© Fox, Pritchard, & McDonald Introduction to Fluid Mechanics Chapter 9 External Incompressible...

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Introduction to Fluid Mechanics

Chapter 9

External Incompressible Viscous Flow

Main TopicsThe Boundary-Layer ConceptBoundary-Layer ThicknessesLaminar Flat-Plate Boundary Layer: Exact

SolutionMomentum Integral EquationUse of the Momentum Equation for Flow with

Zero Pressure GradientPressure Gradients in Boundary-Layer FlowDragLift

The Boundary-Layer Concept

Boundary Layer Thicknesses

Disturbance Thickness,

Displacement Thickness, *

Momentum Thickness,

Laminar Flat-PlateBoundary Layer: Exact Solution

Governing Equations

Boundary Conditions

Equations are Coupled, Nonlinear, Partial Differential EquationsBlasius Solution:

• Transform to single, higher-order, nonlinear, ordinary differential equation

Results of Numerical Analysis

Momentum Integral Equation

Provides Approximate Alternative to Exact (Blasius) Solution

Momentum Integral Equation

Equation is used to estimate the boundary-layer 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 equation2. Assume a reasonable velocity-profile shape inside the boundary layer

3. Derive an expression for w using the results obtained from item 2

Use of the Momentum Equation for Flow with Zero Pressure Gradient

Simplify Momentum Integral Equation(Item 1)

The Momentum Integral Equation becomes

Laminar Flow• Example: Assume a Polynomial Velocity Profile

(Item 2)

• The wall shear stress w is then (Item 3)

Laminar Flow Results(Polynomial Velocity Profile)

Compare to Exact (Blasius) results!

Turbulent Flow• Example: 1/7-Power Law Profile (Item 2)

Turbulent Flow Results(1/7-Power Law Profile)

Pressure Gradients in Boundary-Layer Flow

Drag Coefficient

DragPure Friction Drag: Flat Plate Parallel to the FlowPure Pressure Drag: Flat Plate Perpendicular to the FlowFriction and Pressure Drag: Flow over a Sphere and CylinderStreamlining

DragFlow 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

DragFlow over a Flat Plate Parallel to the Flow:

Friction Drag (Continued)

Laminar BL:

Turbulent BL:

… plus others for transitional flow

DragFlow over a Flat Plate Perpendicular to

the Flow: Pressure Drag

Drag coefficients are usually obtained empirically

DragFlow over a Flat Plate Perpendicular to

the Flow: Pressure Drag (Continued)

DragFlow over a Sphere and Cylinder:

Friction and Pressure Drag

DragFlow over a Sphere and Cylinder:

Friction and Pressure Drag (Continued)

StreamliningUsed to Reduce Wake and hence

Pressure Drag

Mostly applies to Airfoils

Note: Based on planform area Ap

LiftExamples: NACA 23015; NACA 662-215

LiftInduced Drag

LiftInduced Drag (Continued)

Reduction in Effective Angle of Attack:

Finite Wing Drag Coefficient:

LiftInduced Drag (Continued)

© Fox, Pritchard, & McDonald Introduction to Fluid Mechanics Chapter 9 External Incompressible...

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