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KULIAH X

Understand boundary-layer laws, exact solutions, momentum equations, and pressure gradients in incompressible viscous flows. Explore laminar and turbulent boundary-layer profiles in the context of drag and lift forces.

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KULIAH X

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  1. KULIAH X EXTERNAL INCOMPRESSIBLE VISCOUS FLOW NazaruddinSinaga

  2. Main Topics • The Boundary-Layer Concept • Boundary-Layer Thickness • 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

  3. The Boundary-Layer Concept

  4. The Boundary-Layer Concept

  5. Boundary Layer Thickness

  6. Boundary Layer Thickness • Disturbance Thickness, d where • Displacement Thickness, d* • Momentum Thickness, q

  7. Boundary Layer Laws The velocity is zero at the wall (u = 0 at y = 0) The velocity is a maximum at the top of the layer (u = um at =  ) The gradient of BL is zero at the top of the layer (du/dy = 0 at y =  ) The gradient is constant at the wall (du/dy = C at y = 0) Following from (4): (d2u/dy2 = 0 at y = 0)

  8. Navier-Stokes EquationCartesian Coordinates Continuity X-momentum Y-momentum Z-momentum

  9. Laminar Flat-PlateBoundary Layer: Exact Solution • Governing Equations • For incompresible steady 2D cases:

  10. Laminar Flat-PlateBoundary Layer: Exact Solution • Boundary Conditions

  11. Laminar Flat-PlateBoundary Layer: Exact Solution • Equations are Coupled, Nonlinear, Partial Differential Equations • Blassius Solution: • Transform to single, higher-order, nonlinear, ordinary differential equation

  12. Boundary Layer Procedure • Before defining and * and are there analytical solutions to the BL equations? • Unfortunately, NO • Blasius Similarity Solutionboundary layer on a flat plate, constant edge velocity, zero external pressure gradient

  13. Blasius Similarity Solution • Blasius introduced similarity variables • This reduces the BLE to • This ODE can be solved using Runge-Kutta technique • Result is a BL profile which holds at every station along the flat plate

  14. Blasius Similarity Solution

  15. Blasius Similarity Solution • Boundary layer thickness can be computed by assuming that  corresponds to point where U/Ue = 0.990. At this point,  = 4.91, therefore • Wall shear stress w and friction coefficient Cf,x can be directly related to Blasius solution Recall

  16. Displacement Thickness • Displacement thickness * is the imaginary increase in thickness of the wall (or body), as seen by the outer flow, and is due to the effect of a growing BL. • Expression for * is based upon control volume analysis of conservation of mass • Blasius profile for laminar BL can be integrated to give (1/3 of )

  17. Momentum Thickness • Momentum thickness  is another measure of boundary layer thickness. • Defined as the loss of momentum flux per unit width divided by U2 due to the presence of the growing BL. • Derived using CV analysis.  for Blasius solution, identical to Cf,x

  18. Turbulent Boundary Layer Black lines: instantaneous Pink line: time-averaged Illustration of unsteadiness of a turbulent BL Comparison of laminar and turbulent BL profiles

  19. Turbulent Boundary Layer • All BL variables [U(y), , *, ] are determined empirically. • One common empirical approximation for the time-averaged velocity profile is the one-seventh-power law

  20. Results of Numerical Analysis

  21. Momentum Integral Equation • Provides Approximate Alternative to Exact (Blassius) Solution

  22. Momentum Integral Equation • Equation is used to estimate the boundary-layer thickness as a function of x: • 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 • Assume a reasonable velocity-profile shape inside the boundary layer • Derive an expression for tw using the results obtained from item 2

  23. Use of the Momentum Equation for Flow with Zero Pressure Gradient • Simplify Momentum Integral Equation(Item 1) • The Momentum Integral Equation becomes

  24. 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)

  25. Use of the Momentum Equation for Flow with Zero Pressure Gradient • Laminar Flow Results(Polynomial Velocity Profile) • Compare to Exact (Blassius) results!

  26. Use of the Momentum Equation for Flow with Zero Pressure Gradient • Turbulent Flow • Example: 1/7-Power Law Profile (Item 2)

  27. Use of the Momentum Equation for Flow with Zero Pressure Gradient • Turbulent Flow Results(1/7-Power Law Profile)

  28. Pressure Gradients in Boundary-Layer Flow

  29. DRAG AND LIFT • Fluid dynamic forces are due to pressure and viscous forces acting on the body surface. • Drag: component parallel to flow direction. • Lift: component normal to flow direction.

  30. Drag and Lift • Lift and drag forces can be found by integrating pressure and wall-shear stress.

  31. Drag and Lift • In addition to geometry, lift FLand drag FD forces are a function of density  and velocity V. • Dimensional analysis gives 2 dimensionless parameters: lift and drag coefficients. • Area A can be frontal area (drag applications), planform area (wing aerodynamics), or wetted-surface area (ship hydrodynamics).

  32. Drag • Drag Coefficient with or

  33. 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

  34. 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

  35. Drag • Flow over a Flat Plate Parallel to the Flow: Friction Drag (Continued) Laminar BL: Turbulent BL: … plus others for transitional flow

  36. Drag Coefficient

  37. Drag • Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag Drag coefficients are usually obtained empirically

  38. Drag • Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag (Continued)

  39. Drag • Flow over a Sphere : Friction and Pressure Drag

  40. Drag • Flow over a Cylinder: Friction and Pressure Drag

  41. Streamlining • Used to Reduce Wake and Pressure Drag

  42. Lift • Mostly applies to Airfoils Note: Based on planform area Ap

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