教程40.流體力學(xué)numerical methods for heat,年-peakchina l

1、Lecture 14: Convection and DiffusionLast TimeIn the last lecture, weDeveloped the least-squares method for finding cell-based gradients on unstructured meshesLooked at influence of secondary gradients in destroying boundednessConsidered implementation issues, including the use of face- and cell-base

2、d data structuresThis Time We will Start looking at adding convection terms to our transport equationLook at two different schemes for discretizing the convection termsCentral difference schemeUpwind difference schemeConsider the properties of these schemes vis-visBoundednessStabilityAccuracySteady

3、2D Convection-Diffusion EquationGoverning equation:Assume flow field knownAssume Cartesian structured meshDiscretizationAs usual, integrate over control volumeApply divergence theorem, linearize source term:Nothing different so far !Discretization (Contd)Area vectors:Flux on east face:Flow rate on e

4、ast face:Units?Diffusion TermWrite as usual assuming linear profile between (E,P) etc:Grid Peclet NumberFace flux*area written in terms of two coefficientsDefine grid Peclet numberMultiplies face gradientMultiplies face value e: How to evaluate?Central Difference Scheme (CDS)Find face value of using

5、 cell average:Convection through face e:Assuming uniform meshSame sign! Trouble aheadCDS: Discrete EquationNote possibility of negative coefficientsNote extra flow rate term in aPCDS: DiscussionConsider V= u i+ v j with u0, v0. When Fe 2De , i.e., if Peclet number Pee 2 , aE 2Dn or if Pen 2For other

6、 configurations of the velocity vector, the other coefficients can also e negativeThis implies that if neighbor values go up, value at point P can go down! This is true even though for S=0 and extra mass flow rate term =0What about Scarborough criterion ?CDS: DiscussionScarborough criterion not sati

7、sfied:In fact aP =0 is possible for zero diffusion and uniform flow how would you do Gauss-Seidel?CDS: DiscussionNotice extra flow rate term in aP:This is the net mass flow rate out of the control volumeIf the assumed flow field is continuity-satisfying, this term would be zero. If not, it can cause

8、 loss of diagonal dominanceSummary: Spatial wiggles are possible because of negative coefficients for uniform mesh, keep Pe2Scarborough criterion not satisfied cant use iterative schemesUpwind Difference Scheme (UDS)Write face value as:“Upwind” in the direction of the mass flowNote asymmetric nature

9、 of the discretization!UDS: Discrete EquationHereWhat are the signs on the neighbor coefficients?Note extra flow rate term in aPUDS: DiscussionNotice that all coefficients are positiveSince for S=0 and extra mass flow rate term =0Solution is boundedScarborough criterion satisfied in the equality for

10、 S=0 and extra mass flow rate term=0Notice extra flow rate term in aP:Zero for continuity-satisfying velocity fieldSummaryCentral difference scheme (CDS)Makes linear profile assumptions between grid points to get face valueCan lead to spatial wiggles in convection-dominated flowsCan lose diagonal do

11、minance difficult to use iterative solversCan show that it is O(x2) accurateSummary (Contd)Upwind difference scheme (UDS)Makes linear profile assumption for diffusion term, but upwinds convective termBounded solutions guaranteed for continuity- satisfying fields regardless of grid Peclet numberSatisfies Scarborough criterion iterative solutions possibleCan show that UDS is O(x) accurateNeither very satisfactory for practical useClosureIn this lecture:We considered the stea