1. Effect of Pressure Gradient on the flow in a Boundary Layer Pressure gradient is found from freestream (external) velocity field Boundary layer equation: x z

2. Effect of Pressure Gradient on the flow in a Boundary Layer In the accelerating part of the stream, At the wall, the boundary layer equation becomes: x z

3. Effect of Pressure Gradient on the flow in a Boundary Layer x z In the decelerating part of the stream, inflection point

4. Effect of Pressure Gradient on the flow in a Boundary Layer x z Velocity distribution suggests that a ∂p/∂x > 0 contributes to thicken the boundary layer, as seen from continuity: Deceleration also adds viscous effects to make the boundary layer grow --- both viscous effects and advection contribute to b.l. growth --- w is directed away from the wall ( ∂u/∂x - ) – increase in b.l. thickness with x

5. Effect of Pressure Gradient on the flow in a Boundary Layer x z ∂p/∂x < 0 pressure gradient is “favorable” ∂p/∂x > 0 pressure gradient is “adverse” or “uphill” Rapid growth of boundary layer and large w field causes “flow separation”

6. from Kundu’s book u = 0 Separation point = boundary between forward flow and backward flow near wall Drag caused by adverse pressure gradient = form drag Boundary layer equations only valid as far as the point of separation

7. Analytical solutions of viscous flows can be found for Re << 1 Negligible inertial forces – Couette & Poiseuille flows

8. Analytical solutions of viscous flows can be found for Re >> 1 Negligible viscous forces, except near a surface -- match irrotational outer (freestream) flow with boundary layer near surface

9. Low ReLow Re << 11×10 3 < Re < 2×10 5 Re > 2×10 5 For intermediate Re, more difficult analytical solutions – experiments and numerical solutions

10. www.soton.ac.uk/ses/outreach/greenpower/boundarylayers.html

11. Another example https://www.arl.psu.edu/ capabilities/fsm.html

12. http://www.iafr.eu/TESI/5.htm#_Toc243930577

13. http://upload.wikimedia.org/wikipedia/commons/b/b4/Vortex-street-animation.gif von Karman Vortex Street

14. Re= 50 Re= 75 Re= 120

15. http://alg.umbc.edu/usaq/archives/001854.html

16. http://www.designyourway.net/blog/inspiration/extraordinary-satellite-photos-of-earth/ Aleutian Islands Application of vortex shedding