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

Outline Outline 4 Filtering 4 Filtering 4 Large Scales and 4 Large Scales and Subgrid Subgrid Scales Scales 4 Subgrid 4 Subgrid Scales Stresses Scales Stresses 4 Leonard Stress 4 Leonard Stress 4 Smogorinski 4 Smogorinski Model Model 4 Cross Stress 4 Cross Stress ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi ∂ i = Decomposition Decomposition v Large Scale Subgrid Scale Mass Mass 0 ∂ = + ′ = + ′ ′ = + x t t t p p p v v v i ij ij ij i i i Filter (Gaussian) ⎛ ⎞ ∂ ∂ ∂ t v v ⎜ ⎟ ( ) ( ) ( ) ρ + = + ρ ji ∫ ′ ′ ′ Momentum i i Filtered Filtered φ = φ Momentum v f ⎜ ⎟ x G x , x x d x ∂ ∂ ∂ j i t x x ⎝ ⎠ Large Scale Large Scale j j D ( ) = − δ + µ + Newtonian Newtonian t p v v φ′ ≠ φ ≠ φ Note That Note That 0 ij ij i , j j , i ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi 1

= + + ∂ ⎛ ⎞ S R L C ∂ ∂ ∂ ∂ i = S t t t t v t t v v ⎜ ⎟ ρ + = + + ρ ji ji ji ji 0 ji ji i i v f ⎜ ⎟ ∂ ∂ ∂ ∂ ∂ x j i ⎝ t x ⎠ x x i j j j Reynolds Reynolds ′ ′ = − ρ R t v v Stresses Stresses ji j i ) ( Subgrid- -Scale Stress Tensor Scale Stress Tensor Subgrid Leonard Leonard = − ρ − L ( ) t v v v v Stresses Stresses ji j i j i = − ρ − ( ) S t v v v v Cross Cross ji j i j i = − ρ + ′ C t v v v v Stresses Stresses ji j i i j ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi ( ) ⎛ ⎞ = ν ∂ ∂ = − ρ − R v c t 2 D 1 v ⎜ ⎟ t c u u u u = + j i ji T ji D ⎜ ⎟ ji r j i j i ∂ ∂ ji 2 x x ⎝ ⎠ i j c r = Speziale Speziale 1 ) ( ) 2 Smagorinsky Smagorinsky 1 Galilean Invariance Galilean Invariance ( ν = ∆ 2 c D D Model Model l l T S k k Leonard Stresses are Evaluated Leonard Stresses are Evaluated Grid Size Smagorinsky Constant ( ) c S ≈ ( ) 3 1 ∆ = ∆ ∆ ∆ 0 . 21 = − ρ − L t v v v v 1 2 3 ji j i j i ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi 2

ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi (a) (b) ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi 3

Conclusions Conclusions 4 Direct Numerical 4 Direct Numerical Simuation Simuation 4 Filtering 4 Filtering and and Subgrid Subgrid Scales Scales 4 Subgrid 4 Subgrid Scales Stresses Scales Stresses 4 Leonard Stress and Cross Stress 4 Leonard Stress and Cross Stress 4 Smogorinski 4 Smogorinski Model Model ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi 4