Rheology and turbulence - Noppa

Viscosity

Newtonian fluid

Shear stress
[
N/m
2
] is directly

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

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

losses

in a
tube

as a
function

of
average

velocity

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

diffusion

w.r.t
.
laminar

flows

causes

mixing
,
increases

momentum

and
heat

transfer
, etc.

Large
-
scale

flow

structures

(
eddies
)
are

the
driving

force

Turbulence

:

Reynolds experiment

Turbulence

Always

three

dimensional

and

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

,
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