MHMT7
Momentum Heat Mass Transfer
D
     source
Dt
Turbulent flows
Turbulent flow. Statistical characteristics, kinetic energy of
fluctuations. Transition laminar/turbulence. Velocity and stress
profiles in self-preserving flows and at walls. Time averaging
of fluctuations applied to NS equations. Boussineq concept of
turbulent viscosity. Turbulent models (Prandtl, k).
Rudolf Žitný, Ústav procesní a
zpracovatelské techniky ČVUT FS 2010
MHMT7
NS equations and Turbulence
Navier Stokes equations are nonlinear even in the case that the viscosity is
constant. Primary source of nonlinearities in the Navier Stokes equations are
convective inertial terms
u
(
  (uu ))  p   2u   f
t
The convection term is a quadratic
function of velocities – source of
nonlinearities and turbulent phenomena
Relative magnitude of inertial (nonlinear) and viscous (linear) terms is
Reynolds number
 (uu )  uD
Re 

2
 u

MHMT7
Turbulence
Increasing Re increases nonlinearity of NS equations. This nonlinearity
leads to sensitivity of NS solution to flow disturbances.
Laminar flow: Re<Recrit
Distarbunce is
attenuated
Turbulent flow: Re>Recrit
Distarbunce is
amplified
Velocity field (controlled by NS equations) depends on boundary and initial
conditions. Let us assume steady boundary conditions (time independent
velocity/pressure profiles at inlets/outlets): There will be only one (or two) steady
solutions in the laminar flow regime, independent of initial conditions (starting
velocity field). On the other hand the flow field will be unsteady and depending
upon initial conditions (sometimes will be sensitive even to infinitely small
disturbances) in the turbulent flow regime.
MHMT7
Turbulence - fluctuations
Turbulence can be defined as a deterministic chaos. Velocity and pressure
fields are NON-STATIONARY (du/dt is nonzero) even if flowrate and
boundary conditions are constant. Trajectory of individual particles are
extremely sensitive to initial conditions (even the particles that are very close
at some moment diverge apart during time evolution).
Velocities, pressures, temperatures… are still solutions of NS and energy
equations, however they are non-stationary and form chaotically oscillating
turbulent vortices (eddies). Time and spatial profiles of transported properties
are characterized by fluctuations
u  u u'
Actual value
at given
time and
space
Mean
value
fluctuatio
n
MHMT7
Turbulence fluctuations
Statistics of turbulent fluctuations
Mean values
u , p,  , T
rms (root mean square)
u '2
Kinetic energy of turbulence
Intensity of turbulence
(remark: mean values of fluctuations are zero)
I (
1
k  (u '2  v '2  w '2 )
2
2
k)/u
3
Reynolds stresses  t , xx   u '2 ,  t , yy   v '2 ,  t , zz   w '2 ,  t , xy   u ' v ',  t , xz   u ' w '
MHMT7
Turbulent eddies - scales
Kinetic energy of turbulent fluctuations is the sum of energies of turbulent

eddies of different sizes
1 / /
k  ui ui   E ( )d 
2
0
Large energetic eddies (size L) break
to smaller eddies. This transformation
is not affected by viscosity
E()
Spectral energy
of turbulent
eddies
Inertial subrange (inertial effects
dominate and spectral energy depends
only upon wavenumber and )
Smallest eddies (size is called Kolmogorov
scale) disappear, because kinetic energy is
converted to heat by friction
1/L
1/
=2f/u
wavenumber (1/size of eddy)
Typical values of frequency f~10 kHz, Kolmogorov scale ~0.01 up to 0.1 mm
Kolmogorov scale  decreases with the increasing Re
MHMT7
Turbulent eddies - scales
Kolmogorov scales (the smallest turbulent eddies) follow from dimensional analysis,
assuming that everything depends only upon the kinematic viscosity  and upon the
rate of energy supply  in the energetic cascade (only for small isotropic eddies, of course)



3
4
Length scale

 

