Momentum i i Filtered Filtered = Momentum v f - - PowerPoint PPT Presentation

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• G. Ahmadi

ME 637-Particle II

• G. Ahmadi

ME 637-Particle II

Outline Outline

4 4Filtering Filtering 4 4Large Scales and Large Scales and Subgrid Subgrid Scales Scales 4 4Subgrid Subgrid Scales Stresses Scales Stresses 4 4Leonard Stress Leonard Stress 4 4Smogorinski Smogorinski Model Model 4 4Cross Stress Cross Stress

• G. Ahmadi

ME 637-Particle II

Mass Mass

x v

i i =

∂ ∂

i j ji j i j i

f x t x v v t v ρ + ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ρ

( )

i , j j , i ij ij

v v p t + µ + δ − =

Momentum Momentum Newtonian Newtonian

• G. Ahmadi

ME 637-Particle II

i i i

v v v ′ + = p p p ′ + =

ij ij ij

t t t ′ + =

( ) ( ) ( )

∫

′ ′ φ ′ = φ

D

d , G x x x x x

φ ≠ φ ≠ φ′

Large Scale Subgrid Scale Decomposition Decomposition

Note That Note That

Filter (Gaussian)

Filtered Filtered Large Scale Large Scale

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• G. Ahmadi

ME 637-Particle II

x v

i i =

∂ ∂

i j S ji j ji j i j i

f x t x t x v v t v ρ + ∂ ∂ + ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ρ

( )

i j i j S ji

v v v v t − ρ − =

Subgrid Subgrid-

• Scale Stress Tensor

Scale Stress Tensor

• G. Ahmadi

ME 637-Particle II

C ji L ji R ji S ji

t t t t + + =

i j R ji

v v t ′ ′ ρ − =

( )

i j i j L ji

v v v v t − ρ − =

( )

j i i j C ji

v v v v t ′ + ρ − =

Reynolds Reynolds Stresses Stresses Leonard Leonard Stresses Stresses Cross Cross Stresses Stresses

• G. Ahmadi

ME 637-Particle II

Smagorinsky Smagorinsky Model Model

ji T R ji

D 2 t ν =

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ =

j i i j ji

x v x v 2 1 D

( ) (

)2

1 k k 2 S T

D D c

l l

∆ = ν

( )3

1 3 2 1

∆ ∆ ∆ = ∆

21 . cS ≈

Smagorinsky Constant Grid Size

• G. Ahmadi

ME 637-Particle II

Speziale Speziale Galilean Invariance Galilean Invariance

( )

i j i j r c ji

u u u u c t − ρ − = 1 cr =

( )

i j i j L ji

v v v v t − ρ − =

Leonard Stresses are Evaluated Leonard Stresses are Evaluated

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• G. Ahmadi

ME 637-Particle II

• G. Ahmadi

ME 637-Particle II

• G. Ahmadi

ME 637-Particle II (a) (b)

• G. Ahmadi

ME 637-Particle II

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• G. Ahmadi

ME 637-Particle II

• G. Ahmadi

ME 637-Particle II

Conclusions Conclusions

4 4Direct Numerical Direct Numerical Simuation Simuation 4 4Filtering Filtering and and Subgrid Subgrid Scales Scales 4 4Subgrid Subgrid Scales Stresses Scales Stresses 4 4Leonard Stress and Cross Stress Leonard Stress and Cross Stress 4 4Smogorinski Smogorinski Model Model

• G. Ahmadi

ME 637-Particle II