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FT I 2015
Alda Simões
INTERACTION WITH SURFACES
Boundary Layer
Development of a velocity distribution
Theory of the boundary layer
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Ludwig Prandtl
• Wall: null velocity (no-slip)
• Ludwig Prandtl (1904): for a fluid flowing, the velocity is disturbed by
the solid surface only on a thin layer near the surface (boundary
layer).
• The boundary layer develops whenever a flow is disturbed by the
presence of a surface.
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Development of a
boundary layer
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Growth of a boundary layer (BL)
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Magnified scale
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Turbulent boundary layer
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Reynolds number for the boundary layer
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• Thickness of the boundary layer increases from the leading edge along
the flow direction.
• Region close to the leading edge: laminar boundary layer
• Reynolds: 𝑅𝑒𝐵𝐿 =
𝜌𝑣∞ 𝑥
𝜇
• After a certain length, the boundary layer becomes turbulent.
Transition depends on:
–
–
–
–
–
Velocity of the flow
Length of the solid surface
Properties of the fluid
Turbulence out of the BL
Rouphness of the surface.
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Thickness of the BL
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Where does the BL end (at what distance from the
surface, δ)?
Where v  0.99  v
x

l
1/ 2

 x 
Laminar:   5 *  
  
1/ 5


Turbulent:
  
  0,37 * 

 x  
Laminar-turbulent transition:
ReBL = 3,5 x 105 - 3x 106
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Boundary layer
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
Formation and growth of a viscous sub-layer inside the
turbulent BL.

Viscous sub-layer: velocity gradient dvx/dy is large:
Viscous forces predominate. Newton’s law of viscosity
applies.

Turbulent region: turbulent transport of momentum
predominates, by eddies. Local velocity becomes nearly
constant inside the BL.

Between the two, a buffer zone develops, in which the
two transport mechanisms act.
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Development of a velocity profile in a pipe
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
The BL gowing in the entry section of a pipe
determines the flow regime along the pipe.
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FT I Alda Simões
FT I 2015
Alda Simões
FLOW EXTERNAL TO
OBJECTS
Outer flow
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Airplanes, rockets, ships,
submarines, cars, trucks,
motorcycles, bridges, buildings,
wind turbines, undersea cables,
offshore platforms…
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Outer flow
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Separation operations:
Sedimentation operations (solid in
a gas or liquid), extraction of
polutants, washing of gases (gas
bubbles going through a liquid)
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Outer flow
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
Flow around an immersed sphere
Forces acting on an immersed particle
 Drag coefficient
 Stokes Law
 Terminal velocity of a sphere falling under the effect of
gravity.

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Boyant force
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Forces acting at rest, Fs:
Gravity and boyant force
FI
z
Fg
Buoyancy (Arquimedes, century III B.C.):
A body immersed or floating in a
fluid at rest is subjected to a force
acting upwards, which is equal to
the weight of the fluid that is
displaced by the body.
F  FI - Fg    fl - c   Vg
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Falling shere
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Dynamic force, Fk : force resulting from
the relative velocity.
FI
z
How does the velocity
of a falling object
vary?
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Sheres: Stokes’ Law
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Laminar flow of viscous fluids
around objects.
Stokes’s law for flow around
spheres, with low Re:
𝐹𝑘 = 3𝜋𝜇𝐷𝑣∞
Components:
Tangential direction
Normal direction
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 xz
 zz
 v z v x 
  