Time scale
u  
4
velocity scale
These expressions follow from dimension of viscosity  [m2/s] and the rate of
energy dissipation  [m2/s3]
   
 m 2  2  
m 2 m 2 
[m]  [  ][ 3  ]    3  
s
s
 s

1  2  2 
0    3
  3/ 4
  1 / 4
MHMT7
Transition laminar-turbulent
There always exists a steady
solution of NS equation
corresponding to the laminar flow
regime, but need not be the only
one. Sometimes even an infinitely
small disturbance is enough for a
jump to unsteady turbulent regime,
sometimes the disturbance must by
sufficiently strong (typical for
viscous instabilities). Therefore the
transition from laminar to turbulent
flow regime cannot be always
specified by a unique Recrit
because it depends upon the level
of disturbances (laminar flow can
be achieved in pipe at Re10000 in
laboratory).
Hopper
MHMT7
Transition laminar-turbulent
Hydrodynamic instabilities are classified as
Inviscid instabilities
y
Inflection – source of
instability
(max.gradient)
characterised by existence of inflection point of velocity profile
- jets
- wakes
- boundary layers wit adverse pressure gradient p>0
velocity
Viscous instabilities
y
Linear eigenvalues analysis (Orr-Sommerfeld equations)
- channels, simple shear flows (pipes)
- boundary layers with p>0
velocity
There is no inflection of velocity profile
in a pipe, however turbulent regime
exists if Re>2100
Poiseuille flow ~ 5700
Couette flow –
stable?
MHMT7
Transition laminar-turbulent
How to indentify whether the flow is laminar or turbulent ?
Experimentally
Visualization, hot wire anemometers, LDA (Laser Doppler Anemometry).
Numerical experiments
Start numerical simulation selected to unsteady laminar flow. As soon as
the solution converges to steady solution for sufficiently fine grid the flow
regime is probably laminar
Recrit
According to the value of Reynolds number using literature data of
critical Reynolds number
MHMT7
Transition laminar-turbulent
Jets
Geometry
Recrit
D
5-10
Circ.pipe
100
Coiled pipe
5000
300
Suspension
in pipe
7000
400
Cavity
1000
Plate
Baffled
channels
Couette
flow
D
D
Cross flow
Planar
channel
D/2
Geometry
Recrit
2000
D
D
D
D
8000
500000
MHMT7
Jets and mixing layer
Free flows (self preserving flows)
Jets
h
Mixing layers
u
b
umax
y
Wakes
umax
u
y
umin
u
umax
u  umin
y
 g( )
umax  umin
b
y
 f( )
h
Jet thickness ~x, mixing length ~x
const
x
const

x
Circular jet
umax 
Planar jet
umax
see Goertler, Abramovic Teorieja turbulentnych struj, Moskva 1984:
x
x
MHMT7
Turbulence at walls
Flow at walls (boundary layers)
y
Log law
Buffer
layer
Laminar sublayer
u
INNER layer
(independent of
bulk velocities)
OUTER layer
(law of wake)
w
u 

Friction
velocity
u

u  *
u
*
y 
yu* 

laminar
y+<5
u+=y+
buffer
5<y+<30
u+=-3+5lny+
Log law 30<y+<500 u+=lnEy+/
umax  u 1
y

ln(
) A
*
u
 
Turbulence at walls
MHMT7
Flow at walls (turbulent stresses)
t
Prandtl’s
model
u 2
 t  l ( )
y
2
m
y
lm   y[1  exp( )]
26
y 
5
30
yu* 

vanDriest model of
mixing length lm

y 2
 t   u ' v '  24  w [1  exp( )]
26
2
MHMT7
Time averaging fluctuation
MHMT7
Time averaging fluctuation
RANS (Reynolds Averaging of Navier Stokes eqs.)
Time
average
1

t
t t

Favre’s
average

t
t t

 (t )dt
   '
t
(t )dt
t t

     ''
 dt
t
 
 ' '

Proof: (    ')(   ')     '  '
Remark: Favre’s average differs only for compressible substances and at high
Mach number flows. Ma>1
MHMT7
Time averaging fluctuation
Continuity equation
 u 0
 u  0  u  0
Navier Stokes equations

(  u )   (  uu )  p   ( u )
t

(  u )   (  u u )  p   ( u )   (  u ' u ')
t
Reynolds stresses
 t   u ' u '
MHMT7
Time averaging fluctuation
Fourier Kirchhoff, mass transport, kinetic energy equations for transport of scalars

