introduction to turbulence and turbulent flows: … · ¥ non-universality and problems in complex...
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Introduction to Turbulence and
its Numerical Simulation
Charles Meneveau
Department of Mechanical Engineering
Center for Environmental and Applied Fluid Mechanics
Johns Hopkins University
Mechanical Engineering
SCAT 1st Latin-American Summer School
January 9th, 2007Objectives of course (I & II):
• Give an introduction to turbulent flow
characteristics and physics,
• Provide a first, basic understanding of standard
turbulence models used in Computational
Fluid Dynamics (CFD)
• Provide basic understanding of Direct Numerical (DNS)
and Large Eddy Simulation (LES)
• Illustrate state-of-the-art subgrid model for LES
and applications.
Prerequisites:
• Basic Fluid Mechanics,
• Tensors and Index Notation
Outline (I):
• Overview of turbulent flow characteristics,
• Reynolds decomposition,
• Turbulence physics and energy cascade,
•Turbulence modeling for CFD:
Eddy-viscosity and k-! model
• Filtering, Large Eddy Simulation (LES)
• Direct Numerical Simulation (DNS)
Outline (II):
• Smagorinsky model and coefficient calibration,
• Non-universality and problems in complex flows,
• Dynamic model and applications
Turbulent flows:
•multiscale,
•mixing,
•dissipative,
•chaotic,
•vortical
•well-defined statistics,
•important in practice
From: Multimedia Fluid Mechanics, Cambridge Univ. Press
Turbulent flows:
•multiscale,
•mixing,
•dissipative,
•chaotic,
•vortical
•well-defined statistics,
•important in practice
From: Multimedia Fluid Mechanics, Cambridge Univ. Press
e.g. 50 m
H
Turbulence in atmospheric boundary layer
Turbulence in reacting flows:
Premixed flame in I.C.
engine, combustion
Numerical simulation of flame
propagation in decaying
isotropic turbulence
Turbulence in aerospace systems:
LES of flow in thrust-reversers
Blin, Hadjadi & Vervisch (2002)
J. of Turbulence.
Turbulence in thermofluid equipment:
From: Multimedia Fluid Mechanics, Cambridge Univ. Press
Turbulence in turbomachines:
Lattice of wakes:
Superimposed Image of 10 Different Rotor Phases
at 500 rpm and at the Mid-span location. x=0 is at
rotor leading edge. The stage length is Ls = 203
mm.
Optical velocity field measurements in index-matched axial pumpPhase-averaged turbulent kinetic energy distribution
(Uzol, Katz & Meneveau, J. Turbomach. 2003)
Simplest turbulence: Isotropic decaying turbulence
Contractions
Active Grid
M = 6 "
1x
2x
Test Section
1u
2u
Flow
Active Grid
M = 6 "
JHU Corrsin wind tunnel
Physical laws governing fluid flow
•Conservation of mass
•Newton’s second law
•First law of thermodynamics
•Equation of state
•Some constraints in closure relations from second law of TD
•Density field
•Velocity vector field
•Pressure field
•Temperature field (or internal energy, or enthalpy etc..)
Physical quantities describing fluid flow
Navier Stokes equations for a Newtonian, incompressible fluid
Navier-Stokes equations, incompressible, Newtonian
!uj
!t+
!uku
j
!xk
= "1
#
!p
!xj
+ $%2uj+ g
j
�
!uj
!x j= 0
aj=Fj
m
Turbulence: Reynolds decomposition
!uj
!t+
!uku
j
!xk
= "1
#
!p
!xj
+ $%2uj+ g
j
�
!uj
!x j= 0
t
Turbulence: Reynolds decomposition
!uj
!t+
!uku
j
!xk
= "1
#
!p
!xj
+ $%2uj+ g
j
�
!uj
!x j= 0
!uj
!t+!u
ju
k
!xk
= "1
#
!p
!xj
+ $%2uj+ g
j"
!
