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Internal Seminar, November 14th 2007.

Effects of non conformal mesh on LES

S. Rolfo

The University of Manchester, M60 1QD, UKSchool of Mechanical, Aerospace & Civil Engineering.

CFD group

The

Uni

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ity

of M

anch

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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

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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:

• 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

PWR lower Plenum

(EDFCode Saturne)

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Span = 1 cell to 64 cells on body)

Embedded refinement strategy

1 to 2 refinement with central differencingleads to spurious oscillations

2 to 3 refinement now systematically used

2 to 3 refinement now systematically used

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Energy conservation in a continuous sense

ij

jiii x

p

x

uu

t

uu

2iiuuK

t

Kuu

tt

uu iiii

2i

i

i

i

ii x

up

x

pu

x

pu

j

j

j

j

j

jiijii

jj

jii x

uK

x

Ku

x

uuuuuu

xx

uuu

22

0

iji

i

i

i upKxx

pu

x

Ku

t

K

Energy Equation

Kinetic energy Convective term

Temporal term Pressure term

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

1 2 1 1

1 2 2

1( ) ( ) ( )

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(( ) ( ) )2

n n n n nI I I I I I II

n nI I I

t t

t

IJ SIJm u ndS mass flux across face between cells I and J

(1 )IJ IJ I IJ J interpolation on IJ face

I contains the non-orthogonality correction

IJ interpolation weighing. If regular grid 1 2IJ

convection term for cell I is I IJ IJC m

1 2 1 2 1 2( (1 ) )n n nI I I I IJ IJ I J IJ IJ

J neighbours

C m m

cancel locally if

IJ is constant

1 2 1 2 1 2( (1 )( ) ( ))n n nJ J J J IJ IJ J I IJ IJ

I neighbours

C m m

1 2nI I

cancel 2x2 if

and

FV conserves mass & momentum,Energy can only be conserved?

1 2IJ

Conservation of convective flux of “energy” between cells I and J ?

Requirements: - centered in space and time, - regular mesh spacing, and no non-orthogonality corrections- mass flux may be explicit

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Energy conservation: Taylor-Green vortices test case.

)22cos()2cos(4

1

)sin()cos(

)cos()sin(

1

212

211

kxkxp

kxkxu

kxkxu

jj

j

ijj

Lerr 20

20

2)x(

)x()x(

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Taylor-Green vortices test case: mesh generation.

Two different resolutions were tested:

1. 40 x 40

2. 60 x 60

AC

ABRR

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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

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 + 5-8 refinement

Error map of U for the Cartesian mesh 60x60 + 1-2 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

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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)

AcknowledgementsAcknowledgements 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.