Transcript Internal_sem-SRolfo.ppt

The University
of Manchester
Internal Seminar, November 14th 2007.
Effects of non conformal
mesh on LES
S. Rolfo
The University of Manchester, M60 1QD, UK
School of Mechanical, Aerospace & Civil Engineering.
CFD group
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Summary
•
Introduction: LES on a complex geometry
• Conservation of mass, momentum and total
energy in the Navier-Stokes equations
• Test case: Taylor-Green vortices
• Results
• Future work
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Scales separation and levels of approximation.
DNS
LES
RANS
3
Various approaches to LES.
• Spectral method:
•High numerical accuracy
•Not suitable for complex geometries
•Finite Difference method:
• based on conservation law in differential form
• easy to implement and obtain high order scheme (on regular grid)
• not conservative method, need a special attention
• historically used only with structured meshes
• Finite Volume method:
• based on the conservation equations in their integral form
• easy formulation in any type of grid => easy implementation of
unstructured meshes
• difficult to implement higher order scheme because of the three
level of approximation (interpolation, differentiation, integration)
• Finite Element method:
PWR lower
Plenum
(EDF
Code Saturne)
• equations are approximated with polynomial functions
• Easy to use on arbitrary geometries and very strong mathematical background
• Linearized matrices not well structured => difficult to have efficient solutions
4
Embedded refinement strategy
1 to 2 refinement with
central differencing
leads to spurious
oscillations
Span = 1 cell to 64 cells on body)
2 to 3 refinement now
systematically used
2 to 3 refinement now
systematically used
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Energy conservation in a continuous sense
Energy Equation
 ui ui u j


p

ui 


 t


x

x
j
i


Kinetic energy
ui ui
K
2
ui
ui u j
x j


x j
u j
 ui ui u j  ui ui u j Ku j 

 

K
x j
x j
 2  2 x j
Temporal term
ui
Convective term
Pressure term
ui   ui ui  K
 

t t  2  t
ui
p ui p
 ui

p
xi
xi
xi
Kui pui
K

K  p ui   0



t
xi
xi
x j
Final Equation
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Colocated unstructured Finite Volumes
- Ferziger & Peric: Computational Fluiid
Dynamics, 3rd edt. Springer 2002.
-“Face based” data-structure => simple
- Fine for convection terms
- Approximations come from
interpolations and Taylor expansions from
cell centres to cell faces
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Energy conservation in a discrete sense
FV conserves mass & momentum,
Energy can only be conserved?
mIJ  
SIJ
1
2
In 1 2 ( I t ) I  (In 1  In )(In 1  In ) I t
1
 ((In 1 )2  (In )2 ) I t
2
un dS mass flux across face between cells I and J
IJ   IJ I  (1   IJ )J
I
 IJ
interpolation on IJ face
contains the non-orthogonality correction
interpolation weighing. If regular grid  IJ  1 2
CI  IJ mIJ
convection term for cell I is
Conservation of convective flux of “energy” between cells I and J ?
In 1 2CI 
Jn 1 2CJ 

(In 1 2I IJ mIJ

(Jn 1 2J (1   IJ )( mIJ )  Jn 1 2I IJ ( mIJ ))
J  neighbours
I  neighbours
 In 1 2J (1   IJ )mIJ )
cancel locally if
cancel 2x2 if
 IJ is constant
I  In 1 2
Requirements:
- centered in space and time,
- regular mesh spacing, and no non-orthogonality corrections
- mass flux may be explicit
and  IJ  1 2
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Energy conservation: Taylor-Green vortices test case.

u1   sin( kx1 )  cos( kx2 )

u2  cos( kx1 )  sin( kx2 )

1
 p  cos( 2kx1 )  cos( 2kx2)
4

errL 2 
 
j
0
j
(x)   (x)
 
i
j
0
j
(x)

2

2
j
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Taylor-Green vortices test case: mesh generation.
Two different resolutions were tested:
AB
RR 
AC
1. 40 x 40
2. 60 x 60
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Taylor vortices 40 x 40 resolution: Energy conservation.
Convective Flux Formulation: CD
Velocity-pressure: SIMPLE 1e-4
Time step: 0.01 => CFL max < 0.2
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Taylor vortices 40 x 40 resolution: Energy conservation.
Convective Flux Formulation: CD
Velocity-pressure: SIMPLE 1e-4
Time step: 0.01 => CFL max < 0.2
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Energy conservation: effects of the mesh resolution.
Convective Flux Formulation: CD
Velocity-pressure: SIMPLE 1e-4
Time step: 0.01 => CFL max < 0.2
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Energy conservation: Taylor-Green vortices test case.
Mesh smoothing for LES see also
Iaccarino & Ham, CTR briefs 05
Energy conservation: Taylor-Green vortices test case.
Error map for the U
velocity component for
the Cartesian mesh
60x60
Error map of U for the
Cartesian mesh 60x60 +
1-2 refinement.
Error map of U for the
Cartesian mesh 60x60 +
5-8 refinement
Max error where the
velocity is min and
the V component is
max.
Max error in the middle
Energy conservation: Taylor-Green vortices test case.
Velocity components are pointing in the wrong
directions.
Taylor- Green vortices for non conformal mesh
ratio 1-2. The legend refers only to this graph.
The time is 19 sec
Taylor vortices 60 x 60 resolution: Effects of the velocity pressure
coupling.
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Taylor vortices 60 x 60 resolution: Effects of the numerical scheme
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Taylor vortices 60 x 60 resolution: Effects of the velocity pressure
coupling.
Compression Factor:
control the accuracy of
the advection scheme.
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Conclusions and future work
 Using of non conformal mesh can introduce spurious oscillation in the solution
(noise).
 Refinement ratio (RR) higher then 0.75 even with-out a specific sub-pattern.
 When the Refinement Ratio is lower then 0.75 a defined sub-pattern is fundamental
to have energy conservation.
 RR = 0.5 (1-2) is affecting the solution also far from the interface, resulting in very
bad energy conservation. Mesh adaptation didn’t produce any improvements.
 Energy conservation is weakly influenced by the velocity-pressure coupling, it is
instead highly influenced by the numerical scheme.
Future work
 Different way of interpolation of the fluid at the cell faces can produce improvements
in the energy conservation properties
 Possibility of new numerical schemes that provide energy conservation in
unstructured meshes.
 Effects of the viscosity.
 Application of non conformal mesh for LES in complex geometries (Heated Rod
Bundle)
Acknowledgements
This work was carried out as part of the TSEC programme KNOO and as such we are grateful to the EPSRC for funding
under grant EP/C549465/1.
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