!xk
uju
k" u
ju
k( )
!uj
!xj
= 0
Reynolds’ equations:
t
• Reynolds stress
• Energy cascade
• Spectral energy tensor
• Isotropic turbulence
• Kolmogorov spectrum (1941)
!uj
!t+!u
ju
k
!xk
= "1
#
!p
!xj
+ $%2uj+ g
j"
!
!xk
uju
k" u
ju
k( )
!uj
!xj
= 0
! jk
R= ujuk " ujuk
Written as velocity co-variance tensor:
! jk
R= (uj + "uj )(uk + "uk ) # ujuk = ujuk + uj "uk + uk "uj + "uj "uk # ujuk = "uj "uk
0 0
0
• Kinematic Reynolds stress (minus): • Kinematic Reynolds stress (minus):
!uj
!t+
!uju
k
!xk
= "1
#
!p
!xj
+ $%2uj+ g
j"!&
jk
R
!xk
!uj
!xj
= 0
! jk
R" #uj #uk
• Deviatoric (anisotropic part):
!uj
!t+
!uju
k
!xk
= "1
#
!p *
!xj
+ $%2uj+ g
j"!&
jk
R
!xk
! jk
R" # jk
R$1
3#mm
R% jk
• Kinematic Reynolds stress (minus):
!uj
!t+
!uju
k
!xk
= "1
#
!p *
!xj
+ $%2uj+ g
j"!&
jk
R
!xk
!uj
!xj
= 0
u1(x1, x
2, x
3,t),
u2(x1, x
2, x
3,t),
u3(x1, x
2, x
3,t),
p(x1, x
2, x
3,t)
Unknowns: mean velocity and
pressure field
Closure required for Reynolds
stress tensor: express stress
in terms of mean velocity field…
!
jk
R= func(u)
• Spectral representation of co-variance tensor:
!uj !uk =1
(2" )3
# jk (k1,k
2,k
3) d
3k$$$
! jk (k1,k2 ,k3) : Spectral tensor of turbulence
(how much energy there is in each wave vector k)
In homogeneous isotropic turbulence (simplest case, with no preferred
directions) the spectral tensor function of a vector can be expressed based on
a single scalar function of magnitude of wavenumber, E(k):
! jk (k) =1
4"k2# jk $
kjkk
k2
%&'
()*
E(k)
• What is the dependence of energy density E(k)
with wavenumber?
We now discuss the energy cascade
and Kolmogorov theory of turbulence
• Turbulence Physics: the energy cascade
(Richardson 1922, Kolmogorov 1941)
From: Multimedia Fluid Mechanics, Cambridge Univ. Press
L
!K
• Turbulence Physics: the energy cascade
(Richardson 1922, Kolmogorov 1941)
Big whorls have little whorls,
which feed on their velocity,
and little whorls have lesser whorls,
and so on to viscosity (in the molecular sense)
• Turbulence Physics: the energy cascade
(Richardson 1922, Kolmogorov 1941)
Injection of kinetic energy into
turbulence (from mean flow)
!u2
Time!
!u2
L / !u!
!u3
L
Dissipation of kinetic energy into
heat (due to molecular friction)
! !"u3
L
! !"u (r1)
3
r1
! = 2"# $ui#x
j
# $ui#x
j
+# $uj
#xi
%
&'
(
)*
! !"u (r2 )
3
r2
L
!K
Co
nsta
nt
Flu
x o
f e
ne
rgy
acro
ss s
ca
les: !
!u "1
3ui!u
i!
• Turbulence Physics: the energy cascade
(Richardson 1922, Kolmogorov 1941)
Injection of kinetic energy into
turbulence (from mean flow)
Dissipation of kinetic energy into
heat (due to molecular friction)
!
Co
nsta
nt
Flu
x o
f e
ne
rgy
acro
ss s
ca
les: !
k
kL=1
L
k! =1
!K
k
E = f (k,!)
Dimensional Analysis (Pi-theorem: 3-2=1):
E k5 /3
!2 /3
= const
E(k) = cK!2 /3k"5 /3
Solid experimental
support for K-41:
Supports (approximately)the notion that ! is the only
relevant physical scale in
the inertial range,
+ L at large scales+ " at small scales
!
E(k) = c
K!
