Treatment of Boundary Conditions Advanced CFD 03
Momentum BCs ( ρv) ( ρvv) = τ p B t semi-discretized form ( ρv) ( ρv) t discretization type element discretization ( ρv) ρv ( ) t oundary terms Ω Ω ( m f v f ) = Ω τ f S f f ~n() ( ) f ~n() m f v f face discretization element discretization = Ω f ~n() ( ) ( ) f ~n() τ f S f face discretization loop over elements loop over oundary faces ( m f v f ) f ~n() face discretization ( τ f S f ) f ~n() face discretization ( ) f ~int erior n() = m f v f m v ( ) f ~int erior n() = τ f S f oundary term τ S = oundary term ( τ f S f ) F f ~int erior n() oundary term loop over interior faces
No-Slip Wall F = F F = F F F = τ wall S v v! v = v ( v e )e v! v! wall wall d! d d e! e = S S! wall τ wall = µ v d! wall S v = v x ( v x e x v y e y )e x v y ( v x e x v y e y )e y F = µ S d v x ( v x e v e x y y )e x v y ( v x e x v y e y )e y
No-Slip Wall treatment for x-component ( ρv x ) ρv x ( ) t Ω ( m f v x, f ) = i f ~int erior n() ( )Ω τ f S f f ~int erior n() ( ) i µ S d ( v x ( v x e x v y e y )e x ) a a µ S d µ S d 2 ( 1 e x ) v y e y e x µ S 2 v x 1 e x 2 d v y 1 e y ( ) v y e y e x ( ) v x e x e y similar treatment for y-component
Inlet
Outlet
ressure BCs ρ t ρv semi-discretized form ( ) = 0 ρ ρ Ω t discretization type ( ρv S) f = 0 f ~n() loop over elements element discretization ρ ρ Ω t oundary terms ρ ρ Ω t ( ) f f ~n() ρv S face discretization ( ρv S) f f ~int erior n() = 0 ( ) ρv S = 0 oundary term loop over oundary faces loop over interior faces
Boundary Terms Contriution ρ ρ ρ t ( ) f ( v ) f S f Ω ρ ρ v ρ ρ v = 0 f ~int erior n() ( ) ( v ) S oundary term V C ρ C ρ, f a p = m (n) Δt ρ,0 f ρ (n) f D f f =n() f f =n() elements contriution int erior faces contriution? oundary face contriution wall p a F = m f,0 C ρ, f ρ (n) (n) f D f ρ f ( ) = ρ (n) ρ V Δt int erior faces contriution elements contriution (n) m f ρ f ( D f p f ) T f f =n() f =n() int erior faces contriution? oundary face contriution S
No-Slip Wall ( ρv S) = 0 oundary contriution = 0 V C ρ C ρ, f a p = m (n) Δt ρ,0 f ρ (n) f D f f =n() f f =n() elements contriution int erior faces contriution wall a F = m f,0 C ρ, f ρ (n) (n) f D f ρ f int erior faces contriution p S ( ) = ρ (n) ρ V Δt elements contriution (n) m f ρ f ( D f p f ) T f f =n() f =n() int erior faces contriution
Boundary Vallues using Rhie-Chow interpolation ( ρv S) = 0 ρ (n) v ρ (n) D ( p p ) = 0 = ρ (n) v ρ (n) D ( p ) = 0 wall D p (n) (n) ( ) = 0 p = D (n) (n 1) ( ) ( D p ) T D p S assuming a profile p = cons tant profile d linear profile
Susonic Inlet Specified Velocity ( ρ ρ ) ( v v ) = ρ v ρ v ρ v ρ v oundary term == m m ρ ρ p =? m = ρv specified S
Contriutions a p = V C ρ Δt C ρ, f m (n) f ρ,0 ρ (n) f D f f =n() f f =n() int erior faces contriution C ρ, m * ρ (n) oundary face contriution p =? m = ρv specified S ( ) = ρ (n) ρ V Δt elements contriution (n) m f ρ f ( D f p f ) T f m f =n() f =n() int erior faces contriution oundary face contriution
Specified Velocity using Rhie-Chow interpolation m = ρ (n) v ρ (n) D ( p p ) p =? = m = ρ (n) v ρ (n) D ( p ) D p (n) (n) ( ) = m ρ v (n) m = ρv specified S p = m ρ v S D p (n) (n 1) (n) ( ) ( D p ) T D assuming a profile p = cons tant profile d linear profile
Susonic Inlet Specified ressure ( ρ ρ ) ( v v ) S = ρ v S ρ v ρ v ρ v = m ρ v oundary term v = ρ v ρ (n) D ( p p ) = = ρ (n) D ( p ) ρ p = 0 = ρ (n) D p ( ) D ( p T ) v =? p = p specified
Contriutions a p = V C ρ Δt C ρ, f m (n) f ρ,0 ρ (n) f D f ρ (n) D f =n() f f =n() int erior faces contriution oundary face contriution v =? a F = m f,0 C ρ, f ρ (n) (n) f D f ρ f int erior faces contriution ( ) = ρ (n) ρ V Δt elements contriution (n) m f ρ f ( D f p f ) T f m D ( p T ) f =n() f =n() oundary face contriution int erior faces contriution p = p specified
Boundary Values m = ρ (n) v ρ (n) D ( p (n) (n) p ) v = m ( (n) e v S )ρ v =? p = p specified
Susonic Outlet m = ρ (n) v ρ (n) D p (n) (n) ( ) V C ρ ( n ρ ) ( n f D f ρ ) D incompressile Δt f =n( ) oundary face contriution int erior faces contriution a p = V C ρ C ρ, f ( n) m * ( n f,0 ρ ) f D f C ρ, ( n) m * ( n ρ ) D compressile Δt f =n( ) ρ f f =n( ) ρ int erior faces contriution oundary face contriution
Susonic Outlet Specified ressure ρ ρ ρ t ( ) f ( v ) f S f Ω ρ ρ v ρ ρ v = 0 f ~int erior n() ( ) ( v ) oundary term = 0 ρ = 0 = ( ) v = v D ( ) v = D = D E D ( S E ) d ( ρ ρ ) ( v v ) = ρ v ρ v = m ρ v oundary term ρ D E ρ D ( S E ) ρ d