Aula Teórica 18
Escoamento Turbulento
Turbulent Flow
• It is a chaotic flow- and thus non-permanent –
and with strong velocity gradients.
• Because the velocity varies in space and time,
the fluid is submitted to high acceleration and
thus the inertial forces are high.
• The velocity gradients being high imply that
the shear stresses are high and so is the
energy dissipation rate when compared with
laminar flow.
How to describe a turbulent flow?
t1
t2
u’
• Although velocity is continously varying in time at every point, one can
define an average velociy at each point and consequently a fluctuation.
Average turbulent flow can be
permanente or not
u
2u’
u
t
u
What should be the
integration time to
get the average
turbulent velocity?
t
How is turbulence generated?
• The laminar flow is disturbed = > a random component of
velocity is created and the disturbed fluid will disturb
neighboring fluid, spreading the disturbance.
• If the kinetic energy associated with the disturbance is
small then disturbances are dissipated and the turbulence
does not spread and the flow remains laminar.
• As disturbances are continuously created, and thus when
turbulent flow is created, turbulence remains.
• The turbulent kinetic energy is proportional to the kinetic
energy of the average flow.
• The velocity disturbance generates accelerations and thus
give rise to pressure disturbances, which produce turbulent
kinetic energy. This is the mechanism for turbulence to
spread.
How to characterise turbulence?
• Random velocity field (3D, non-permanent).
• Occurs when viscous forces are not enough to
eliminate disturbances. i.e. when inertia
forces are larger than viscous forces, i.e. at
high Reynolds numbers.
• Because it depends on Re, it depends on the
flow and not on the fluid.
• Originates high mixing.
http://www.youtube.com/watch?v=oOGXEfgKttM
Laminar velocity profil vs turbulent
profile
u
Laminar
Turbulent
t
• Turbulent profile is more uniform than laminar
profile.
Laminar Sub - layer
• Close to the wall shear is higher than in inner
layers.
• Close to the wall the speed tends to zero and
inertial forces too. So there will always be an
region where viscous forces dominate over the
forces of inertia and where the flow will keep in
laminar regime. Is the laminar sub-layer.
• On laminar sub-layer the Newton's law of
Viscosity is valid and we can affirm that in
turbulent flow shear stresses on the wall are
higher (for the same average speed) than in the
laminar flow.
Averave values and fluctuations
1
ui 
T
t T
 u dt
i
t
ui  ui  u
'
i
Taking the average value of this equation we see that the
average value of fluctuation is zero (as we already knew), as the
average value of the average value is the average value itself.
ui  ui  ui' 
Properties of the average values
1
ui 
T
t T
 u dt
i
t
• One can get them using the properties of the integral calculation:
– The average value of the average value is the average value itself,
– The average value of the addition is the addition of the average values,
– The average value of a constant times a variable is the product of the
contant times the average value of the property.
– The average value of a derivative is the derivative of the average value.
– The average value of an integral is the integral of the average value.
The Reynolds stresses
ui
ui
p

ui

 u j
 


 gi
t
x j
x i
x j x j
ui  ui  u'i
substituting :
u'i u'j
ui
ui
p

ui

 u j
 



 gi
t
x j
x i
x j x j
x j
The meaning of the Reynolds’
Stresses
u'i u'j
ui
ui
p

ui

 u j
 



 gi
t
x j
xi
x j x j
x j
• They were originated by the non linear term,
i.e. by the inertia term (advective term)
• They are proportional to the inertia force, i.e.
to the square of the velocity, while the viscous
forces are proportional to the velocity.
 visc
ui
 
x j
 turb   u'i u'j
Friction coefficient
 tot   visc   turb
ui
 
 u'i u'j
x j
ui
U

 u'i u'j
  UU
x j
 tot

D
f 



 C te
1
1
1
UD
2
2
U
U
U 2
2
2
2
1
f 
 C te
Re
Physically, when does the flow became completely rough? When
the length of the rugosity is longer than the thickness of the viscous
sub-layer!