z 
 x
v z
 v z v z 
  

  2
z 
z
 z
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http://faraday.fc.up.pt/Faraday/Recursos/multimedia/stokes_anim.gif
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Terminal velocity of a falling sphere
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Sphere falling in a viscous fluid at rest, with
uniform velocity:
FI
Fk
FI  Fk - Fg  0
4 3
4 3
 R  fl g  6 R v -  R  s g  0
3
3
gD   s   fl 
2
v 
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Fg
• Viscometers
• Sedimentation
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Reynolds number immersed objects
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The drag coefficient depends on the
Reynolds number, defined for the
object-fluid system:
𝑣∞ 𝜌𝐿
𝑅𝑒𝑃 =
𝜇
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𝑣∞ : velocity of approach
L: reference length
, : properties of the fluid
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Projected area
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Sphere:
𝜋𝐷2
𝐴𝑃 =
4
Cylinder:
Cross flow
𝐴𝑃 = 𝐿𝐷
Axial flow
𝜋𝐷2
𝐴𝑃 =
4
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Drag Coefficient
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Objects
𝐷𝑟𝑎𝑔 𝑓𝑜𝑟𝑐𝑒
𝐹𝑘
𝐶𝑤 =
=
𝑃𝑟𝑜𝑗𝑒𝑐𝑡𝑒𝑑 𝑎𝑟𝑒𝑎 × 𝐾 𝐴𝑝 𝐾
Cw: Drag coefficient,
Re < 0,1: Creeping flow
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𝐶𝑤 =
𝑅𝑒𝑝
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Exemple
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Determine the uniform velocity for a plastic sphere falling freely in
a liquid at rest.
D= 5 cm; (plastic) =1,1 g cm-3
Liquid: =0,98 g cm-3 ; viscosity= 90 cP? Assume creeping flow.
R: 182 cm s-1
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Drag coefficient vs. Friction factor
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Objects
𝐷𝑟𝑎𝑔 𝑓𝑜𝑟𝑐𝑒
𝐶𝑤 =
𝑃𝑟𝑜𝑗𝑒𝑐𝑡𝑒𝑑 𝑎𝑟𝑒𝑎 × 𝐾
𝐹𝑘
𝐶𝑤 =
𝐴𝑝 𝐾
Re < 0,1: creeping flow (spheres)
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𝐶𝑤 =
𝑅𝑒𝑝
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Flow in ducts
𝑓=
𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑓𝑜𝑟𝑐𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑤𝑎𝑙𝑙
𝑊𝑒𝑡 𝑎𝑟𝑒𝑎 × 𝐾
𝑓=
𝐹𝜏
𝐴𝑤𝑒𝑡 𝐾
Re < 2100, laminar flow
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𝑓=
𝑅𝑒
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Drag Coefficient diagram
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Drag coefficient
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Drag coefficient values at high Re
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CW,1
CW,2
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Cw values
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Example
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A chimney 20 m high and with D= 2 m suffers the action of lateral
wind with 80 km/h.
Determine the lateral force that the chimney suffers.
R:
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Flow pattern vs. Re
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Fluxo em torno de objectos
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Drag coefficient for sheres
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Fk  Fg  FI
4 3
Fk   R g   s   fl 
3
4 3
 R g   s   fl 
Fk
Cw 
3
1
Ap  K
2
 R   fl v2
2
4 Dg   s   fl 
Cw  2
 fl
3v
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FI
Fk
Fg
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Calculation of terminal velocity
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In the absence of data
for calculation of Re:
• Iterative method
• Graphical method
Iterative method:
•
•
•
•
•
•
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We assume a starting velocity v∞
Calculate Re
Enter the diagrama and extract Cw
Re-calculate v∞
Use v∞ in the subsequent iteration
Stop the process when two
identical readings are made in the
diagram
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Graphical method
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Non-dimensional friction group:
3
gD
 fl
2 4
Cw Re 
esf   fl
2
3 




log Cw  log Cw  Re2  2 log Re
Plot this correlation in the diagram
Determine the interseption of the
straight line with the Cw curve.
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Determination of diameter of a shere
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Non-dimensional friction group:
3
gD
 fl
2 4
Cw Re 
esf   fl
2
3 




log Cw  log Cw  Re2  2 log Re
Plot this correlation in the diagram
Determine the interseption of the
straight line with the Cw curve.
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A drop of water approximately sherical and with a diameter of 0,5 mm
falls in air at rest. The distance made is sufficiently long for the velocity
to reach uniform movement.
What is the terminal velocity of the drop, assuming constant
dimensions?
. ρair= 1,21 kg/m3, μair = 0,0181 mPas.
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