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Understanding Boundary Layer Effects on Pressure Gradients

Exploring displacement and momentum thickness impact on boundary layers. Discusses displacement thickness, momentum law, skin friction, and drag coefficients. Covers laminar and turbulent flows with practical applications.

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Understanding Boundary Layer Effects on Pressure Gradients

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  1. Section 8, Lecture 1, Supplemental Effect of Pressure Gradients on Boundary layer • Not in Anderson

  2. Displacement and Momentum Thickness • Fluid sticks to wall .. Creating boundary layer • Local thickness is a function of downstream distance • Local effects (at each x) -- External streamlines are displaced (displacement thickness, ) -- Momentum is list to friction (momentum thickness, )

  3. Displacement Thickness • How are external streamlines displaced by boundary layer? “0” • Steady Mass Flow Across Control Volume

  4. Displacement Thickness (2) • How are external streamlines displaced by boundary layer?

  5. Displacement Thickness (3) • How are external streamlines displaced by boundary layer? • At leading edge of plate u(y) = U --->

  6. Displacement Thickness (4) • How are external streamlines displaced by boundary layer? • Simplifying …

  7. Displacement Thickness (5) • How are external streamlines displaced by boundary layer? • Defining …  = yx-y0 @ yx=... edge of the boundary layer “Displacement Thickness”

  8. Momentum Thickness • Look at the flow near the aft end of a flat plate with span, b Start with incompressible flow for now ( r= const ) U e • Mass flow into the small segment dy: • Momentum defect from freestream across segment dy:

  9. Momentum Thickness (2) • Assuming no pressure gradient along plate Drag = loss in momentum U d • Integrating across the depth of the boundary layer

  10. Momentum Thickness (3) • Integrating across the depth of the boundary layer U d “momentum thickness”

  11. Displacement and Momentum Thickness (2) • Displacement Thickness … -- How far external streamlines are displaced by local boundary layer (*) • Momentum Thickness … -- Momentum Loss in Boundary Layer from front of plate to local station, x()

  12. Displacement and Momentum Thickness (3) d • Valid for Both Laminar and Turbulent Flows

  13. Displacement and Momentum Thickness (3) d • Useful parameter … “Shape Parameter” • More on This later

  14. Local Skin Friction Coefficient andTotal Skin-Drag Coefficient • Per Earlier Discussion • Start with Wall Shear Stress U “dynamic viscosity” dx “incremental drag /unit surface area”

  15. Local Skin Friction Coefficient andTotal Skin-Drag Coefficient (2) •From Earlier derivation of momentum thickness U “equate and collect” dx

  16. Local Skin Friction Coefficient andTotal Skin-Drag Coefficient (3) •Von Karman Momentum law for boundary layer • Valid for flat Plate with no pressure gradient U • Contrast with “Total Plate Skin Drag Coefficient” dx Plate length “c”

  17. Local Skin Friction Coefficient andTotal Skin-Drag Coefficient (4) • Valid for flat Plate with no pressure gradient “Total Plate Skin Drag Coefficient” Plate length “c ”… logically …

  18. Local Skin Friction Coefficient andTotal Skin-Drag Coefficient (5) Laminar

  19. Local Skin Friction Coefficient andTotal Skin-Drag Coefficient (6) Laminar Flow • Similar process for turbulent flow

  20. Incompressible Flat Plate Summary Laminar Turbulent Velocity profile Total Skin Drag Coefficient Local Skin Friction Coefficient For most applications n = 7

  21. Incompressible Flat Plate Summary (2) Laminar Turbulent Boundary Layer Thickness Displacement Thickness Momentum Thickness H = 2.592 Shape Parameter

  22. What Happens when pressure (or velocity) along plate is not constant? • Modified Von-Karman Momentum Equation -- Flat Plate (no gradient) -- with pressure (velocity) gradient

  23. What Happens when pressure (or velocity) along plate is not constant? (2) • Write in terms of pressure gradient …. Bernoulli …

  24. What Happens when pressure (or velocity) along plate is not constant? (3) • Express in terms of local skin friction coefficient -->Let

  25. What Happens when pressure (or velocity) along plate is not constant? (4) • Collect Terms, solve for cfx With Pressure gradient Flat Plate (no pressure gradient)

  26. What Happens when pressure (or velocity) along plate is not constant? (5) • Apply to Diamond -wedge airfoil Along lower surface

  27. Apply to Diamond Airfoil • Along lower surface

  28. Apply to Diamond Airfoil (2) • Evaluate Integrals

  29. Apply to Diamond Airfoil (3) • Distance from c/2(-) to c/2(+) .. Almost infinitesimal • Which is exactly the flat plate formula … ! • So we see that the pressure gradient due to The expansion wave has negligible effect on the skin drag coefficient for this airfoil!

  30. Apply to Diamond Airfoil (4) • Calculate c for lower surface, Turbulent flow • To “map” to flat plate we base calculation on “run length” along surface …. xrun = c/cos(wedge) c c”effective” = c/cos(wedge)

  31. Apply to Diamond Airfoil (5) • To calculate drag we apply to BOTH upper and lower surfaces • To “map” to flat plate we base calculation on “run length” along surface …. xrun = c/cos(wedge) c

  32. Apply to Diamond Airfoil (6) • Calculate c for lower surface, Turbulent flow • To account for angle Of attack we use only Velocity component Along wing centerline xrun = c/cos(wedge) U sin() U a c U cos() U ~ freestream

  33. Apply to Diamond Airfoil (7) • We have to account for BOTH sides of the wing xrun = c/cos(wedge)

  34. Apply to Diamond Airfoil (8) • Normalize by planform area (b x c) • Apply compressibility correction

  35. Summary for Turbulent Flow, n=7 Referenced To planform

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