Effect of Pressure Gradient on the flow in a Boundary Layer
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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
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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
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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
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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
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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”
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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
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Analytical solutions of viscous flows can be found for Re << 1 Negligible inertial forces – Couette & Poiseuille flows
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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
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Low Re << 1
1×103 < Re < 2×105
Re > 2×105
For intermediate Re, more difficult analytical solutions – experiments and numerical solutions
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www.soton.ac.uk/ses/outreach/greenpower/boundarylayers.html
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Another example
https://www.arl.psu.edu/capabilities/fsm.html
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http://upload.wikimedia.org/wikipedia/commons/b/b4/Vortex-street-animation.gif
von Karman Vortex Street
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Re= 50
Re= 75
Re= 120
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http://alg.umbc.edu/usaq/archives/001854.html
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http://www.designyourway.net/blog/inspiration/extraordinary-satellite-photos-of-earth/
Aleutian Islands
Application of vortex shedding