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Unformatted text preview: 1 Atms 502 Numerical Fluid Dynamics Tue., Nov. 28, 2006 12/4/06 Atms 502  Fal 2006  Jewett 2 Advection and diffusion • Upstream advection, centered diffusion Differentialdifference expression: Modified equation: Unless Péclet number P e << 1, modified equation shows that this scheme gives poor answer How do we get small P e ?? Small ∆ x, or Large diffusion so diffusion dominates advection If low viscosity flow: need better d φ /dx (advection) approximation "# " t + c # j $ # j $ 1 % x & ’ ( ) * + = M # j + 1 $ 2 # j + # j $ 1 % x 2 & ’ ( ) * + "# " t + c "# " x = M 1 + P e 2 $ % & ’ ( ) " 2 # " x 2 where P e = c * x M = ratio of advection, diffusion terms 12/4/06 Atms 502  Fal 2006  Jewett 3 Advection and diffusion • Centered advection, centered diffusion Modified equation is: () term is negative if M<(c 2 ∆ t/2) (i.e. r < ν 2 /2 ) … again , diffusion must dominate advection; M can’t be too small here ! If ν =1 and r =1/2, FTCS reduces to: Shift condition not ok for Burger’s equation!! " t + c " x = M # c 2 $ t 2 % & ’ ( ) * " xx + () " xxx + () " xxxx + ... " j n + 1 = " j # 1 n 12/4/06 Atms 502  Fal 2006  Jewett 4 Time differencing • Back to the general differential equation: Substitute Fourier mode: For stability: " t + c " x = M " xx d ˆ " dt + c ik ( ) ˆ " = M ik ( ) 2 ˆ " # d ˆ " dt = $ ikc ˆ " $ Mk 2 ˆ " % i & ˆ " + ’ ˆ " " ( t ) # " (0) for $ < 0 2 12/4/06 Atms 502  Fal 2006  Jewett...
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