Chapter 3.pdf

1
Chapter 3:
Turbulence and its modeling
Ibrahim Sezai
Department of Mechanical Engineering
Eastern Mediterranean University
Fall 2015-2016
I. Sezai – Eastern Mediterranean University
ME555 : Computational Fluid Dynamics 2
Is the Flow Turbulent?
External Flows
Internal Flows
Natural Convection
5
10
5

x
Re along a surface
around an obstacle
where

UL
ReL 
where
Other factors such as free-stream
turbulence, surface conditions, and
disturbances may cause earlier
transition to turbulent flow.
L = x, D, Dh, etc.
,300
2

h
D
Re
20,000

D
Re
is the Rayleigh number

2
Introduction
For Re > Recritical => Flow becomes Turbulent.
Chaotic and random state of motion develops.
Velocity and pressure change continuously with
time
The velocity fluctuations give rise to additional
stresses on the fluid
=> these are called Reynolds stresses
We will try to model these extra stress terms
3.1 What is turbulence?
( ) ( )
u t U u t

 

3
Two Examples of Turbulent Flow
Larger Structures
Smaller Structures
In turbulent flows there are rotational flow structures called
turbulent eddies, which have a wide range of length scales.
In turbulent flow:
A streak of dye which is introduced at a point will rapidly
break up and dispersed  effective mixing
Give rise to high values of diffusion coefficient for;
- Mass
- Momentum
- and heat
All fluctuating components contain energy across a wide
range of frequencies or wave numbers
2 f
wavenumber f frequency
U

 

4
Fig.3.3 Energy
Spectrum of
turbulence behind a
grid
Scales of turbulence
8
 Largest eddies break up due to inertial forces
 Smallest eddies dissipate due to viscous forces
 Richardson Energy Cascade (1922)

5
Large Eddies:
- have large eddy Reynolds number,
- are dominated by inertia effects
- viscous effects are negligible
- are effectively inviscid
Small Eddies:
- motion is dictated by viscosity
- Re ≈ 1
- length scales : 0.1 – 0.01 mm
- frequencies : ≈ 10 kHz
Energy associated with eddy motions is dissipated and
converted into thermal internal energy
 increases energy losses.
- Largest eddies  anisotropic
- Smallest eddies  isotropic
vl

3.2 Transition from laminar to turbulent flow
Can be explained by considering the stability of
laminar flows to small disturbances
 Hydrodynamic instability
Hydrodynamic stability of laminar flows;
a) Inviscid instability: flows with velocity profile having a
point of inflection.
1. jet flows
2. mixing layers and wakes
3. boundary layers with adverse pressure gradients
b) Viscous instability: flows with laminar profile having no
point of inflection
- occurs near solid walls

6
Fig. 3.4 Velocity profiles
susceptible to
a) Inviscid instability and
b) Viscous instability
Transition to Turbulence
Jet flow: (example of a flow with point of inflection in velocity
profile)
Fig. 3.5. transition in a
jet flow

7
Boundary layers on a flat plate (Example with no
inflection point in the velocity profile)
The unstable two-dimensional disturbances are called Tolmien-
Schlichting (T-S) waves
Fig. 3.6 plan view
sketch of transition
processes in boundary
layer flow over a flat
plate
Fig. 3.7 merging
of turbulent spots
and transition to
turbulence in a
natural flat plate
boundary layer

8
Common features in the transition process:
(i) The amplification of initially small disturbances
(ii) The development of areas with concentrated rotational
structures
(iii) The formation of intense small scale motions
(iv) The growth and merging of these areas of small scale motions
into fully turbulent flows
Transition to turbulence is strongly affected by:
- Pressure gradient
- Disturbance levels
- Wall roughness
- Heat transfer
The transition region often comprises only a very small fraction of
the flow domain
 Commercial CFD packages often ignore transition entirely
(classify the flow as only laminar or turbulent)
3.3Effect of turbulence on time-averaged Navier-Stokes eqns
In turbulent flow there are eddying motions of a wide range
of length scales
A domain of 0.1×0.1m contains smallest eddies of 10-100
μm size
We need mesh points
The frequency of fastest events ≈ 10 kHz Δt ≈ 100μs
needed
DNS of turbulent pipe flow of Re = 105 requires a computer
which is 10 million times faster than CRAY supercomputer
Engineers need only time-averaged properties of the flow
Let’s see how turbulent fluctuations effect the mean flow
properties
9 12
10 10


9
Descriptors of Turbulent Flow
Time average or mean
First we define the mean Φ of a flow property φ as follows:
The property φ can be thought of as the sum of a steady mean component
Φ and fluctuation component φ'
The time average of the fluctuations φ' is , by definition, zero:
0
1
( )
t
t dt
t


 
  (3.2)
( ) ( )
t t
 
  
0
1
( ) 0
t
t dt
t
 

 
 
  (3.3)
Variance, r.m.s. and turbulence kinetic energy
The spread of the fluctuations φ' about the mean Φ are measured by
   
1/2
2 2
0
1
t
rms dt
t
  

 
 
   

 

The rms values of velocity components can be measured (e.g. by hot-
wire anemometer)
The kinetic energy k (per unit mass) associated with the turbulence is
defined as
2 2 2
1
( )
2
k u v w
  
  
The turbulence intensity Ti is linked to the kinetic energy and a
reference mean flow velocity Uref as follows:
1/ 2
(2/3 )
i
ref
k
T
U

(3.4a)
(3.5)
(3.6)
   
2 2
0
1
t
dt
t
 

 

 
variance
and root mean square (r.m.s.) (3.4b)

10
Moments of different fluctuating variables
The variance is also called the second moment of the fluctuations.
If and with 0
     
   
       
Their second moment is defined as
0
1
t
dt
t
   

   

 
If velocity fluctuations in different directions were independent
random fluctuations, then would be equal to zero.
However, although are chaotic, they are not independent.
As a result the second moments are non-zero.
, and
u v u w v w
     
, and
u v w
  
, and
u v u w v w
     
(3.7)
Higher order moments
Third moment is related to skewness (asymmetry) of the
distribution of the fluctuations:
Third moment:
3 3
0
1
( ) ( )
t
dt
t
 

 

 
Fourth moment:
4 4
0
1
( ) ( )
t
dt
t
 

 

 
Fourth moment is related to kurtosis (peakedness) of the
distribution of the fluctuations:
(3.8)
(3.9)

11
Correlation functions – time and space
The autocorrelation function based on two
measurements shifted by time τ is defined as
( )
R 
  ξ
0
1
( ) ( ) ( ) ( ) ( )
t
R t t t t dt
t
        

     
   
 
Similarly, the autocorrelation function based on two
measurements shifted by a certain distance in function space
is defined as
1
( ) ( , ) ( , ) ( , ) ( , )
t t
t
R t t t t dt
t
      

        
   
 
x x ξ x x ξ
( )
R  
 
(3.10)
(3.11)
When τ = 0, or ξ = 0 and it is maximum because the
two correlations are perfectly correlated.
Since φ' is chaotic we expect that the fluctuations become
increasingly decorrelated as τ or ξ → ∞ so, Rφ'φ'(∞) → 0.
The eddies at the root of turbulence cause a certain degree of local
structure in the flow, so there will be a correlation between the values
of φ' at time t and a short time later or at a given location x and a
small distance away.
The decorrelation process will take place gradually over the lifetime
(or size scale) of a typical eddy.
The integral time and length scale represent concrete measures of the
average period or size of a turbulent eddy. They can be computed
from integrals of the autocorrelation functions Rφ'φ'(τ) or Rφ'φ'(ξ).
By analogy it is also possible to define cross-correlation functions
Rφ'ψ'(τ) or Rφ'ψ'(ξ) between pairs of different fluctuations.
2
(0)
R  
  


12
Probability density function
Probability density function P(φ*) is related to the fraction of
time that a fluctuating signal spends between φ* and φ*+d φ*:
* * * * *
( ) Prob( )
P d d
     
   
The average, variance and higher moments of the variables
and its fluctuations are related to the probability density
functions as follows:
( )
P d
   


 
( ) ( ) ( )
n n
P d
   


   
 
If n = 2  we get variance, if n = 3, 4,… we get higher order
moments.
(3.12)
(3.13a)
(3.13b)
3.4. Characteristics of simple turbulent flows
In turbulent thin shear layers,
The following 2D incompressible turbulent flows with constant P will be
considered:
Free turbulent flows
- Mixing layers
- Jet
- Wake
Boundary layer near solid walls
- Flat plate boundary layer
- Pipe flow
Data for U, will be reviewed.
These can be measured by 1) Hot wire anemometry 2) Laser doppler
anemometers
This image cannot currently be displayed.
1
x y
L

 

 

2 2 2
, ,
u v w and u v
   
    
   

13
3.4.1 Free turbulent flows
b= cross sectional layer width, y=distance in cross-stream direction
x=distance downstream the source
max
min
max max min max min
for jets for mixinglayers for wakes
U U
U U
U y y y
g f h
U b U U b U U b


     
  
     
 
     
Eddies of a wide range of
length scales are visible
Fluid from surroundings is
entrained into the jet.