2/3k"5/3, c
K~ 1.6
Solid line is equivalent of a 3D radial spectrum equal to
Back to Closure problem for Reynolds equations:
!T ! (velocity scale) " (length scale)
!uj
!t+
!uju
k
!xk
= "1
#
!p *
!xj
+ $%2uj+ g
j"!&
jk
R
!xk
!
jk
R= func(u)
In analogy with molecular friction:
!jk
R= "#
T
$uj
$xk
+$u
k
$xj
%
&'
(
)*
“Eddy-viscosity”
Kolmogorov:
Characterization of turbulence: minimum 2 variables
! !"u3
L
Turbulent kinetic energy: K =1
2!ui!ui=1
2!u2
! !K3/2
L
Need (K,L) or (!,L) or (K,!)
Eddy-viscosity scaling:
!T ! (velocity scale) " (length scale)
~ !u
~ K1/2
~ L
~K
3/2
!
L
!K
!u
!T= Cµ
K2
"
!T(x,t) = Cµ
K2(x,t)
"(x,t)
Transport equations for K(x,t) and !(x,t)
Empirical calibrations:
5 adjustable coefficients
!K
!t+ u
k
!K
!xk
= "#ij
R!u
i
!xj
+!!x
j
$T
%k
+ $&
'()
*+!K
!xj
&
'(
)
*+ " ,
!,!t
+ uk
!,!x
k
= C,1"#
ij
R!u
i
!xj
&
'(
)
*+,K
+!!x
j
$T
%,
+ $&
'()
*+!,!x
j
&
'(
)
*+ " C, 2
,,K
Launder & Spalding (1972) - but earlier (1940s) Kolmogorov K-# model
Cµ = 0.09
C!1= 1.44
C! 2= 1.92
"K= 1.0
"!= 1.3
!uj
!t+!u
ju
k
!xk
= "1
#!p
!xj
+ gj+
!!x
k
$T+ $( )
!uj
!xk
+!u
k
!xj
%
&'
(
)*
+
,--
.
/00
!uj
!xj
= 0
K-! model for turbulence mean flow predictions:
+ boundary (& initial) conditions
!K
!t+ u
k
!K
!xk
= "#ij
R!u
i
!xj
+!!x
j
$T
%k
+ $&
'()
*+!K
!xj
&
'(
)
*+ " ,
!,!t
+ uk
!,!x
k
= C,1"#
ij
R!u
i
!xj
&
'(
)
*+,K
+!!x
j
$T
%,
+ $&
'()
*+!,!x
j
&
'(
)
*+ " C, 2
,,K
Direct Numerical Simulation:
N-S equations:
!uj
!t+!uku j
!xk= "
!p
!x j+ #$
2uj + gj
�
!uj
!x j= 0
Moderate Re
(~ 103),
DNS possible
High Re
(~ 107),
DNS impossible
From: Multimedia Fluid Mechanics, Cambridge Univ. Press
Regions of large vorticity in isotropic turbulence
World-record DNS (Nagoya
group in Japan):
Source: Kaneda & Ishihara,
J. of Turbulence, 2006 © Taylor & Francis
On Earth Simulator (2003)
4,0963 grid points
~ 2 Terabytes at each time-step
Thousands of time-steps
$
G(x): Filter
Large-eddy-simulation (LES) and filtering:
�
!˜ u j
!t+ ˜ u k
!˜ u j
!xk
= "!˜ p
!x j
+ #$2˜ u j "
!
!xk
% jk
$
$
G(x): Filter
Large-eddy-simulation (LES) and filtering:
N-S equations:
�
!˜ u j
!t+ ˜ u k
!˜ u j
!xk
= "!˜ p
!x j
+ #$2˜ u j "
!
!xk
% jk
Filtered N-S equations:
�
!uj
!t+!ukuj
!xk= "
!p
!x j
+ #$2
u j
�
!˜ u j
!t+!ukuj
!xk
= "!˜ p
!x j
+ #$2
˜ u j
where SGS stress tensor is:
�
! ij = uiu j " ˜ u i ˜ u j
�
!uj
!x j= 0