Effect of Pressure Gradient on the flow in a Boundary Layer

Pressure gradient is found from freestream (external) velocity field x

UUxp

0xp

0xp

U

zxU ,

02

2

wallzu 02

2

wallzu

2

21zu

xp

zuw

xuu

Boundary layer equation:

xz

Effect of Pressure Gradient on the flow in a Boundary Layer

In the accelerating part of the stream, 00 2

2

wallzu

xp

0xp

0xp

U

zxU ,

02

2

wallzu 02

2

wallzu

2

210zu

xp

At the wall, the boundary layer equation becomes:

xz

wallzu

xp

2

2

Effect of Pressure Gradient on the flow in a Boundary Layer

0xp

0xp

U

zxU ,

02

2

wallzu 02

2

wallzu

xz

wallzu

xp

2

2

In the decelerating part of the stream, 02

2

wallzu

inflection point

Effect of Pressure Gradient on the flow in a Boundary Layer

0xp

0xp

U

zxU ,

02

2

wallzu 02

2

wallzu

xz

Velocity distribution suggests that a ∂p/∂x > 0 contributes to thicken the boundary layer, as seen from continuity:

z

dzxuzw

zw

xu

0

0

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

0xp

0xp

U

zxU ,

02

2

wallzu 02

2

wallzu

xz

∂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

0

wallzu

Drag caused by adverse pressure gradient = form dragBoundary 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

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

Another example

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

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

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

von Karman Vortex Street

Re= 50

Re= 75

Re= 120

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

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

Aleutian Islands

Application of vortex shedding