14
If x is large enough  functions f, g and h are independent
of x.  such flows are called self – preserving.
The turbulence structure also reaches a self-preserving state
albeit after a greater distance from the flow source than the
mean velocity. Then
Uref= Umax – Umin for a mixing layer and wakes
Uref= Umax for jets
Form of functions f, g, h and fi varies from flow to flow
See Fig.3.10
2 2 2
1 2 3 4
2 2 2 2
ref ref ref ref
u y v y w y uv y
f f f f
U b U b U b U b
    
       
   
       
       
Fig. 3.10 mean velocity
distributions and turbulence
properties for
(a) two- dimensional mixing layer,
(b) planar turbulent jet and (c)
wake behind a solid strip

15
In Fig.3.10:
are maximum when is maximum
u' gives the largest of the normal stresses (v´and w´) .

Fluctuating velocities are not equal anisotropic structure of
turbulence
As mean velocity gradients tend to zero turbulence quantities tend
to zero turbulence can’t be sustained in absence of shear
The mean velocity gradient is also zero at the centerline of jets and
wakes.No turbulence there.
The value of is zero at the centerline of a jet and wake since
shear stress must change sign here.
u
y


2 2 2
,
u v and w and u v
    

2
max
0.15 0.40
u
u

 
u v
 

3.4.2 Flat plate boundary and pipe flow
y: distance away from the wall
Near the wall (y small) is small viscous forces dominate
Away from the wall (y large) is large inertia forces dominate.
Near the wall U only depends on y, ρ, μ and τ (wall shear stress), so
Dimensional analysis shows that
Formula (3.18) is called the law of the wall
Re
inertia forces
viscous forces

Re
Uy
v

Rey
Rey
( , , , )
w
U f y   

( )
u y
U
u f f y
u




 
 
  
 
 
(3.16)
 
1/2 1/2
( / 2) friction velocity
w f
u V C
   
 

16
Far away from the wall:
Dimensional analysis yields
The most useful form emerges if we view the wall shear stress
as the cause of a velocity deficit Umax – U which decreases
the closer we get to the edge of the boundary layer or the
pipe centerline. Thus
This formula is called the velocity – defect law.
( , , , )
w
U g y independent of
   

U y
u g
u 
  
   
 
max
U U y
g
u 
  
  
 
(3.17)
Linear sub layer – the fluid layer in contact with smooth wall
Very near the wall there is no turbulent (Reynolds) shear stresses
 flow is dominated by viscous shear
For shear stress is approximately constant,
Integrating and using U = 0 at y = 0,
After some simple algebra and making use of the definitions of u+
and y+ this leads to
Because of the linear relationship between velocity and distance
from the wall the fluid layer adjacent to the wall is often known
as the linear sub-layer
( 5)
y

( ) w
U
y
y
  

 

w y
U



u y
 
 (3.18)

17
Log-law layer – the turbulent region close to smooth wall
Outside the viscous sublayer (30 < y+ < 500) a region exists
where viscous and turbulent effects are both important.
The shear stress τ varies slowly with distance; it is assumed to
be constant and equal to τw
Assuming a length scale of turbulence lm = κy where lm is
mixing length, the following relationship can be derived:
κ = 0.4, B=5.5, E=9.8; (for smooth walls)
B decreases with roughness
κ and B are universal constants valid for all turbulent flows past smooth
walls at high Reynolds numbers.
(30 < y+ < 500)  log – law layer
1 1
ln ln( )
u y B Ey
 
  
   (3.19)
Outer layer – the inertia dominated region far from the wall
Experimental measurements show that the log-law is valid in
the region 0.02 < y ⁄ δ < 0.2.
For larger values of y the velocity-defect law (3.19) provides
the correct form.
In the overlap region the log-law and velocity-defect law have
to become equal.
Tennekes and Lumley (1972) show that a matched overlap is
obtained by assuming the following logarithmic form:
where A is constant
See Fig.3.11
max 1
ln
U U y
A Law of the Wake
u  
  
 
 
 
(3.20)

18
- The inner region: 10 to 20% of the total thickness of the wall layer; the shear stress is
(almost) constant and equal to the wall shear stress τw. Within this region there are three
zones
- the linear sub-layer: viscous stresses dominate the flow adjacent to the surface
- the buffer layer: viscous and turbulent stresses are of similar magnitude
- the log-law layer: turbulent (Reynolds) stresses dominate.
- The outer region or law-of-the-wake layer: inertia dominated core flow far from wall; free
from direct viscous effects.
Fig.3.11
For y ⁄ δ > 0.8 fluctuating velocities become almost equal
isotropic turbulence structure here. (far away the wall)
For y ⁄ δ < 0.2 large mean velocity gradients
high values of .(high turbulence
production) .
Turbulence is anisotropic near the wall.
2 2 2
,
u v and w and u v
    


19
Rules which govern the time averages of fluctuation properties
and :
.
0; ; ;
; ; ; 0
ds ds
s s

  
      
 
 
       
 
  
            
 
 
     
  
Control volume in a 2D turbulent
shear flow
Fluid entering from top into the CV
will bring in higher momentum fluid
and will accelerate the slower
moving layer. As a result, the fluid
layer will experience additional
turbulent shear stresses which are
known as the Reynolds stresses.
Since div and grad are both differentiations the above rules can be
extended to a fluctuating vector quantity a = A + a' and its
combinations with a fluctuating scalar :
To illustrate the influence of turbulent fluctuations on mean flow we consider
Substitute
       
; ;
div div div div div div
div grad div grad
  

 
    
 
a A a a a
(3.22)
 
 
 
0
1
1
1
div
u p
div u v div grad u
t x
v p
div v vdiv grad v
t y
w p
div w vdiv grad w
t z




 
   
 
 
   
 
 
   
 
u
u
u
u
(3.24a)
(3.23)
(3.24b)
(3.24c
)
; ; ; ;
u U u v V v w W w p P p
    
         
u U u
 
  

20
Consider continuity equation: note that this yields the
continuity equation
A similar process is now carried out on the x-momentum equation (3.9a).
The time averages of the individual terms in this equation can be
written as follows:
Substitution of these results gives the time-average x-momentum
equation
   
; ( )
1 1
;
u U
div u div U div u
t t
p P
vdiv grad u vdiv gradU
x x
 
 
 
  
 
 
   
 
u U u
0
div 
U (3.25)
div div

u U
1
( ) ( )
( ) ( ) ( ) ( ) ( )
U P
div U div u vdiv gradU
t x
I II III IV V

 
 
    
 
U u (3.26a)
Repetition of this process on equations (3.9b) and (3.9c) yields the
time-average y-momentum and z-momentum equations
The process of time averaging has introduced new terms (III).
Their role is additional turbulent stresses on the mean velocity
components U, V and W.
1
( ) ( )
(I) (II) (III) (IV) (V)
1
( ) ( )
(I) (II) (III) (IV) (V)
V P
div V div v vdiv gradV
t y
W P
div W div w vdiv gradW
t z


 
 
    
 
 
 
    
 
U u
U u
(3.26b)
(3.26c)

21
Placing these terms on the rhs:
2
1
( )
U P u u v u w
div U v div gradU
t x x y z

 
    
    
       
 
    
 
 
U
(3.27a)
2
1
( )
V P u v v v w
div V v div gradV
t y x y z

 
    
    
       
 
    
 
 
U
(3.27b)
2
1
( )
W P u w v w w
div W v div gradW
t z x y z

 
    
    
       
 
    
 
 
U
(3.27c)
Reynolds
equations
The extra stress terms result from six additional stresses, three normal stresses
and three shear stresses:
-These extra turbulent stresses are termed the Reynolds stresses.
-The normal stresses are always non-zero
-The shear stresses are also non-zero
If, for example, u' and v' were statistically independent fluctuations, the time
average of their product would be zero
Similar extra turbulent transport terms arise when we derive a transport
equation for an arbitrary scalar quantity. The time average transport
equation for scalar φ is
2 2 2
xx yy zz
xy yx xz zx yz zy
u v w
u v u w v w
     
        
  
     
     
        
(3.28a)
*
( ) ( )
u v w
div div grad S
t x y z
  
 
 
     
   
         
 
   
 
U (3.29)
2 2 2
, and
u v w
  
  
  
, and
u v u w v w
  
     
  
u v
 
(3.28b)

22
In compressible flow density fluctuations are usually negligible
The density-weighted averaged (or Favre-averaged) form of the compressible
turbulent flow are:
Closure problem – the need for turbulence modeling
In the continuity and Navier – Stokes equations (3.8)
and (3.9 a-c) there are 6 additional unknowns
 the Reynolds stresses
Similarly in scalar transport eqn (3.14) there are 3
extra unknowns:
Main task of turbulence modeling:
is to develop computational procedures to predict the
9 extra terms. (Reynolds stresses and scalar
transport terms).
,
u v and w
  
     

23
3.5 Turbulence models
We need expressions for :
- Reynolds stresses:
- Scalar transport terms :
2 2 2
' , , , , ,
u u v u w v v w w
       
,
u v and w
  
     
Classical models: use the Reynolds eqn’s.(all commercial CFD codes)
Large eddy simulation: the flow eqn’s are solved for the
- mean flow and largest eddies
- but the effect of the smaller eddies are modeled
- are at the research state
- calculations are too costly for engineering use.

