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

© Fox, Pritchard, & McDonald

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


The Boundary-Layer Concept


Boundary Layer Thicknesses


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

Drag Coefficient

with

or


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


Lift

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)

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