Viscosity
–
Newtonian fluid
Shear stress
[
N/m
2
] is directly
proportional to the velocity gradient
where
[Ns/m
2
=Pa.s] is dynamic
viscosity.
Kinematic viscosity is derived by dividing
dynamic viscosity by density:
Usually viscosity depends on the
temperature, e.g. for water and lubrication
oils it decreases as the temperature
increases.
Shear thinning material is an example
of generalized Newtonian materials
(Power law):
where
k
2
and n are the Power Law
parameters (n<1)
For example, paints are usually shear
thinning liquids
Ketchup
is
Bingham
plastic
In
Figure
is apparent viscosity
Viscosity
–
Generalized
Newtonian
Classification acc. to viscosity
Newtonian
Generalized Newtonian
Non

Newtonian
Rheology
Science which studies deformation and flow under stresses (
E.C.Bingham
and
M.Rainer
in 1920)
Between traditional fluid mechanics and solid mechanics
Basic concepts
Continuum assumed
Kinematics
Conservation laws
Constitutive models
Classification
Rheology
and
shear
stress
Stress
tensor
for a
Newtonian
fluid
(
water
)
For a
fully
developed
1D
flow
:
Viscosity
is
constant
,
especially
independent
on the
shear
For a
Generalized
Newtonian
fluid
(in 1D):
i.e.
depends
on
shear
rate
,
where
the
shear
stress
is
Power

law
model
Carreau
model
Cross
model
Rheological
models
in ELMER
(
here
in 1D)
General
dimensions
2D and 3D
Example
the Power
law
:
In general
form
where
Analytical
solution
for a
laminar
flow
in a
tube
Axisymmetric
Navier

Stokes
results
in:
Power

law
viscosity
:
Pressure
gradient
from
linear
decrease
of P:
1
2
z
r
L
( 1 )
( 2 )
( 3 )
R
Analytical
solution
for a
laminar
flow
in a
tube
Boundary
conditions
:
The
resulting
analytical
solution
of the
equations
(1):
Major
head
losses
in a
tube
as a
function
of
average
velocity
U, radius R and the
Power

law
model
parameters
k and n:
( 4 )
( 5 )
( 6 )
•
k
and
n
are
the
Power

law
model
parameters
Analytical
solution
for a
laminar
flow
in a
tube
Velocity
profile
(
left
) and
stresses
(
right
)
change
as a
function
of n
Turbulence
Random
and
transient
(
fluctuating
)
All
flow
properties
(
velocity
,
pressure
,
temperature
,
concentration
, etc.)
fluctuate
Almost
impossible
to
simulate
accurately
(
ref
. DNS =
Direct
Numerical
Simulation
based
on
transient
Navier

Stokes
equations
)
Diffusivity
Added
diffusion
w.r.t
.
laminar
flows
causes
mixing
,
increases
momentum
and
heat
transfer
, etc.
Large

scale
flow
structures
(
eddies
)
are
the
driving
force
Turbulence
Link
:
Reynolds experiment
Turbulence
Always
three
dimensional
and
unsteady
Dissipation

Smallest
turbulent
eddies
are
dissipated
to
heat
Continuum
phenomenon
The
smallest
turbulent
scales
are
larger
than
the
molecular
length
scale
Flow
property
,
not
a
material
property
Flow
of the
same
fluid
can
be
laminar
of
turbulent
Turbulence
modelling

RANS
The
Reynolds

Averaged
Navier

Stokes
equations

RANS
Turbulence
modelling

RANS
2D
Navier

Stokes
equations
:
Reynolds
(1895) idea to
time

average
the N

S
equations
based
on Reynolds
decompostion
:
Time

averaged
velocity
and
pressure
Turbulence
modelling

RANS
By
substituting
the
fluctuating
velocities
to the
original
N

S
equations
and
by
time

averaging
,
we
get
New
unknown
variables
in the
equations
called
as the
Reynoulds
stresses
:
Symmetrical
stresses
,
,
totally
six
(in 3D)
Turbulence
modelling

RANS
Turbulence
modelling

RANS
Note
,
that
time

averaged
velocity
component
is
zero
,
but
the Reynolds
stresses
are
not
!
Turbulence
modelling

RANS
RANS
equations
:
Four
equations
and
ten
unknowns
in 3D
closure
problem
in
turbulence
modelling
Extra
models
are
needed
in
order
to
get
closed
system
Turbulence
transport
models
Most
RANS
models
are
based
on
Boussinesq
(1877)
eddy
viscosity
concept
It
is
assumed
that
Reynolds
stresses
are
proportional
to the
velocity
gradient
,
like
the
stresses
due
to
molecular
viscosity
Boussinesq
concept
:
where
is the
eddy
viscosity
(
turbulent
viscosity
)
Now
the RANS
equations
are
Two

equation
model
–
k

epsilon
The eddy viscosity
modelled
by using two new unknowns
The
most
popular
is the
k

ε
model
:
,
where is an experimental coefficient
,
Other models
Zero

equations models (e.g. mixing

length model)
Boundary layer simulations in aerodynamics by using
Cebeci

Smith
model or Baldwin

Lomax model
One

equation models (
Spalart

Allmaras
model)
Two

equation models (k

epsilon, RNG k

epsilon, k

omega,…)
Reynolds Stress Models
–
RSM
Algebraic models
Differential models
–
all the stresses have their own PDEs
Large Eddy Simulation
–
LES
Large

scale
phonomena
like DNS, small

scales by RANS

type
Direct Numerical Simulation
–
DNS
All scales based on transient
Navier

Stokes equations
Boundary
layers
Solid
boundaries
affect
strongly
to
turbulence
Always
(
very
narrow
)
laminar
layer
on the
wall
,
which
cause
problems
for
turbulence
models
assuming
fully
turbulent
flow
See the animation..
Structure
of the
boundary
layer
Boundary
conditions
for the
k

epsilon
model
,
where
Y
+
is the
limit
between
laminar
and
turbulent
boundary
layers
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