24
energy cascade
(Richardson, 1922)
Turbulence Scales and Prediction Methods
The mixing length and k-ε models are the most widely used and
validated.
They are based on the presumption that
“there exists an analogy between the action of viscous stresses
and Reynolds stresses on the mean flow”
Viscous stresses are proportional to the rate of deformation. For
incompressible flow:
Notation:
i = 1 or j = 1  x-direction
i = 2 or j = 2  y-direction
i = 3 or j = 3  z-direction
2
j
i
ij ij
j i
u
u
s
x x
  
 


  
 
 
 
 
(2.31)

25
For example
Turbulent stresses are found to increase as the mean rate of
deformation increases.
It was proposed by Boussinesq in 1877 that
Reynolds stress could be linked to mean rate of deformation
where
This is similar to viscous stress equation
except that μ is replaced by μt, where μt = turbulent (eddy) viscosity
1 2
12
2 1
xy
u u u v
x x y x
   
   
   
    
   
   
 
 
2
3
j
i
ij i j t ij
j i
U
U
u u k
x x
    
 


 
    
 
 
 
 
(3.33)
2
j
i
ij ij
j i
U
U
s
x x
  
 


  
 
 
 
 
2 2 2
0.5( )
k u v w
  
  
Similarly; kinematic turbulent or eddy viscosity.
Turbulent momentum transport is assumed to be proportional to mean
gradients of velocity
By analogy turbulent transport of a scalar is taken to be proportional to
the gradient of the mean value of the transported quantity. In suffix
notation we get
where Γt is the turbulent diffusivity.
We introduce a turbulence Prandtl / Schmidt number as
/
t t
v  

i t
i
u
x
 

 
  

(3.34)
t
t
t

 

(3.35)

26
Experiments in many flows have shown that
Most CFD procedures use
Mixing length models:
Attempts to describe the turbulent stresses by means of simple
algebraic formulae for μt as a function of position
The k-ε models:
Two transport eqn’s (PDE’s) are solved:
1. For the turbulent kinetic energy, k
2. For the rate of dissipation of turbulent kinetic energy, ε.
Both models assume that μt is isotropic (same for u, v, and w equations)
(i.e. the ratio between Reynolds stresses and mean rate of deformation is the same in all
directions)
1
t
 
1
t
 
2
/ ( constant)
t C C k C
 
   
  

This assumption fails in many categories of flow.
It is necessary to derive and solve transport eqn’s for the 6
Reynolds stresses themselves. ( ).
These eqn’s contain:
1. Diffusion
2. Pressure strain
3. Dissipation
terms whose individual effects are unknown and cannot be measured
Reynolds stress equation models:
Solves the 6 transport equations (PDE’s) for the Reynolds stress terms
together with k and ε eqn’s., by making assumptions about
- diffusion
- pressure strain
- and dissipation terms.
2 2 2
' , , , , ,
u u v u w v v w w
       

27
Algebraic stress models:
Approximate the Reynolds stresses in terms of
algebraic equations instead of PDE type transport
equations.
Is an economical form of Reynolds stress model.
Is able to introduce anisotropic turbulence effects
into CFD simulations
3.5.1 Mixing length model
where C is a dimensionless constant of proportionality.
Turbulent viscosity is given by
This works well in simple 2-D turbulent flows where the only significant
Reynolds stress is and only significant velocity
gradient is
For such flows
where c is dimensionless constant
xy yx u v
    
  
t
v C
  (3.36)
t C
 
 
U
y


U
c
y




 (3.37)
2
: turbulent velocity scale
: length scale of largest eddies
t
m m
m
s s
 
  

 

28
Combining (3.26) and (3.27) and absorbing C and c into a new length scale
This is Prandtl’s mixing length model.
The Reynolds stress is
(3.38)
2
t m
U
v
y




m

2
xy yx m
U U
u v
y y
   
 
 
   
 
 (3.39)
The transport of scalar quantity φ is modeled as
where Γt = μt ⁄ σt and vt is found from (3.28).
σt = 0.9 for near wall flows
σt = 0.5 for jets and mixing layers Rodi(1980)
σt = 0.7 in axisymmetric jets
In Table 3.3:
y : distance from the wall
κ = 0.41 von Karman’s constant
(3.40)
t
v
y
 

 
  


29

30
3.5.2. The k – ε model
If convection and diffusion of turbulence properties
are not negligible (as in the case of recirculating
flows), then the mixing length model is not
applicable.
The k-ε model focuses on the mechanisms that affect
the turbulent kinetic energy.
Some preliminary definitions:
 
 
2 2 2
2 2 2
1
mean kinetic energy
2
1
turbulent kineticenergy
2
( ) instantaneous kinetic energy
K U V W
k u v w
k t K k
  
  
  
 
To facilitate the subsequent calculations it is common to write the components
of the rate of deformation sij and the stresses τij in tensor (matrix) form:
- Using gives
and
xx xy xz xx xy xz
ij yx yy yz ij yx yy yz
zx zy zz zx zy zz
s s s
s s s s
s s s
  
   
  
   
   
 
   
   
   
( )
ij ij ij
s t S s
 
( ) ; ( ) ;
( ) ;
1 1
( ) ( )
2 2
1
( ) ( )
2
xx xx xx yy yy yy
zz zz zz
xy xy xy yx yx yx
xz xz xz zx zx zx
U u V v
s t S s s t S s
x x y y
W w
s t S s
z z
U V u v
s t S s s t S s
y x y x
U W
s t S s s t S s
z x
 
   
 
       
   

 

   
 
 
   
   
 
        
   
   
   
 
 
 
       
 
 
 
1
2
1 1
( ) ( )
2 2
yz yz yz zy zy zy
u w
z x
V W v w
s t S s s t S s
z y z y
 
 
 

 
 
 
 
   
   
 
        
   
   
   

31
The product of a vector a and a tensor bij is a vector c
The scalar product of two tensors aij and bij is evaluated as follows
Suffix notation:
1 x – direction
2 y – direction
3 z – direction
 
11 12 13
1 2 3 21 22 23
31 32 33
1 11 2 21 3 31 1
1 12 2 22 3 32 2
1 13 2 23 3 33 3
ij i ij
T T
j
b b b
b a b a a a b b b
b b b
a b a b a b c
a b a b a b c c
a b a b a b c
 
 
   
 
 
 
   
   
     
   
   
 
   
a
c
11 11 12 12 13 13 21 21 22 22 23 23
31 31 32 32 33 33
ij ij
a b a b a b a b a b a b a b
a b a b a b
      
  
Governing equation for mean flow kinetic energy K
Reynolds equations: (3.27a), (3.27b), (3.27c)
Multiply x – component Reynolds eqn. (3.27a) by U
Multiply y – component Reynolds eqn. (3.27b) by V
Multiply z – component Reynolds eqn. (3.27c) by W
Then add the result together.
After some algebra the time-average eqn. for mean
kinetic energy of the flow is

32
Or in words, for the mean kinetic energy K, we have
 
( )
( ) 2 2
( ) ( ) ( ) ( ) ( ) ( ) ( )
ij i j ij ij i j ij
K
div K div P S u u S S u u S
t
I II III IV V VI VII

    

   
        

U U U U
(3.41)
Governing equation for turbulent kinetic energy k
1- Multiply x, y and z momentum eqns (3.24a-c) by u´, v´ and w´,respectively
2- Multiply x, y and z Reynolds eqns (3.27a-c) by u´, v´ and w´, respectively
3- subtract the two of the resulting equations
Rearrange:
 yields the equation for turbulent kinetic energy k:
( ) 1
( ) 2 2
2
( ) ( ) ( ) ( ) ( ) ( ) ( )
ij i i j ij i j ij
ij
k
div k div p s u u u s s u u S
t
I II III IV V VI VII

    
  
          
         
 
  
U u u
(3.42)

33
The viscous term (VI)
Gives a negative contribution to (3.32) due to the appearance of the sum
of squared fluctuating deformation rate s'ij
The dissipation of turbulent kinetic energy is caused by work done by the
smallest eddies against viscous stresses.
The rate of dissipation per unit mass (m2/s3) is denoted by
The k – ε model equations:
It is possible to develop similar transport eqns. for all other turbulence
quantities
The exact ε – equation, however, contains many unknown and unmeasurable
terms
The standard k – ε model(Launder and Spalding,1974) has two model eqns
(a) one for k and (b) one for ε
 
2 2 2 2 2 2
11 22 33 12 13 23
2 2 2 2 2
ij ij
s s s s s s s s
 
       
        
2 ij ij
vs s
  
  (3.43)
We use k and ε to define velocity scale and length scale
representative of the large scale turbulence as follows:
Applying the same approach as in the mixing length model
we specify the eddy viscosity as follows
where Cμ is a dimensionless constant

3/ 2
1/ 2 k
k


 


2
t
k
C C
  

 
 (3.44)

34
The standard model uses the following transport equations used for k and ε
In words the equations are
The equations contain five adjustable constants . The
standard k – ε model employs values for the constants that are arrived at by
comprehensive data fitting for a wide range of turbulent flows:
( )
( ) 2
k
t
t ij ij
k
P
k
div k div grad k S S
t


  

 

    
 
  
U




(3.45)
2
1 2
( )
( ) 2
t
t ij ij
div div grad C S S C
t k k
 


  
   

 

    
 
  
U (3.46)
1 2
0.09; 1.00; 1.30; 1.44; 1.92
k
C C C
   
 
    
(3.47)
1 2
, , , and
k
C C C
   
 
To compute the Reynolds stresses with the k – ε model (3.44-3.46),
Boussinesq relationship is used:
δij = 1 if i = j, δij = 0 if i ≠ j
Contraction gives
Using
Then which is the definition of k.
So an equal 1/3 is allocated to each normal stress component in eqn
(3.48) to have –2ρk when summed. This implies an isotropic
assumption for the normal Reynolds stresses.
2 2
2
3 3
j
i
i j t ij t ij ij
j i
U
U
u u k S k
x x
      
 


 
     
 
 
 
 
(3.48)
2 2
2 2
3 3
i i
i i t ii t
i i
U U
u u k k
x x
     
 
 
    
 
3
1 2
1 2 3
0 due to continuity
i
i
U U
U U
x x x x
 
 
   
   
3
1
2
i i
i
u u k

  

 
1 1 2 2 3 3
1
where
2
i
i i i
k u u u u u u u u u u
 
         
   
 
 

35
B.C.’s for high Re model k and ε-equations
The following boundary conditions are needed
For inlet, if no k and ε is available crude approximations can be obtained
from the
- Turbulence intensity, Ti
- Characteristic length, L (equivalent pipe diameter) :
The formulae are closely related to the mixing length formulae in section
3.5.1 and the universal distributions near a solid wall given below.
 
