說明40.流體力學(xué)numerical methods for heat,年_peakchina l

1、Lecture 5: A First Look at the Diffusion EquationLast TimeWe completed an overview of the numerical discretization and solution process Domain discretizationDiscretization of governing equationsSolution of linear algebraic setProperties of discretization and path to solutionAccuracy, consistency, co

2、nvergence, stabilityThis TimeWe willApply the finite volume scheme to the steady diffusion equation on Cartesian structured meshesExamine the properties of the resulting discretizationDescribe how to discretize boundary conditions Consider steady diffusion with a source term: Here Integrate over con

3、trol volume to yield2D Steady Diffusion2D Steady DiffusionApply divergence theorem to yield Writing integral over control volume:Compactly: Discrete Flux BalanceDiscrete Flux Balance (contd)Area vectors given by:Fluxes given byDiscretizationAssume varies linearly between cell centroidsNote:Symmetry

4、of (P, E ) and (P,W) in flux expressionOpposite signs on (P,E) and (P,W) termsSource LinearizationSource term must be linearized as:Assume SP 0More on this later!Final Discrete EquationPNSEWCommentsDiscrete equation reflects balance of flux*area with generation inside control volumeAs in 1-D case, w

5、e need fluxes at cell facesThese are written in terms of cell-centroid values using profile assumptions.Comments (contd)Formulation is conservative: Discrete equation was derived by enforcing conservation. Fluxes balance source term regardless of mesh densityFor a structured mesh, each point P is co

6、upled to its four nearest neighbors. Corner points do not enter the formulation.Properties of DiscretizationaP, anb have same sign: This implies that if neighbor goes up, P also goes upIf S=0:Thus is bounded by neighbor values, in keeping with properties of elliptic partial differential equationsPro

7、perties of Discretization (contd)What about Scarborough Criterion ?Satisfied in the equalityWhat about this?Boundary ConditionsFlux BalanceDifferent boundary conditions require different representations of JbDirichlet BCsDirichlet boundary condition: b = givenPut in the requisite flux into the near-

8、boundary cell balanceDirichlet BCs (contd)For near-boundary cells:Satisfies Scarborough Criterion !Also, P bounded by interior neighbors and boundary value in the absence of source termsNeumann BCsNeumann boundary conditions : qb givenReplace Jb in cell balance with given fluxNeumann BCs (contd)For

9、Neumann boundariesSo inequality constraint in Scarborough criterion is not satisfied Also, P is not bounded by interior neighbors and boundary value even in the absence of source terms this is is fine because of the added flux at the boundaryBoundary Values and FluxesOnce we solve for the interior v

10、alues of , we can recover the boundary value of the flux for Dirichlet boundary conditions usingSimilarly, for Neumann boundary conditions, we can find the boundary value of usingClosureIn this lecture we Described the discretization procedure for the diffusion equation on Cartesian meshesSaw that the resulting discretization process preserves the properties of elliptic equationsSince we get diagonal dominance with Dirichlet bc, the discretiza