数值传热学-Patankar01.pptVIP

Numerical Transfer Processes E4386 Methods of prediction Experimental investigation: full scale expensive and often impossible measurement errors on a small scale model simplified difficult to extrapolate results measurement errors Theoretical calculation: analytical solutions exist only for a few cases sometimes complex numerical solutions for almost any problem Modeling vs. experimentation Advantages of modeling: cheaper more complete information can handle any degree of complexity as long as… Disadvantages of modeling: deals with a mathematical description not with reality mathematical description can be inadequate multiple solutions can exist … and the winner is… Mathematical description of transfer processes Conservation principle leads to a general differential equation: Nature of coordinates with respect to xj : 1D, 2D, 3D with respect to t : steady unsteady(transient, time dependent) one-way coordinates = parabolic problem two-way coordinates = elliptic problem Discretization methods analytical solution gives a function (continuous) numerical solution gives a set of numbers (discrete) Example: steady 1D heat conduction Integration Discretization Discretization equation Source term linearization Guiding principles Solution should always be physically realistic Overall balance of fluxes must be maintained Four basic rules Rule 1. Flux consistency at control volume faces “when a face is common to two adjacent control volumes, the flux across it must be represented by the same expression in the discretization equations for the two control volumes” Rule 2. Positive coefficients “all coefficients a must always be positive ” Four basic rules Rule 3. Negative slope linearization of the source term “When the source term is linearized as S=SC+SPTP, the coefficient SP must always be less than or equal to zero. ” Rule 4. Sum of the neighbour coefficients “it is required that aP=? anb for situations where the differential