3/ 2
2 3/4
2
; ; 0.07
3
ref i
k
k U T C L


  


Wall boundary conditions for high Re models
Near the wall, at high Reynolds numbers, equations (3.44-3.46) of the
standard k – ε model are not integrated right to the wall. Instead,
universal behaviour of near wall flows is used.
In this region measurements of the budgets indicates that:
the rate of turbulence production = rate of dissipation
Using this assumption and , the following wall functions can be derived
for a grid point P, which is nearest to the wall, in the region 30 < y+ < 500:
where
κ = 0.41 (Von Karman’s constant)
E = 9.8 (wall roughness parameter) for smooth walls
1
ln for 30 500
u Ey y

  
   (3.19)
 
2 3
1
ln ; ; for 30 500
P
P P
u u
U
u Ey k y
u y
C
 
 

 
  
      (3.49)
2
/
t C k

  

 
1/2
friction velocity,
w
u y
u y 


 


 

36
Wall b.c.’s for high Re models: Momentum Eqns
Using k from Eqn. (3.49),
2
1/2 1/4
P P
u
k u k C
C

 

  
1/2 1/4
P P
P
P
k C y
u y
y 



 

 
In the viscous sublayer, near the wall (y+ < 5) the flow is laminar and
the velocity is given by
u y
 

The position of the interface between the laminar sublayer and the
log law layer can be found by equating Eqns. (3.18) and (3.19):
(3.18)
int int int
1
ln( ), 11.63
y Ey y

  
  
if 11.63
1
ln( ) if 11.63
P P
P
P P
y y
u
Ey y

 

 
 

 



Then, the velocity at a point P near the wall is found from
Wall b.c.’s for high Re models: k and ε-Eqns
k-equation, Eqn. (3.45), is integrated right to the wall.
However, the source term of the k-equation given by
2
k t ij ij k
s S S P
  
    
is calculated at a near wall point P by assuming local
equilibrium. Under this assumption,
the rate of turbulence production = rate of dissipation.
Thus, the rate of turbulence production is calculated from
2 1/2 1/4
1/4 1/2
, where ,
w
k w w w P
P P
U
P u u k C
y C k y
  


   


   

and ε is calculated from
3/4 3/2
P
P
P
C k
y





37
Wall b.c.’s for high Re models: k and ε-Eqns
The boundary condition for k imposed at the wall is
/ 0
k n
  
where n is normal to the wall.
The ε-equation is not solved at the wall-adjacent cells but is computed
from
3/4 3/2
P
P
P
C k
y




Wall b.c.’s for high Re models: Energy equation
The wall heat flux is given by
where T+ is calculated from the universal near wall temperature
distribution valid at high Reynolds numbers (Launder and Spalding,
1974):
Finally P is the “pee-function”, a correction function dependent on the
ratio of laminar to turbulent Prandtl numbers (Launder and Spalding,
1974)
  ,
,
,
,
,
with temperature at near wall point turbulent Prandtl number
wall temperature / Prandtl number
wall heat flux thermal conducti
T l
w P
T t
w T t
p P T t
w T l P T
w T
T T C u
T u P
q
T y
T C
q
 




 
 
 
 

   
 
 
 
 
 
 
 
   
   vity
fluid specific heat and constant pressure
p
C 
(3.50)
1/4 1/2
( ) /
w P P P w
q C C k T T T

 
  

38
Low Reynolds number k-ε models
At low Reynolds numbers (which is the case near the wall) the constants
Cμ, C1ε and C2ε in Eqns. (3.51–3.53) are not valid so these equations cannot
be integrated right to the wall. To integrate the governing equations right to
the wall, modifications to the standard k-ε model is necessary.
The equations of the low Reynolds number k – ε model, which replace
(3.44-3.46), are given below:
( )
( ) 2
t
t ij ij
k
k
div k div grad k S S
t


   

 
 

     
 
 
 
  
 
U
2
t
k
C f
 
 


2
1 1 2 2
( )
( )
2
t
t ij ij
div div grad
t
C f S S C f
k k

 


  

 
 
 
 

  
 
 
 
  
 
  
U
(3.51)
(3.53)
(3.52)
Modifications:
A viscous contribution is included
Cμ, C1ε and C2ε are multiplied by wall damping functions fμ, f1 , f2
which are functions of turbulence Reynolds number or similar
functions
As an example we quote the Lam and Bremhorst (1981) wall-
damping functions which are particularly successful:
In function the parameter is defined by . Lam and
Bremhorst use as a boundary condition.
0
y

  
2
3
2
1 2
20.5
1 exp( 0.0165Re ) 1 ;
Re
0.05
1 ; 1 exp( Re )
y
t
t
f
f f
f


 
 
   
 
 
 
 
    
 
 
 
(3.54)
1/2
/
k y v
f Rey

39
Assessment of performance
Table 3.4 k-ε model assessment
k-ε Model Implementation Details
The flow equations (3.27) can be written in tensor notation as
 
   
j
i i
i j i j
j j j i j i
j
i
i j
j j i i
U
U U P
U U u u
t x x x x x x
U
U P
u u
x x x x

  
 
 
 

 
   
 
    
 
 
 
      
 
 
 
 
 


 
 
   
 
 
 
   
 
 
 
Inserting the Boussinesq relation (3.38)
2
3
j
i
ij i j t ij
j i
U
U
u u k
x x
    
 


 
    
 
 
 
 
 
 
*
( ) j
i i
i j t
j j j i i
U
U U P
U U
t x x x x x

  
 
 

 
  
    
 
 
 
     
 
 
 
where P* = P + (2/3)ρk
(3.54a)
(3.54b)
Note that Eqn. (3.54b) is the same as that of Navier-Stokes Eqns.
given by (2.32a,b,c) with μ replaced by (μ + μt) and p by P*.

40
Rearranging
 
 
*
*
( )
( ) ( )
(
j
i i
i j t
j j j i i
j
i
t t
j j j i i
t
j
U
U U P
U U
t x x x x x
U
U P
x x x x x
x

  
   
 
 
 

 
  
    
 
 
 
     
 
 
 
  
 

  
    
   
    
   
 

 

*
*
source term
) ( ) ( )
( ) ( )
j j
i
t
j j i j i i
j
i
t t
j j j i i
U U
U P
x x x x x x
U
U P
x x x x x
 
  
   
   
   
  
     
     
     
 
  
 

  
   
   
    
   
  
Note that we have used:
 
( ) 0 0
(due to continuity if = constant, and the fluid is incompressible)
j j j
j i j i i j i
U U U
x x x x x x x
   

 
  
   
   
   
 
 
   
      
     
(3.54c)
The terms after the first parenthesis on the RHS of Eqn. (3.54c) are treated as a source .
k-ε Model Equations in Generic Form
*
( )
( ) ( )
Equation
Continuity 1 0 0
x-Momentum
y-Momentum
eff eff eff eff
eff eff eff
div div grad s
t
s
U V W P
U
x x y x z x x
U V
V
x y y



  

   
  

   


      
     
  
     
      
     
 
   

 
   
 
U
*
*
2
1 1 2 2
z-Momentum
Energy 0
Turbulence ke
Dissipation of ke
eff
eff eff eff eff
t
t
t
k
k
t
k
ij ij
W P
y z y y
U V W P
W
x z y z z z z
T
k P
C f P C f
k k
U
S S
x
 


   


 

 

  
  

   
  
 
   
  
   
      
     
  
     
      
     

 
 




 
2 2 2
2 2 2
3
* 2
1
1 1 1 1
,
2 2 2 2
(2 / 3) , , / , 1 0.05 / , 2
j
i
ij
j i
eff t t k t
U
U
V W U V W U V W
S
y z y x x z z y x x
P P k C f k f f P
  
       
 

      
       
    
         
 
     
       
         
     
       
       
   
2
2 2
2
1/2
1 exp 0.0165Re 1 20.5 / Re , 1 exp( Re ), Re / , = minimum distance to the nearest wall
Re / , 0.09, 1.00, 1.30,
j
i i
ij ij t
j i j
y t t t
y k t
U
U U
S S
x x x
f f k y
k y C

 

 
    
 

 
 
 
 
  
 
 
       
 
     1 2
0.9, 1.44, 1.92, Pr
C C
  
  
The low Reynolds number k-ε model equations can be written in generic form as

41
Source term linearization in k-ε models
When the source term is linearized as
C P P
s s s 
 
1) sP must be negative for a convergent iterative solution,
which ensures diagonal dominance of the coefficient
matrix. This is a requirement for the boundedness
criteria discussed in chapter 5.
2) sC must be positive (and sP must be negative ) to obtain
all positive  values. (Patankar, 1980)
Both k and ε are strictly positive quantities. So, to obtain
always positive results for k and ε, the source terms should
be formulated such that sC is positive and sP is negative for
k and ε equations.
Source term linearization in k equation
k-equation:
Consider first term, ρPk. Inserting into Pk
k k
s P 
 
2
/
t C f k
 
  

2
2
3
2 2
k t ij ij ij ij
k
P S S C f S S C k
 
 

  
Linearization of ρPk in the form of sk = sPP + sC would
give . However, since C3 > 0 and k > 0,
then sP > 0, which violates the boundedness criteria. We can
conclude that positive terms (e.g. Pk) cannot be linearized
using such an implicit manner.
As a result, Pk should be put into sC.
Consider the second term, –ρε. From
. Inserting Pk and ε into the source term
, gives:
3 and 0
P C
s C k s
 
2
/ t
C f k
 
  

2
/
t C f k
 
  

k k
s P 
 

42
Source term linearization in k-equation
k-equation: 2 2
2 2
, ,
k k k k
t t
C f C f
s P P k b ak b P a
   
 

 
       
For the second term, ak2, the linearization proposed by Patankar (1980) can be used:
     
*
* * * 2 * * * 2 *
( ) 2 ( ) 2
P P P P P P P P P
ds
s s k k b a k ak k k b a k ak k
dk
 
 
         
   
 
2 2
* 2 * 2 * *
( ) ( ) and 2 2
C P k P P P P
t t
C f C f
s b a k P k s ak k
   
 
 
       
Then,
where the superscript * refers to the previous iteration values. However, this
linearization does not yield a robust algorithm. A better approach is to put the
coefficient of k to the sP term. Comparing sk relation with gives:
C P P
s s s
 
