2. Types of flow depend on the Reynold number ,
Re =ρVd/µ
• Re < 2000 – flow is laminar
• Re > 2000 – flow is turbulent
• 2000 < Re < 4000 – flow changes from laminar to turbulent.
INTRODUCTION
3. INTRODUCTION
• The fluid particles are in extreme state of disorder
• Their movement is haphazard and large scale eddies are developed,
which results in a complete mixing of the fluid
• The velocity and pressure do not remain constant with time
• In case pipe flows- if Reynolds number is greater than 2000 the flow is
turbulent
4. Some characteristics of turbulent flows
are:
Randomness: Turbulent flows seem irregular, chaotic, and
unpredictable.
Nonlinearity: Turbulent flows arc highly nonlinear.
Diffusivity: Due to the macroscopic mixing of fluid particles, turbulent
flows are characterized by a rapid rate of diffusion of momentum and
heat.
Vorticity: Turbulence is characterized by high levels of fluctuating
vorticity. The identifiable structures in a turbulent flow are vaguely called
eddies.
Dissipation: The vortex stretching mechanism transfers energy and
vorticity to increasingly smaller scales, until the gradients become so
large that they are smeared out (i.e., dissipated) by viscosity. Turbulent
5. HISTORICAL NOTES
• The first systematic work on turbulence was carried out by Osborne
Reynolds
in 1883. His experiments in pipe flows, showed that the
flow becomes turbulent or irregular when the no dimensional ratio Re =
UL/v,
later named the Reynolds number by Sommerfeld, exceeds a certain critical
value.
• The discovery of the significance of Reynolds number and turbulent stress
has proved to be of fundamental importance in our present knowledge of
turbulence.
• In 1921 the British physicist G. I. Taylor, in a simple and elegant study of
turbulent
diffusion, introduced the idea of a correlation function.
• During the 1920s Prandtl and his student von Karman, , developed the
semicmpirical theories of turbulence. The most successful of these was the
mixing length theory.
• Meteorologist Lewis Richardson, he proposed that the turbulent kinetic
energy is transferred from large to small eddies, until it is destroyed by
6. Shear stress in turbulent flow
Velocity fluctuations cause a continuous interchange of fluid
masses between the neighboring layers ,which accompanied by a
transfer of momentum.
J. Boussinesq, a French mathematician in 1877developed an
expression for the turbulent shear stress which may be expressed
as
When viscous action is also included the total shear stress may be expressed as
7. In 1886 Reynolds developed an expression for the turbulent shear stress
which may be derived as follows.
• It is assumed that VA > VB and the relative velocity of layer A with
respect of layer B is ( VA – VB ) = vx; where vx is the fluctuating
component of velocity in the x -direction due to turbulence .
8. The corresponding turbulent shear stress exerted on the fluid
layers is equal to
by taking the time average on both sides of Eq. becomes
The stress represented by Eq. is usually termed as Reynolds stress.
9. In 1925 L. Prandtl, a German Engineer made an important
advance in this direction by presenting mixing length hypothesis,
According to Prandtl, mixing length is that distance in the transverse direction
which must be covered by a lump of fluid particles travelling with its original mea
velocity in order to make the difference between its velocity and the velocity of th
new layer
equal to the mean transverse fluctuation in turbulent flow.
the expression for turbulent shear stress becomes
10. the total shear stress at any point is the sum of the
viscous shear stress and turbulent shear stress and it
may be expressed as
11. The variables in a turbulent flow are not deterministic in details and have to
be treated as random variables
12. Reynolds stresses
• These equations contain an additional stress tensor.
These are called the Reynolds stresses.
• In turbulent flow, the Reynolds stresses are usually large compared
to the viscous stresses.
• The normal stresses are always non-zero because they contain
squared velocity fluctuations. The shear stresses would be zero if the
fluctuations were statistically independent. However, they are
correlated (amongst other reasons because of continuity) and the
shear stresses are therefore
usually also non-zero.
13. Magnitude of Turbulence :
It is the degree of turbulence, and measures how strong, violent
or intense the turbulence.
The mean square value of a variable is called the variance. The
square root of variance is called the root-mean-square (rms)
value:
Magnitude of Turbulence = Arithmetic mean of root mean
square of turbulent fluctuations
14. Intensity of turbulence :
It is the ratio of the magnitude of turbulence to the average
flow velocity at a point in the flow field
So, Intensity of Turbulence =
The time series [u(t) – u bar], obtained after subtracting the mean U of the serie
represents the fluctuation of the variable about its mean. The rms value of the
fluctuation is called the standard deviation, defined a
15. Hydrodynamically Smooth and Rough Pipe Boundaries
Hydronamically smooth pipe :
• The hight of roughness of pipe is less
than thickness of laminar sub layer of flowing
fluid.
• K < δ′
Hydronamically rough pipe :
• The hight of roughness of pipe is greater
than the thickness of laminar sub layer of
flowing fluid.
• K > δ′
16. REFERENCES
• Hydraulics and fluid mechanics including hydraulics machines – P.N MODI, S.M
• Fluid mechanics-Pijush K. Kundu, Ira M. Cohen