( )   ( u )   ( )
t

( )   (  u  )   ( )   (   ' u ')
t
Turbulent fluxes
qt    ' u '
MHMT7
Turbulent viscosity
It was Boussinesq who suggested:
Why not to calculate the turbulent
Reynolds stresses and turbulent
heat fluxes in the same way like in
laminar flows, just replacing actual
viscosity by an artificially increased
turbulent viscosity and to replace
actual driving potential (rate of
deformation) by driving potential
evaluated from averaged
velocities, temperatures,…
this is self portrait of Hopper
…and this is Joseph Boussinesq
MHMT7
Turbulent viscosity
Analogy to Fourier
law q  T
  ' u '  t 

  ' ui '  t
xi
 u ' u '  t (u  (u ) )
T
Analogy to Newton’s
law    (u  (u )T )
u j
ui
 ui ' u j '   t (

)
xi
x j
 t  2t E
1
E  (u  (u )T )
2
Rate of deformation
based upon gradient
of averaged velocities
MHMT7
Turbulent viscosity - models
Prandtl’s model of mixing length. Turbulent viscosity is derived from
analogy with gases, based upon transport of momentum by molecules (kinetic
theory of gases). Turbulent eddy (driven by main flow) represents a molecule, and
mean path between collisions of molecules is substituted by mixing length.
Excellent model for jets, wakes, boundary layer flows. Disadvantage: it fails in
recirculating flows (or in flows where transport of eddies is very important).
2 u
t   lm
y
Mixing length
h
u(y)
lm  0.075h
lm  0.09h
lm  0.07h
h
y
Circular jet
Planar jet
Mixing layers
y
lm   y[1  exp(
)]
26
Currently used algebraic models are Baldwin Lomax, and Cebecci Smith
MHMT7
Turbulent viscosity - models
Two equation models calculate turbulent viscosity from the
pairs of turbulent characteristics k-, or k-, or k-l (Rotta 1986)
k (kinetic energy of turbulent fluctuations)
 (dissipation of kinetic energy)
 (specific dissipation energy)
t  
k
m2
kg 2
kg
[ 3 s 
]
m 1
m.s
s

t   C
k2

m4
kg 4
kg
[ 3 s2 
]
m m
m.s
s3
 u3 / l
[m2/s2]
[m2/s3]
[1/s]
Wilcox (1998) k-omega model,
Kolmogorov (1942)
Jones,Launder (1972), Launder
Spalding (1974) k-epsilon model
C = 0.09 (Fluent-default)
MHMT7
Turbulent viscosity - models
Standard k- model
Transport equation for kinetic energy of turbulence (see also lecture 9)
t
 k
  (  ku )   ( k )  2t E : E  
t
k
convective transport
diffusive transport by
fluctuating turbulent
eddies
production term
(energy of main flow
converted to energy of
turbulent eddies)
dissipation kinetic
energy of turbulence to
heat
Similar transport equation for dissipation of kinetic energy
t


  (  u )   (  )  (C1 2 t E : E  C2  )
t

k
this term follows
from inspection
analysis
m2
3

1
[  s2  ]
k m
s
2
s
MHMT7
CFD modelling - FLUENT
This is example of 2 pages in Fluent’s manual (Fluent is the most frequently used program for Computer Fluid Dynamics modelling)
MHMT7
EXAM
Turbulence
MHMT7
What is important (at least for exam)
Definition of turbulence – deterministic chaos
(stability of solution of NS equations, explain role of
convective acceleration)
Statistical characteristics (what is it the second correlation?)
Kinetic energy
1
k  (u '2  v '2  w '2 )
2
Intensity of turbulence I  (
2
k)/u
3
Reynolds stresses  t , xx   u '2 ,  t , yy   v '2 ,  t , zz   w '2 ,  t , xy   u ' v ',  t , xz   u ' w '
MHMT7
What is important (at least for exam)
Transition laminar-turbulent flows. Critical Reynolds number
Re 
uD 

10
D
D
D
2100
D
400
500000
What is important (at least for exam)
MHMT7
Velocity profile at wall u+(y+)
u
u  *
u


y 
Friction velocity
yu* 

w
u 

*
Laminar sublayer y+<5
(how to calculate w?)
MHMT7
What is important (at least for exam)
Boussinesq hypotheses of turbulent viscosity
 t  2t E

(what is the difference between E and  ?)
1
E  (u  (u )T )
2
Prandtl’s model of turbulent viscosity
u
t   l
y
2
m
(what is it lm?)
Principles of k-epsilon models. Derive turbulent viscosity as a
function of k and .