Day28-slides - 1 Atms 502 Numerical Fluid Dynamics Tue.,...

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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 Differential-difference expression: Modified equation: Unless Pclet 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 /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 < 2 /2 ) again , diffusion must dominate advection; M cant be too small here ! If =1 and r =1/2, FTCS reduces to: Shift condition not ok for Burgers 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 " dt + c ik ( ) " = M ik ( ) 2 " # d " dt = $ ikc " $ Mk 2 " % i & " + " " ( t ) # " (0) for $ < 0 2 12/4/06 Atms 502 - Fal 2006 - Jewett...
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Day28-slides - 1 Atms 502 Numerical Fluid Dynamics Tue.,...

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