1. 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

2. 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,

3. 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,

4. 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

5. ∂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”

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 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

11.
Another example

13. von Karman Vortex Street

14. Re= 50
Re= 75
Re= 120

16. Aleutian Islands