 
2
* *
( )
( )
C k k
P k
t
s b P
C f
s ak k
 


 
   
Positivity of sC term in k-equation
For consistency, the turbulent viscosity, t, should be a positive
quantity.
However,
t may become negative if ε becomes negative during iterations.
t should be enforced to take positive values during iterations. The
limiter used for enforcing positive values for t is given later in Eqn.
(3.51b).
2
t
C f k
 





43
Source term linearization in ε-equation
ε-equation:
2
2
1 1 2 2
1 1 2 2
where , /
k
k
s C f P C f b a
k k
b C f P a C f k
k
  
 
 
 


   
 
Note that, since b > 0, it cannot be linearized impicitly as it gives a
positive sP.
The second term, aε2, can be linearized by transferring the coefficient
of the variable ε to the sP term, in accordance with
Then, for  =  we obtain:
where the superscript * refers to the previous iteration values.
C P P
s s s
 
 
*
1 1 *
*
2 2 *
( )
( )
C k
P
s b C f P
k
s a C f
k
 
 


 
 
   
Source term linearization in ε-equation
The source terms sP and sC of the ε-equation can be written
in terms of eddy viscosity μt using
2
t
t
C f k C f k
k
   
 


 
  
which yields
*
1 1
*
2 2
( )
( )
C k
t
P
t
C f k
s b C f P
C f k
s a C f
 
 
 
 



 

 
   

44
Positivity of sC term in k- and ε-equations
sC term should be positive to obtain positive values during the
iteration of a general variable . Rewriting the sC term for the k-
equation:
We observe that, (sC)k is positive if μt is enforced to take positive
values since Pk is always positive.
Rewriting the sC term for the -equation:
Again, we observe that, (sC) is positive if μt is enforced to take
positive values.
Then, enforcing μt to take positive values is necessary to obtain
positive results during iterations of both k- and -equations.
( ) 2
C k k t ij ij
s P S S

 
*
1 1
( )
C k
t
C f k
s C f P  
 



Limiting of eddy viscosity μt
Note that both k and ε are strictly positive quantities.
However, during iterations, undershoots may develop which
will result in negative values for either k or ε which will
give negative μt values in accordance with Eqn. (3.51):.
(3.51)
Also if ε → 0, division by zero will occur in accordance
with Eqn. (3.51).
Also, sP and sC terms of both k and ε-equations contain 1/μt
term so that if μt → 0, division by zero will occur. To
overcome these difficulties μt is limited to a small fraction
of laminar viscosity μl by using
(3.51b)
2
/
t C f k
 
  

2
4
max ,10
t l
k
C f
 
  


 
  
 

45
3.5.3 Reynolds stress equation models
The most complex classical turbulence model is Reynolds
stress equation model (RSM),
Also called :
- second order
- or second – moment closure model
Drawbacks of k – ε model emerge when it is attempted to
predict
- flows with complex strain field
- flows with significant body forces
The exact Reynolds stress transport eqn can account for the
directional effects of the Reynolds stress field.
Let (Reynolds stress)
ij
ij i j
R u u


 
  
The exact solution for the transport of takes the following form
Equation (3.45) describes six partial differential equations: one for
the transport of each of the six independent Reynolds stresses
( )
Two new terms appear compared with ke eqn (3.32)
1. . : pressure strain correction term
2. . : rotation term
ij
ij ij ij ij ij
DR
P D
Dt

       (3.55)
2 2 2
1 2 3 1 2 1 3 2 3 2 1 1 2 3 1 1 3 3 2 2 3
, , , , ,since ,
u u u u u u u and u u u u u u u u u u and u u u u
                    
  
ij

ij

ij
R

46
The convective term is
The production term is
The rotation term is
where
( )
( )
k i j
ij i j
k
U u u
C div u u
x


 

 
 

U (3.56)
(3.57)
(3.58)
j i
ij im jm
m m
U U
P R R
x x

 

  
 
 
 
2 ( )
ij k j m ikm i m jkm
u u e u u e
    
   
rotation vector
1 if , and are different and in cyclic order
1 if , and are different and in anti-cyclic order
0 if any two indices are the same
k
ijk
ijk
ijk
e i j k
e i j k
e
 
 
 

The diffusion term Dij can be modelled by the assumption that the rate of
transport of Reynolds stresses by diffusion is proportional to gradient of
Reynolds stress.
The dissipations rate εij is modeled by assuming isotropy of the small
dissipative eddies. It is set so that it effects the normal Reynolds stresses (i =
j) only and in equal measure. This can be achieved by
where ε is the dissipation rate of turbulent kinetic energy defined by (3.43).
The Kronecker delta, δij is given by δij = 1 if i = j and δij = 0 if i ≠ j
 
2
with ; 0.09 1.0
ij
t t
ij ij
m k m k
t k
R
v v
D div grad R
x x
k
v C C and
 
 



   

 
   
 
   
  
(3.59)
2
3
ij ij
 
 (3.60)

47
For Πij term:
The pressure – strain interactions are the most difficult to model
Their effect on the Reynolds stresses is caused by two distinct
physical processes:
1. Pressure fluctuations due to two eddies interacting with each
other
2. Pressure fluctuations due to the interactions of an eddy with a
region of flow of different mean velocity
Its effect is to make Reynolds stresses more isotropic and to
reduce the Reynolds shear stresses
Measurements indicate that;
The wall effect increases the isotropy of normal Reynolds
stresses by damping out fluctuations in the directions normal to
the wall and decreases the magnitude of the Reynolds shear
stresses.
A comprehensive model that accounts for all these effects is given in
Launder et al (1975). They also give the following simpler form favoured by
some commercial available CFD codes:
Turbulent kinetic energy k is
1 2
1 2
2 2
3 3
with 1.8; 0.6
ij ij ij ij ij
C R k C P P
k
C C

 
   
     
   
   
 
   
2 2 2
11 22 33 1 2 3
1 1
2 2
k R R R u u u
  
     
(3.61)

48
The six equations for Reynolds stress transport are solved along with a
model equation for the scalar dissipation rate ε. Again a more exact form is
found in Launder et al (1975), but the equation from the standard k – ε
model is used in commercial CFD for the sake of simplicity
(3.62)
2
1 1 2
1 2
2
where 1.44 and 1.92
t
t ij ij
v
D
div grad C f v S S C
Dt k k
C C
 

 
  


 
   
 
 
 
Boundary conditions for the RSM model
The usual boundary conditions for elliptic flows are required for the
solution of the Reynolds stress transport equations:
inlet: specified distributions of Rij and ε
outlet and symmetry: ∂Rij /∂n = 0 and ∂ ε/∂n = 0
free stream: Rij = 0 and ε = 0 are given
or, ∂Rij /∂n = 0 and ∂ ε/∂n = 0
solid wall: use wall functions relating Rij to either k or (uτ)2
e.g.
In the absence of any information, inlet distributions are calculated from
where ℓ is the characteristic length of the equipment (e.g. pipe diameter)
2 2 2
1 2 3 1 2
1.1 , 0.25 , 0.66 , 0.26
u k u k u k u u k
    
    
 
3 / 2
2 3 / 4
2 2 2
1 2 3
2
; ; 0.07 ;
3
1
; ; 0 ( )
2
ref i
i j
k
k U T C L
u k u u k u u i j


  
    
    



49
Boundary conditions for RSM model
For computations at high Reynolds numbers wall-function-type boundary
conditions can be used which are very similar to those of the k-ε model .
Near wall Reynolds stress values are computed from formulae such as
where cij are obtained from measurements.
Low Reynolds number modifications to the model can be incorporated to add
the effects of molecular viscosity to the diffusion terms and to account for
anisotropy in the dissipation rate in the Rij-equations.
Wall damping functions to adjust the constants of the ε-equation and a modified
dissipation rate variable
give more realistic modeling near solid walls.
ij i j ij
R u u c k
 
 
1/ 2 2
( 2 ( / ) )
v k y
 
   

Similar models exist for the 3 scalar transport terms of eqn (3.32)
in the form of PDE’s.
In commercial CFD codes a turbulent diffusion coefficient
is added to the laminar diffusion coefficient, where
for all scalars.
/
t t 
 
 
0.7

 
i
u
 

50
Assessment of RSM model
Table 3.6 Reynolds stress equation model assessment.
Asssesment of performance of RSM model
Figure 3.16
Comparison of predictions of RSM and standard k-ε model with measurements on a
high-lift Aerospatiale aerofoil: (a) pressure coefficient; (b) skin friction coefficient
Source: Leschziner, in Peyret and Krause (2000)

51
Incorrect usage of the terms: High and low Reynolds number
High or low Reynolds number turbulence models do not necessarily
refer to models which are used to simulate flows having high or low
speeds. In this usage of the term, the Reynolds number is based on the
distance from the wall. In the near-wall region the flow velocity and
distance to the wall is low so that the Reynolds number is low. Then,
low Reynolds number models refer to turbulence models which can
simulate the flow in near-wall region where the viscous effects
dominate, by integrating the equations right to the wall, without
resorting to wall functions.
High Reynolds number models on the other hand refer to models
which can simulate the flow in the fully turbulent region only, where
the Reynolds number is high. In this region the turbulence effects
dominate over the viscous effects. The model equations are not valid
in the near wall region. As a result, high Reynolds number model
equations are not integrated right to the wall. These models use wall
functions to find the variables in the near wall region.
Shortcomings of two-equation models such as k-ε model
Low Reynolds number flows: Very rapid changes occur in the
distribution of k and ε as we reach the buffer layer between the fully
turbulent region and the viscous sublayer. To force the rapid changes
to k and ε, the constants of the high Reynolds number model are
multiplied by exponential damping functions which require large
number of grid points to resolve the changes. As a result, the system
of equations become stiff which makes the numerical solution difficult
to converge.
Rapidly changing flows: The Reynolds stress is proportional
to Sij in two-equation models. This only holds in equilibrium flows
where the rates of production and dissipation of k are roughly in
balance. In rapidly changing flows this is not true.
i j
u u
  


52
Stress anisotropy: Two equation model predicts normal stresses
which are all approximately equal to –⅔ρk if a thin shear layer is
simulated. Experimental data presented in section 3.4 showed that this
is not correct. In spite of this the k-e model performs well in such
flows because the gradients of normal turbulent stresses are
small compared with the gradient of the dominant turbulent shear
stress . In more complex flows the gradients of normal
turbulent stresses are not negligible and can drive significant flows.
These effects can not be predicted by the two equation models.
Strong adverse pressure gradients and recirculation regions: This
problem is also attributable to the isotropy of the predicted normal
stresses of the k-ε model. The k-ε model overpredicts the shear stress
and suppresses separation in flows over curved walls. This is a
significant problem in flows over airfoils, e.g. in aerospace
applications.
2
i
u
 

2
i
u
 

u v
  

Extra strains: Streamline curvature, rotation and extra body forces all
give rise to additional interactions between the mean strain rate and
the Reynolds stresses. These physical effects are not captured by
standard two-equation models.
RSM model addresses most of these problems adequately, but at the
cost of additional storage and computer time.
Below we consider some of the more recent advances in turbulence
modeling that seek to address some or all of the above problems.

