Transcript TURBULENCE MODELLING - Kettering University

TURBULENCE MODELING
A Discussion on Different Techniques used in Turbulence Modeling
-Reni Raju
Topics Covered
Concept
Definition
Methods of Solving Turbulent Equations
Navier Stokes Equation
Models
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Turbulence
Examples:
Wake of a water near the columnn of a
bridge.
Dispersion of Smoke in the atmosphere.
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Definition
 A Fluid motion in which velocity,pressure, and other flow quantities
fluctuate irregularly in time and space.
 “Turbulent Fluid motion is an irregular condition of flow in which the
various quantities show a random variation with time and space
coordinates, so that statistically distinct average values can be
obsevered.”
- Hinze
 “Turbulence is due to the formation of point or line vortice on which
some component of the velocity becomes infinite.:”
-Jean Leray
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Methods for Solving
Turbulent Equations
DIRECT NUMERICAL SIMULATION
(DNS)
LARGE-EDDY SIMULATION
(LES)
REYNOLDS AVERAGED NAVIER-STOKES
(RANS)
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Navier Stokes Equation
For a Steady, Incompressible Fluid the
Continuity and x-momentum equations
u v w


0
x y z
u u
u
1 dp  2u  2u  2u
u v w  
 2 2 2
x x
x
 dx x y z
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For turbulent flow,
u(t )  u  u' (t )
The time averages,
1
u' 
T
 
t T
t

1
u (t )  u dt 
T

t T
t
u ' (t )dt  0
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Time averaged Navier Stokes Equation
u
u
u
1 dp   2 u  2 u  2 u   u '2 u ' v' u ' w' 
u
v
w

 v 2  2  2  



x
y
z
 dx  x
y
z   x
y
z 
For all the Three Momentum Equation
 ( xx  u '2 ) ( xy  u ' v') ( xz  u ' w')


2
 ij   ( yx  v' u ') ( yy  v' ) ( yz  v' w') 

2 
(


u
'
w
'
)
(


w
'
v
'
)
(


w
'
)
zy
zz
 zx

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Turbulence Models
Integral Method
Eddy-Viscosity Models
Zero-Equation Models
One-Equation Models
Two- Equation Models
Reynolds Stress Models
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1.Integral Method
Advantages
-Computational Simplicity and Ease.
-Useful for same kind of flow.
-Easy to interpolate with experimental bench marks.
Disadvantages
-Lack of Flexibility.
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2.Eddy Viscosity models
For 2-D incompressible boundary layer
equation
u
 u ' v'  
y
or
 u ' v'

u
y
Momentum Equation,
u
u
u
1 dp
 
  u
v

v
1  
x
y
 dx
y 
v  y
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(a) ZERO-EQUATION MODELS
 u
 i  l .l 
 y






 0  0.0168 (ue  u )dy
0
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(a) ZERO-EQUATION MODELS
Advantages
-Simplest of Models satisfying the requirements.
Disadvantages
-Some ad hoc assumptions have to be made regarding boundary layer and
velocity.
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(b) ONE-EQUATION MODELS




ui
 q2 2
 q2 2

 q2 2
q2
u
v
v
1   (r )
 v Sij
 Cv 1   (r ) 2
x
y
y
y
xi
2l
where
q 2 u '2  v'2  w'2

,
2
2
Turbulence Kinetic Energy

Dimensionless Turbulent Viscosity


1  ui u j 
Sij 


2  x j xi 
Mean Strain Rate
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(b) ONE-EQUATION MODELS
Advantages
-Additional assumptions can be avoided.
-Break from the equilibrium concepts in a practical consderation.
Disadvantages
-The length scale is still a algebraic quantity.
-Computationally more difficult.
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(c) TWO-EQUATION MODELS
Turbulence K.E.
u i
k
 
T

u j
  ij

  
x j
x j x j 
k
 k 1/ 2
 k  

      2  
 x j  
 x j




2




Dissipation Rate

 u i
 
T

u j
 C1  ij




x j
k
x j x j 

  
  2 2T


  C2
k

 x j 
 u
 2 
 x2 
2
2
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(c) TWO-EQUATION MODELS
Advantages
-Overcomes the short comings of zero and one equation model.
Disadvantages
-Not appropriate to use in a viscous sublayer.
-Still need to make assumptions.
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3.Reynolds Stress Models
Advantages
-More General than Eddy-Viscosity Models.
-Better Prediction for flow with sudden changes.
-Possible Ultimate turbulence models.
Disadvantages
-None of the equations can be solved exactly.
-Computational difficulty because of more no. of PDE.
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Causes of Turbulent Motion .
Steady State.
Mass Weighted averaging.
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