Introduction to Fluid Mechanics Chapter 9 External Incompressible

Introduction to Fluid Mechanics Chapter 9 External Incompressible Viscous Flow © Fox, Pritchard, & Mc. Donald

Main Topics ü The Boundary-Layer Concept ü Boundary-Layer Thicknesses ü Laminar Flat-Plate Boundary Layer: Exact Solution ü Momentum Integral Equation ü Use of the Momentum Equation for Flow with Zero Pressure Gradient ü Pressure Gradients in Boundary-Layer Flow ü Drag ü Lift © Fox, Pritchard, & Mc. Donald

The Boundary-Layer Concept © Fox, Pritchard, & Mc. Donald

Boundary Layer Thicknesses © Fox, Pritchard, & Mc. Donald

Boundary Layer Thicknesses ü Disturbance Thickness, d ü Displacement Thickness, d* ü Momentum Thickness, q © Fox, Pritchard, & Mc. Donald

Laminar Flat-Plate Boundary Layer: Exact Solution ü Governing Equations © Fox, Pritchard, & Mc. Donald

Laminar Flat-Plate Boundary Layer: Exact Solution ü Boundary Conditions © Fox, Pritchard, & Mc. Donald

Laminar Flat-Plate Boundary Layer: Exact Solution ü Equations are Coupled, Nonlinear, Partial Differential Equations ü Blasius Solution: • Transform to single, higher-order, nonlinear, ordinary differential equation © Fox, Pritchard, & Mc. Donald

Laminar Flat-Plate Boundary Layer: Exact Solution ü Results of Numerical Analysis © Fox, Pritchard, & Mc. Donald

Momentum Integral Equation ü Provides Approximate Alternative to Exact (Blasius) Solution © Fox, Pritchard, & Mc. Donald

Momentum Integral Equation ü 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 © Fox, Pritchard, & Mc. Donald

Use of the Momentum Equation for Flow with Zero Pressure Gradient ü Simplify Momentum Integral Equation (Item 1) ü The Momentum Integral Equation becomes © Fox, Pritchard, & Mc. Donald

Use of the Momentum Equation for Flow with Zero Pressure Gradient ü Laminar Flow • Example: Assume a Polynomial Velocity Profile (Item 2) • The wall shear stress tw is then (Item 3) © Fox, Pritchard, & Mc. Donald

Use of the Momentum Equation for Flow with Zero Pressure Gradient ü Laminar Flow Results (Polynomial Velocity Profile) Compare to Exact (Blasius) results! © Fox, Pritchard, & Mc. Donald

Use of the Momentum Equation for Flow with Zero Pressure Gradient ü Turbulent Flow • Example: 1/7 -Power Law Profile (Item 2) © Fox, Pritchard, & Mc. Donald

Use of the Momentum Equation for Flow with Zero Pressure Gradient ü Turbulent Flow Results (1/7 -Power Law Profile) © Fox, Pritchard, & Mc. Donald

Pressure Gradients in Boundary-Layer Flow © Fox, Pritchard, & Mc. Donald

Drag ü Drag Coefficient with or © Fox, Pritchard, & Mc. Donald

Drag ü Pure Friction Drag: Flat Plate Parallel to the Flow ü Pure Pressure Drag: Flat Plate Perpendicular to the Flow ü Friction and Pressure Drag: Flow over a Sphere and Cylinder ü Streamlining © Fox, Pritchard, & Mc. Donald

Drag ü 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 © Fox, Pritchard, & Mc. Donald

Drag ü Flow over a Flat Plate Parallel to the Flow: Friction Drag (Continued) Laminar BL: Turbulent BL: … plus others for transitional flow © Fox, Pritchard, & Mc. Donald