53
Advanced turbulence models
Advanced treatment of the near-wall region: two-layer k-ε
model
The numerical instability problems associated with the wall damping
functions used in low Reynolds number k-ε models are avoided by
subdividing the boundary layer into two regions (Jongen,1997):
1) Fully turbulent region:
In this region the standard k-ε model is used where the eddy
viscosity is defined by Eqn. (3.44);
2) Viscous region, Red < Red
*:
Only the momentum equations and the k-equation is solved
(Eqn.3.45). Turbulent viscosity in this viscous region is calculated
from
and ε is calculated from
where
* *
Re / Re , Re 50 200
d d d
y k 
   
2
, /
t t C k

  

3/2
/
k 
  
1/2
,
t v C k
 
  
Two-layer k-ε model
The length scales ℓμ and ℓε contain necessary damping effects in the
near wall region and are taken from the one–equation model of
Wolfshtein (1969):
where
y is the normal distance to the wall.
In order to avoid instabilities associated with differences between μt,t
and μt,v at the join between the fully turbulent and viscous regions, a
blending formula is used to evaluate the eddy viscosity in
 
1 exp( Re /
1 exp( Re /
l d
l d
C y A
C y A
 
 
 
  
 
  


3/4 1/2
, 2 , Re /
l l d
C C A C k d
 
 

  
2 2 / 3
ij i j t ij ij
u u S k
    
 
   

54
The turbulent viscosity is calculated from
where
The blending function λε is zero at the wall and tends to 1 in the fully
turbulent region when Red >> Red
*.
The constant A can be adjusted in order to control the sharpness of the
transition from one model to the other. For example A =1,…,10 leads
to transitions occurring within a few cells, smaller values giving
sharper transitions.
, ,
(1 )
t t t t v
 
    
  
*
Re Re
1
1 tanh
2
d d
A


 
 

 
 
 
 
 
Comparison of two-layer k-ε model with various
turbulence models in predicting Cp = (P2–P1) / (ρU1
2)
in a 2-D diffuser, (Jongen, 1997).
Geometry of the diffuser.
(Jongen, 1997)
Profiles of ε at the exit section of the
diffuser (θ = 10), for different values of the
switching parameter Red
* between the two
layers, (Jongen, 1997).

55
Strain sensitivity: RNG k-ε model
RNG k - ε model (Renormalization Group). The RNG procedure:
systematically removes the small scales of motion from the governing
equations by expressing their effects in terms of large scale motions and a
modified viscosity. (Yakhot et al 1992):
RNG k-ε model equations for high Reynolds number flows:
 
( )
( ) k eff ij ij
k
div k div grad k S
t

    

    

U
 
2
*
1 1 2 2
2
1
( )
( )
with 2 2 / 3
and ;
and 0.0845; 1.39; 1.42;
eff ij ij
ij i j t ij ij
eff t t
k
div div grad C f S C f
t k k
u u S k
k
C
C C
  

  
  
     
    
    

 

    

 
   
  
   
U
2 1.68
C  
(3.65)
(3.67)
(3.66)
 
 
1/2
0
*
1 1 0
3
1 /
and ; 2 ; 4.377; 0.012
1
ij ij
k
C C S S
 
  
  
 

     

Only the constant β is adjustable; the above value is
calculated from near wall turbulent data. All other constants
are explicitly computed as part of the RNG process.
The ε-equation has long been suspected as one of the main
sources of accuracy limitations for the standard version of
the k-ε model and the RSM.
It is, therefore, interesting to note that the model contains a
strain-dependent correction term in the constant C1ε of the
production term in the RNG model ε-equation.
Its performance is better than the k-ε model for an
expanding duct, but actually worse for a contraction with
the same ratio.

56
Spalart-Allmaras model
Has only one transport equation for kinematic eddy viscosity
parameter . Turbulent eddy viscosity is computed from
(3.68)
where fv1 is the wall damping function given by
which tends to unity for high Reynolds number flows ( ).
and fv1 → 0 at the wall.
The Reynolds stresses are computed from
(3.69)

1
t v
f
 
 
3
1 3 3
1
;
v
v
f
c
 

 
 


t
 


1
2 j
i
ij i j t ij v
j i
U
U
u u S f
x x
   
 


 
    
 
 
 
 

The transport equation for is as follows:

2
2 1 1
( )
( ) ( ) ( )
(I) (II) (III) (IV) (V) (VI)
b b w w
k k
div div grad C C C f
t x x y
   
      

   
  
      
   
    
 
U
   

   
Transport Transport of Rate of Rate of
Rate of change of by byturbulent production dissipation
of convection diffusion of of
 
  
   
 
  
(3.70)
2
2
( )
where 2 mean vorticity
1
mean vorticity tensor
2
v
ij ij
j
i
ij
j i
f
y
U
U
x x


   
    
 


   
 
 
 
 


1/6
6
3
2 6 6
1 3
1
1 , wall damping functions
1
w
v w
v w
c
f f g
f g c


 

    
 
 

57
In the k-ε model the length scale is ℓ = k3/2/ε.
In a one-equation model ℓ cannot be computed since there is no
transport equation for k and ε. However, ℓ should be specified to
determine the dissipation rate. Inspection of Eqn. (3.70) reveals that ℓ
= κy has been used as a length scale. This is also the mixing length
used in developing the log-law for wall boundary layers.
The constants are:
The model gives good performance for flows with adverse pressure
gradients. In complex geometries it is difficult to determine the length
scale, so the model is unsuitable for more general internal flows.
2 2
1 2 1 1
6
2 2 3
2 2
1
2 / 3, 0.4187, 0.1355, 0.622,
( ), min ,10 , 0.3, 2
b
v b b w b
v
w w w
C
C C C C
g r C r r r C C
y
  




     
 
     
 

 


Wilcox k-ω model
In the k-ε model the kinematic eddy viscosity is expressed as
However, ε is not the only possible length scale determining variable.
In fact, k-ω model (Wilcox,1994) uses ω = ε / k as the second
variable. Then, the length scale is ℓ = (k)1/2 / ε and
(3.71)
The Reynolds stresses are computed from Boussinesq expression:
(3.72)
The transport equation for k and ω for turbulent flow at high Reynolds
is:
3/2
, velocity scale, / length scale
t k k
   
  
 
/
t k
  

2 2
2
3 3
j
i
ij i j t ij ij t ij
j i
U
U
u u S k k
x x
       
 


 
      
 
 
 
 
*
( )
( ) ( )
t
k
k
k
div k div grad k P k
t


    

 
 

    
 
 
  
 
U

58
where
and
(3.74)
The model constants are
2
2
3
i
k t ij ij ij
j
U
P S S k
x
  

  

2
1 1
( )
( ) ( )
2
2
3
t
i
ij ij ij
j
div div grad
t
U
S S
x



  

     
 
 

  
 
 
  
 
 

   
 
 

 
U
Transport Transport of Rate of Rate of
Rate of change of or by or by turbulent production dissipation
of or convection diffusion of or of or
k
k
k k k
 
  
   
*
1 1
2.0, 2.0, 0.553, 0.075, 0.09
k 
    
    
The k-ω model initially attracted attention because integration to the
wall does not require wall-damping functions in low Reynolds
number applications.
The value of k is set to zero at the wall.
ω tends to infinity at the wall, → use very large value or use
at the near wall grid point P.
At inlet boundaries k and ω must be specified.
At outlet boundaries zero gradient conditions are used.
In a free stream k→0 and ω→0, then μt→∞ from Eqn. (3.71). This
causes problems in free stream regions. So a non-zero value of ω
should be specified. Unfortunately, results of the model tend to be
dependent on the assumed free stream value of ω.
2
1
6 / ( )
P P
y
  


59
Menter SST k-ω model
Menter (1992) noted that the results of the k-ε model are much less
sensitive to the (arbitrary) assumed values in the free stream, but its
near-wall performance is unsatisfactory for boundary layers with
adverse pressure gradients. He suggested a hybrid model using
(i) a transformation of the k-ε model into k-ω model in the near-wall
region
(ii) the standard k-ε model in the fully turbulent region far from the
wall.
The calculation of τij and the k-equation are the same as in Wilcox’s
original k-ω model, but the ε-equation is transformed into an ω-
equation by substituting ε = kω. This yields
2
2 2
,1 ,2
( ) 2
( ) ( ) 2 2
3
t i
ij ij ij
j k k
U k
div div grad S S
t x x x
 

  
        
  
   
  
  
       
   
 
   
   
 
   
 
