Fluid Mechanics - II

External Incompressible Viscous Flow

Boundary layer and Applications

The Boundary-Layer Concept

Boundary Layer Thicknesses

Boundary Layer Thicknesses

• Disturbance Thickness,

Displacement Thickness, *

Momentum Thickness,

Laminar Flat-PlateBoundary Layer: Exact Solution

• Governing Equations

Laminar Flat-PlateBoundary Layer: Exact Solution

• Boundary Conditions

Laminar Flat-PlateBoundary Layer: Exact Solution

• Equations are Coupled, Nonlinear, Partial Differential Equations• Blasius Solution:

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

Laminar Flat-PlateBoundary Layer: Exact Solution• Results of Numerical Analysis

Momentum Integral Equation

• Equation is used to estimate the boundary-layer thickness as a function of x:1. Obtain a first approximation to the free stream 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 layer3. 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

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 w is then (Item 3)

Use of the Momentum Equation for Flow with Zero Pressure Gradient

• Laminar Flow Results(Polynomial Velocity Profile)

Compare to Exact (Blasius) results!

Use of the Momentum Equation for Flow with Zero Pressure Gradient

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

Use of the Momentum Equation for Flow with Zero Pressure Gradient

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

Pressure Gradients in Boundary-Layer Flow

Drag

• Drag Coefficient

with

or

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

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

Drag

• Flow over a Flat Plate Parallel to the Flow: Friction Drag (Continued)

Laminar BL:

Turbulent BL:

… plus others for transitional flow

Drag

• Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag

Drag coefficients are usually obtained empirically

21

22

23

24

Drag• Flow over a Flat Plate Perpendicular to the

Flow: Pressure Drag (Continued)

Drag• Flow over a Sphere and Cylinder: Friction

and Pressure Drag on a Sphere

Drag• Flow over a Sphere and Cylinder: Friction

and Pressure Drag (Continued) on Cylinder

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Note that the lift coefficient strongly depends on the rate of rotation, especially at low angular velocities. The effect of the rate of rotation on the drag coefficient is small. Roughness also affects the drag and lift coefficients. In a certain range of Reynolds number, roughness produces the desirable effect of increasing the lift coefficient while decreasing the drag coefficient. Therefore, golf balls with the right amount of roughness travel higher and farther than smooth balls for the same hit.

Streamlining• Used to Reduce Wake and hence Pressure

Drag

Lift

• Mostly applies to Airfoils

Note: Based on planform area Ap

Lift• Examples: NACA 23015; NACA 662-215

Lift• Induced Drag

Lift• Induced Drag (Continued)

Reduction in Effective Angle of Attack:

Finite Wing Drag Coefficient:

Lift• Induced Drag (Continued)