
Flow with Zero Pressure Gradient
 Turbulent Flow Results
(1/7-Power Law Profile)
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Pressure Gradients in
Boundary-Layer Flow
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Drag
Drag Coefficient
with
or
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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
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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
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Drag
 Flow over a Flat Plate Perpendicular to
the Flow: Pressure Drag
Drag coefficients are usually obtained empirically
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Drag
 Flow over a Flat Plate Perpendicular to
the Flow: Pressure Drag (Continued)
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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)
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Streamlining
 Used to Reduce Wake and hence
Pressure Drag
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Lift
Mostly applies to Airfoils
Note: Based on planform area Ap
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Lift
 Examples: NACA 23015; NACA 662-215
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Lift – Magnus Effect
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