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

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

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

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

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

Another example capabilities/fsm.html

von Karman Vortex Street

Re= 50 Re= 75 Re= 120

Aleutian Islands Application of vortex shedding