U
(3.75)
Comparison with Eqn. (3.74) shows that (3.75) has an extra source
term (VI): the cross-diffusion term, which arises during the ε = kω
transformation of the diffusion term in the ε-equation.
,1
(II)
(I)
(III)
2
2 2
,2
(V)
(VI)
(IV)
( )
( ) ( )
2
2 2
3
t
i
ij ij ij
j k k
div div grad
t
U k
S S
x x x




  

 
     
 
 
 

  
 
 
 
  
 
 
 
  
    
 
 
  
 
U




 




 



 

60
Revised Menter SST k-ω model (2003)
Model constants:
Blending functions:
The differences between the μt values computed from the standard k-ε
model in the far field and the k-ω model near the wall may cause
instabilities. To prevent this, blending functions are introduced in the
equation to modify the cross diffusion term. Blending functions are
also used for model constants such as σω, γ and β:
(3.76)
where
C1 = constants in original k-ω model (inner constants)
C2 = constants in Menter’s transformed k-ε model (outer constants)
F1 = a blending function
*
,1 ,2 2 2
1.0, 2.0, 1.17, 0.44, 0.083, 0.09
k  
     
     
1 1 1 2
(1 )
C FC F C
  
e.g. FC = FC(ℓt / y, Rey) is a function of the ratio of turbulence ℓt =
(k)1/2/ω and distance y to the wall and a turbulence Reynolds number
Rey = y2ω/ν.
The functional form of FC is chosen such that
(i) it is zero at the wall
(ii) tends to 1 in the far field
(iii) produces a smooth transition around a distance half way between
the wall and the edge of the boundary layer.
In this way, the model combines the good near-wall behaviour of the
k-ω model with the robustness of the k-ε model in the far field.

61
Limiters
Eddy viscosity models tend to overpredict k in regions where
Sij is high. This problem is sometimes referred to as
“stagnation point anomaly” [Durbin, 1996]. To avoid such
overprediction, turbulent viscosity μt should be limited. The
limiter proposed by Medic and Durbin [2002] is
min ,
6
t k
C k C S
 
 
 
 
  
 
 
where α = 0.6
1/2
( )
ij ij
S S S

Limiter in SST k-ω model
Limiters:
The eddy viscosity in SST k-ω model is limited to give improved
performance in flows with adverse pressure gradients and wake
regions:
(3.77a)
The k-production, Pk, is limited to prevent the build-up of turbulence
in stagnation regions:
(3.77b)
1
1 2
1 2
max( , )
where 2 , a constant, a blending function
t
ij ij
a k
a SF
S S S a F




  
* 2
min 10 ,2
3
i
k t ij ij ij
j
U
P k S S k
x
     
 

  
 
 

 

62
Menter SST k-ω model: summary
C1 = 5/9, C2 = 0.44

63
Each of the constants is a blend of an inner (1) and outer (2) constants,
blended via:
where C1 → inner (1) constants, and C2 → outer (2) constants.
Note that it is generally recommended to use a production limiter
where P in the k-equation is replaced with
The boundary conditions recommended are:
1 1 1 2
(1 )
C FC F C
  
 
*
min ,10
P P k
 

2
1 1
6
10
( )
0
wall
wall
y
k






Assessment of turbulence models for aerospace applications
External Aerodynamics:
The Spalart-Allmaras, k-ω and SST k-ω models are all suitable
The SST k-ω model is most general; it gives superior performance for
zero pressure gradient and adverse pressure gradient boundary layers.
General purpose CFD:
The Spalart-Allmaras model is unsuitable, but the k-ω and SST k-ω
models can both be applied.
They both have a similar range of strengths and weaknesses as the k-ε
model and fail to include accounts of more subtle interactions between
turbulent stresses and mean flow compared with RSM.

64
The k-ε-v2-f turbulence model
In the k-ε-v2-f model of Durbin, (1991, 1993, 1995) two additional
equations, apart from the k and ε-equations, are solved:
(i) the normal stress,
(ii) a relaxation function f for the production of k, (f = Pk /k)
In usual eddy-viscosity models wall effects are accounted for through
wall functions.
In the k-ε-v2-f model the problem of accounting for the wall damping
of is simply resolved by solving its transport equation similar to
RSM model:
2
2
v
2
2
v
2 2
2 2
2 2
2 2
( )
( ) t
k
v v
div v div grad v kf
t k


    

 
 
 

 
    
 
 
  
 
U
An equation for the production of k is formulated in terms of a
function f = Pk /k:
where
Note that:
The eddy viscosity is given by:
Constants:
Boundary conditions at the wall:
2
2
2
1 2
2 / 3 /
( ( )) (1 ) k
v k P
L div grad f f C C
T k
 


 
   
1/2
max ,6 time scale
k
T

 
 
 
  
 
 
 
  1/2
3 3
2 2
2
, max , , length scale
L
k
L C C

 
 
 
 
  
 
 
 
 
  
2
2
/
t C v T

   
 
1 2
0.19, 1.4, 0.3, 0.3, 70.0
L
C C C C C
 
    
2 2
2 2 2
2 4
20
0, 0, 2 / , , normal distance to the wall
v
k v k y f y
y

 



     
2 2 2 2
/ / /
divgrad x y z
            

65
A variant of the k-ε-v2-f turbulence model: ς-f model
In the k-ε-v2-f model of Durbin (1995), f is proportional to y4 near the
wall. As a result, the boundary condition for f makes the equation
system numerically unstable.
An eddy-viscosity model based on k-ε-v2-f model of Durbin is
proposed by Hanjalic et.al. (2004), which solves a transport equation
for the velocity scales ratio instead of
Thus, the resulting model is more robust and less sensitive to grid
non-uniformities.
Eddy viscosity is defined as
where
can be interpreted as the ratio of the wall-normal velocity time scale Tv
and the general (scalar) turbulence time scale T:
2
2 /
v k
 
 2
2
v
t C kT


 

2
2 /
v k
 

2
2 / and /
v
T v T k
 

 
A variant of the k-ε-v2-f turbulence model: ς-f model
The complete set of the model equations are:
 
 
 
( )
( ) 0
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
eff Mx
eff My
eff Mz
t
k
t
k
div
t
U
div U div gradU s
t
V
div V div gradV s
t
W
div W div gradW s
t
div div grad f P
t k
k
div k div
t




 

 

 

 
   



 


 


  


  


  

 
 

    
 
 
 
  
 
 
 

  

 
U
U
U
U
U
U
1 2
2
1 2
( )
( )
1 2
( ( )) 1
3
k
t k
k
grad k P
C P C
div div grad
t T
P
L div grad f f C C
T
 


  

  



 
 
 


 
 
  

   
 
 
  
 
  

    
 
 
 
 
U

66
where
1/2
1/4
3/2 1/2 3
1 2 2
max min , ,
6
max min , ,
6
0.6, =0.22, =1.4(1+0.012/ ), =1.9, =0.65, =1, =1.3,
1
t
T
L
k
C kT
k a
T C
C S
k k
L C C
C S
a C C C C








   

  

 


 

  


 
   
 
    
   
 
 
 
 
   
 
    
 
 
 
 
 



*
*
.2, 6.0, 0.36, 85
T L
Mx eff eff eff
My eff eff eff
Mz eff eff
C C C
U V W P
s
x x y x z x x
U V W P
s
x y y y z y y
U V
s
x z y z

  
  
 
  
      
     
   
     
      
     
     
      
   
     
      
     
    
   
  
   
    
   
*
*
2 2 2
2 2 2
, (2 / 3) , 2 , 2
1 1 1
2 2 2
eff
eff t k t ij ij ij ij
ij ij
W P
z z z
P P k P S S S S S
U V W U V W U V W
S S
x y z y x x z z y

    
 
 

 
 
 
     
     
        
     
        
     
     
        
     
     
Boundary conditions at the wall:
Note that both the nominator and the denominator of fw ( f at the wall)
are proportional to y2 instead of y4 as in the Durbin’s v2-f model.
This brings improved stability to computational scheme.
2
2
2
0, 0, 2 / , , normal distance to the wall
k k y f y
y

  
     

67
3.5.4 Algebraic stress equation models (ASM)
- ASM is an economical way of accounting for the
anisotropy of Reynolds stresses.
- Avoids solving the Reynolds stress transport eqns.
- Instead, uses algebraic eqns to model Reynolds
stresses.
The Simplest method is:
To neglect the convection and diffusion terms, Cij and
Dij in the Reynolds stress equations (3.55).
A more generally applicable method is:
To assume that the sum of the convection and diffusion
terms of the Reynolds stresses is proportional to the
sum of the convection and diffusion terms of turbulent
kinetic energy
Introducing approximation (3.78) into the Reynolds stress transport equation
(3.55) with production term Pij (3.57), modeled dissipation rate term (3.60)
and pressure – strain correction term (3.61) on the right hand side yields
after some arrangement the following algebraic stress model:
- appear on both sides (on rhs within Pij ). For swirling flows, CD=0.55,
C1= 2.2
- eqn(3.79) is a set of 6 simultaneous algebraic eqns. for 6 unknowns,
- solved iteratively if k and ε are known.
- The standard k-ε model eqns has to be solved also (3.44 - 3.47)
i j
u u
 
 
 
transport of ( . . )
i j i j
ij
i j
i j ij
Du u u u Dk
D k i e div terms
Dt k Dt
u u
u u S
k

     
   
 
 
 
 
    
(3.78)
1
2 2
3 1 / 3
D
ij i j ij ij ij
C k
R u u k P P
C P
 
 
  
 
   
  
   
 
(3.79)
i j
u u
 

68
Algebraic stress model assessment
Non-linear k-ε models
The standard k-ε model uses the Boussinesq approximation (3.33)
(3.33)
and eddy viscosity expression (3.44).
(3.44)
Hence (8.80)
This relationship implies that the turbulence adjusts itself
instantaneously as it is convected through the flow domain. However,
the viscoelastic analogy holds that the adjustment does not take place
immediately. → τij should also be a function DSij / Dt. So,
In RSM, τij is actually regarded as a transported quantity.
Bringing in a dependence on DSij /Dt can be regarded as a partial
account of Reynolds stress transport, which recognises that the state
of turbulence lags behind the rapid changes.
2
3
j
i
ij i j t ij
j i
U
U
u u k
x x
    
