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lectfv2d - Chapter 19 Finite Volume Method for Scalar...

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Chapter 19 Finite Volume Method for Scalar Advection in 2D 19.1 Introduction The purpose of this exercise is to code a program to integrate the scalar advection equation in two-dimensional flows. 19.2 The 2D equation The 2D scalar advection equation in conservative form is: T t + ( uT ) x + ( vT y ) = 0 . (19.1) where vectoru = ( u,v ) is a two-dimensional flow field that obeys the mass conservation equation of an incompressible fluid ∇ · vectoru = u x + v y = 0. The formulation of this partial differential equation into a finite volume formulation follows the steps outlined in chapter 17. Here we concern ourselves primarily with the FV 2D discretization of equation 19.1. 19.3 The 2D spatial discretization The spatial discretization proceeds by deviding the domain into rectangular cells as depicted in figure 19.1. We are now concerned with assigning the different terms appearing in equation 17.7. Here we presume that diffusion is non-existent ( α = 0). We have the following remarks: The cell volume is actually an area in 2D and we denote by δA = Δ x Δ y . where Δ x × Δ y are the cell sizes in the x × y directions. 171
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172 CHAPTER 19. FINITE VOLUME METHOD FOR SCALAR ADVECTION IN 2D a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a117 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a45 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 a54 j 1 j j + 1 k + 1 k k 1 Figure 19.1: Cartesian finite volume grid with rectangular cells. The solid point represents the average of T over a cell while the arrows denote the x y advective fluxes into a cell through its 4 edges. The cells can be referenced by a pair of indices ( j,k ) along the ( x,y ) direc- tions. The cell center has coordinate ( x j ,y k ) and the cell walls are located at ( x j ± Δ x 2 ,y k ± Δ y 2 ). Each cell has four edges with constant normal along each. The outward unit normal to cell ( j,k ) at x = x j + Δ x/ 2 points in the positive x -direction, and we have vectoru · vectorn = u ; whereas at x = x j Δ x/ 2 it points in the negative x -direction and we have vectoru · vectorn = u . Similarly along y = y k + Δ y/ 2 we have vectoru · vectorn = v , whereas along y = y k Δ y/ 2 we have vectoru · vectorn = v . With these remarks we can now write down the finite volume equation for cell ( j,k ): δA d T j,k d t + integraldisplay y k + Δ y 2 y k - Δ y 2 bracketleftBig u parenleftBig x j + 1 2 ,y parenrightBig T parenleftBig x j + 1 2 ,y parenrightBig u parenleftBig x j - 1 2 ,y parenrightBig T parenleftBig x j - 1 2 ,y parenrightBigbracketrightBig d y + integraldisplay x j + Δ x 2 x j - Δ j 2 bracketleftBig v parenleftBig x,y k + 1 2 parenrightBig T parenleftBig x,y k + 1 2 parenrightBig v parenleftBig x,y k - 1 2 parenrightBig T parenleftBig x,y k - 1 2 parenrightBigbracketrightBig d x = 0 (19.2) where x j ± 1 2 = x j ± Δ x 2 , and y k ± 1 2 = y k ± Δ y 2 . In this form we see that the equation is nothing but an accounting of the flux entering/leaving cell j . For ease of notation
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