Boundary layer equation: Effect of Pressure Gradient on the flow in a Boundary Layer x z Boundary layer equation: Pressure gradient is found from velocity field

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

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

Deceleration adds viscous effects to make the boundary layer grow. 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 adds viscous effects to make the boundary layer grow. w is directed away from the wall – increase in boundary layer thickness with x

∂p/∂x < 0 pressure gradient is “favorable” 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”

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

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

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

Low Re << 1 1×103 < Re < 2×105 Re > 2×105 For intermediate Re, more difficult analytical solutions – experiments and numerical solutions Low Re << 1 1×103 < Re < 2×105 Re > 2×105

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

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

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

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

Re= 50 Re= 75 Re= 120

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

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