 


 
    
 
 
 
 
2
/
t C C k

   
 

( , , , )
i j ij ij ij
u u S k
    
 
  
( , , , , )
ij
i j ij ij ij
DS
u u S k
Dt
    
 
  

69
Elaborating this idea, a non-linear k-ε model is proposed. This
approach involves the derivation of asymptotic expansions for the
Reynolds stresses which maintain terms that are quadratic in
velocity gradients.
The nonlinear k-ε model (Speziale, 1987):
The nonlinear k - ε model accounts for anisotropy.
Accounts for the secondary flow in non-circular duct flows.
2
3
2
2
2
2
3
1 1
4
3 3
where ( ) and 1.68
ij i j ij ij
o o
D im mj mn mn ij ij mm ij
ij j
o i
ij ij mj mi D
m m
k
u u k C S
k
C C S S S S S S
S U
U
S grad S S S C
t x x


    

 

 
    
 
     
 
 
 
 

       
 
  
 
U
(3.82)
The simplest non-linear eddy viscosity model
The simplest non-linear eddy viscosity model relates the Reynolds
stresses to quadratic tensor products of Sij and Ωij:
(3.83)
where
Quadratic terms allow Reynolds stresses to be different.
The model has the potential to capture anisotropy effects.
C1, C2 and C3 are constants in addition to the 5 constants of the
original k-ε model.
 
1
2
3
2
2
3
1
3
quadratic terms
1
3
ij i j t ij ij
t ik jk kl kl ij
t ik jk jk ik
t ik jk kl kl ij
u u S k
k
C S S S S
k
C S S
k
C
    
 



 

 
   

 
   
  
  

    


 
     
  
  
1/ 2( / / ), 1/ 2( / / )
ij i j j i ij i j j i
S U x U x U x U x
            

70
Cubic k-ε model
Craft et al. (1996) demonstrated that it is necessary to introduce cubic
tensor products to obtain the correct sensitising effect for interactions
between Reynolds stress production and streamline curvature.
They also included:
Variable Cμ with functional dependence on Sij and Ωij
Ad hoc modification of the ε-equation to reduce the overprediction of
the length scale, leading to poor shear stress predictions in separated
flows.
Wall damping functions to enable integration of the k- and ε-equations
to the wall through the viscous sub-layer.
The performance of cubic k-ε model is very close to that of RSM.
Large Eddy Simulation (LES)
Up to now, developing a general-purpose RANS turbulence model
suitable for a wide range of practical applications have not been
possible.
This is due to differences in the behaviour of large and small eddies.
Small eddies are nearly isotropic, but large eddies are anisotropic and
their behaviour is dictated by the geometry, boundary conditions and
body forces.
Problem dependence of the largest eddies complicates the search for
widely applicable models using RANS.
An alternative approach to the problem is large eddy simulation
(LES) which computes the larger eddies with a time dependent
simulation, but tries to model only the smaller eddies.
The universal behaviour of the smaller eddies is easier to capture with
a compact model.

71
Instead of time averaging, LES uses a spatial filtering operation to
separate the larger and smaller eddies.
To resolve all those eddies with a length scale greater than a certain
width, the method selects a filtering function and a certain cutoff
width.
Then, the filtering operation is performed on the time-dependent flow
equations.
During spatial filtering, information relating to the smaller, filtered-
out turbulent eddies is destroyed.
This, and interaction effects between the larger, resolved eddies and
the smaller unresolved ones, gives rise to sub-grid-scale stresses or
SGS stresses.
Their effect on the resolved flow must be described by means of an
SGS model.
In LES,
- the time-dependent, space filtered flow equations
- together with the SGS model of the unresolved stresses
are solved on a grid of control volumes
The solution gives:
- the mean flow and
- all turbulent eddies at scales larger than the cutoff width.

72
Spatial filtering of unsteady Navier-Stokes equations
Filters are familiar separation devices in electronics that are designed
to split an input into a desirable, retained part and an undesirable,
rejected part.
Filtering Functions:
In LES spatial filtering is done by using a filter function G(x, x′,Δ):
(3.84)
In this section overbar indicates spatial filtering, not time averaging.
Only in LES, the integration is not carried out in time but in three-
dimensional space.
1 2 3
( , ) ( , , ) ( , )
where ( , ) filtered function
( , ) original (unfiltered) function
filter cutoff width
t G t dx dx dx
t
t
 


  
  
    
 


 
  
x x x x
x
x
Common filtering functions in LES
Top hat filter:
(3.85a)
Gaussian filter:
(3.85b)
Spectral cutoff:
(3.85c)
3
1/ if / 2
( , , )
0 if / 2
G
 
   

   

  


x x
x x
x x
2
3/2
2 2
( , , ) exp , 6
G

 

 


 
  
   
   
 
   
x x
x x
3
1
sin[( ) /
( , , )
( )
i i
i i i
x x
G
x x


 
  



x x

73
The top-hat filter is used in finite volume implementations of LES.
Gaussian and spectral cutoff filters are preferred in research.
Δ is intended as an indicative measure of the size of eddies that are
retained in the computations and the eddies that are rejected.
In principle we can choose Δ to be any size, but in finite volume
method it is pointless to choose Δ < grid size, since only a single
nodal value of the variable is retained over the control volume, so all
finer detail is lost anyway.
The most common selection is
(3.86)
where Δx, Δy and Δz are the length of a cubic cell in x, y and z-
directions.
1/3
3 ( )
cell
x y z V
     
Filtering in 1-D
In LES we filter (volume average) the equations. In 1-D we get
0.5
0.5
1
( , ) ( , )
x x
x x
x t t d
x
   
 
 

 

74
Filtered unsteady Navier-Stokes equations
The unsteady Navier-Stokes equations for a fluid with constant
viscosity μ are
(2.4)
(2.37a)
(2.37b)
(2.37c)
If the flow is also incompressible → div(u) = 0, → Su, Sv, Sw = 0
 
 
 
( ) 0
( )
( ( ))
( )
( ( ))
( )
( ( ))
u
v
w
div
t
u p
div u div grad u S
t x
v p
div v div grad v S
t y
w p
div w div grad w S
t z



 

 

 

 

 
    
 
 
    
 
 
    
 
u
u
u
u
Filtering of the equations (2.4) and (2.37) yields:
(3.87)
(3.88a)
(3.88b)
(3.88c)
The overbar indicates a filtered flow variable.
The problem is, how to compute
The second term on the rhs is modeled.
 
 
 
( ) 0
( )
( ( ))
( )
( ( ))
( )
( ( ))
div
t
u p
div u div grad u
t x
v p
div v div grad v
t y
w p
div w div grad w
t z



 

 

 

 

 
   
 
 
   
 
 
   
 
u
u
u
u
( )
div u
( ) ( ) [ ( ) ( )]
div div div div
   
  
u u u u

75
Substitution of the modeled into (3.88a-c) yields the LES
momentum equations:
(3.89a)
(3.89b)
(3.89c)
The filtered momentum eqns. are similar to RANS momentum eqns.
(3.26a-c) or (3.27a-c).
The last terms (V) are caused by the filtering operation similar to the
Reynolds stresses in RANS equations that arose as a result of time
averaging.
They can be considered as a divergence of a set of stresses τij
( )
div u
 
 
 
( )
( ( )) ( ( ) ( ))
( )
( ( )) ( ( ) ( ))
( )
( ( )) ( ( ) ( ))
(I) (II) (III)
u p
div u div grad u div u div u
t x
v p
div v div grad v div v div v
t y
w p
div w div grad w div w div w
t z

   

   

   
 
     
 
 
     
 
 
     
 
u u u
u u u
u u u
(IV) (V)
The i-component of these terms can be written as follows:
(3.90a)
(3.90b)
τij are termed as sub-grid-scale stresses.
Remembering that a flow variable (x, t) can be decomposed as the
sum of
(i) the filtered function with spatial variations that are larger
than the cutoff width and are resolved by the LES computations and
(ii) ′(x, t), which contains unresolved spatial variations at length
scales smaller than the filter cutoff width,
we can write:
( ) ( ) ( )
( )
where
i i i i i i
i i
ij
j
ij i i i j i j
u u u u u v u v u w u w
div u u
x y z
x
u u u u u u
     
 

    
     
   
  



   
u u
u u
( , )
t
 x

76
(3.91)
Using this decomposition in eqn. (3.90b) we can write the first term
on the far rhs as
Then, we can write SGS stresses as
(3.92)
Leonard stresses, Lij, are due to effects at resolved scale. They are
caused due to the fact that for space filtered variables, unlike in
time averaging, where
( , ) ( , ) ( , )
t t t
  
 
x x x
( )( )
( )
i j i i j j i j i j i j i j
i j i j i j i j i j i j
u u u u u u u u u u u u u u
u u u u u u u u u u u u
     
     
     
      
   
     

LES Reynolds
Leonard stresses, cross-stresses,
stresses,
( )
ij ij
ij
ij i j i j i j i j i j i j i j
L C
R
u u u u u u u u u u u u u u
       
   
      



 



 

( ) ( )
t t
 
    
The cross stresses Cij are due to interactions between the SGS eddies
and the resolved flow.
LES Reynolds stresses Rij are caused by convective momentum
transfer due to interactions of SGS eddies and are modeled with a so-
called SGS turbulence model.
The SGS stresses (3.92) must be modeled.

Chapter 3.pdf

Recommended

Recommended

More Related Content

Similar to Chapter 3.pdf

Similar to Chapter 3.pdf (20)

More from ssusercf6d0e

More from ssusercf6d0e (20)

Recently uploaded

Recently uploaded (20)

Chapter 3.pdf