convective_boundary_value_problems

Haghighi as for t n t n da un vn conv a x y

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Unformatted text preview: T ⎛ D ∂φ cos θ + D ∂φ sin θ ⎞ dΓ ⎟ I y ∫Γ ⎜ x ∂x ⎜ ⎟ ∂y ⎝ ⎠ T T (e ) = ⎛ ⎛ D ∂[N] ∂[N] + D ∂[N] ∂[N] ⎞ dA ⎞ + ⎜⎜ k ∫A ⎜ x ∂x ∂x y ∂y ∂y ⎟ ⎟ ⎟⎟ ⎜ ⎠⎠ ⎝⎝ ⎛⎛T ∂[N] ⎞ ⎞ ∂[N] T ⎜ ∫ ⎜ [N] ρ φu ⎟ dA ⎟ + [N] ρ φ v ⎜ A⎜ ∂y ⎟ ⎟ ∂x ⎠⎠ ⎝⎝ T + ∫ G[N] [N] dA A New Chapter Page 6 K. Haghighi and {f (e )}= ∫A Q[N]T dA where e) [k (e) ]= [kD(e) ]+ [k (conv ]+ [k G(e) ] New Chapter Page 7 K. Haghighi • Element Matrices: Triangular Elements [ φ (e ) = Ni N j Nk {Φ} [ [ The matrices k D(e ) , k G(e ) and {f e } after integration yields (similar to Ch 7 in Segerlind) New Chapter Page 8 K. Haghighi ⎡ b2 ⎢i (e ) = D x ⎢ b b kD ij 4A ⎢ ⎢b ib k ⎣ b ib j b2 j b jb k ⎡ c2 b ib k ⎤ ⎥ Dy ⎢ i ⎢ c ic j b jb k ⎥ + ⎥ 4A ⎢ 2 bk ⎥ ⎢c ic k ⎦ ⎣ c ic j c2 j c jc k c ic k ⎤ ⎥ c jc k ⎥ ⎥ ck ⎥ ⎦ ⎡2 1 1⎤ GA ⎢ 1 2 1⎥ k (e ) = G ⎥ 12 ⎢ ⎢ 1 1 2⎥ ⎣ ⎦ and ⎧1⎫ (e ) = QA ⎪1⎪ f ⎨⎬ 3⎪⎪ ⎩1⎭ {} New Chapter Page 9 K. Haghighi As for [ ⎛ T ∂[N] T ∂[N] ⎞ ⎟ dA = ∫ ⎜ ρ φu[N] + ρ φ v[N] Conv A⎜ ∂x ∂y ⎟ ⎝ ⎠ k (e ) Selecting the first term yields ⎧ Ni ⎫ ⎧ Ni ⎫ ρ φu ⎪ ⎪ ⎪ ⎪ ⎧ ∂Ni ∂N j ∂Nk ⎫ ∫A ρ φu⎨ N j ⎬ ⎨ ∂x ∂x ∂x ⎬dA = ∫A 2A ⎨ N j ⎬ bi b j bk dA ⎭ ⎪N ⎪ ⎩ ⎪N ⎪ ⎩ k⎭ ⎩ k⎭ { } New Chapter Page 10 K. Haghighi Using the factorial equation and applying it to the whole term results in: ⎡b ρ φu ⎢ i ⎢ bi k (e ) = Conv 6⎢ ⎢ bi ⎣ [ bj bj bj bk ⎤ ⎥ ρv ⎥+ φ bk 6 ⎥ bk ⎥ ⎦ ⎡c ⎢i ⎢ ci ⎢...
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