ME 540

Numerical Solutions Applied To Heat Transfer And Fluid Mechanics Problems


Taylor Series Method for forming finite difference equation (DEM)
 

Basic Mathmatical Requirements:

  • f(x) is a single valued function and if in the interval a<x<b
  • Its first (N-1) derivatives are continuous
  • Its Nth derivative exists, fN(x)

  • Taylor Series in Cartesian coordinates:


    where


     

    Example:

    Heat conduction two dimensional - steady state with internal heat generation

    Approximate each differential using Taylor Series:


    Add and truncate

    thus

    A similar expression is obtained for 
    We've been using the obvious subscript abbreviation To for T(xo.yo). We'll simplify notation further with the notation:

    Then the finite difference equation at a given point becomes
    with an error

    The error is abbreviated as:


    Improvement of the approximation

    Simple option is to decrease the mesh size (increase number of mesh points)
    Another option is to Obtain a higher order approximation


    High order approximation

    We must consider more points on the mesh when working with the equations. Introduce extended notation:

    We want an approximation to the second derivative that only involves To, TE, TW, TEE, and TWW. Look at the Taylor expansions giving the values of T at these 4 points.




    Now multiply the equation for TE by the unknown coefficient "A", multiply the equation for TW by the unknown "B", multiply the equation for TEE by the unknown "C", and multiply the equation for TWW by the unknown "D". Sum these modified equations. We want the summed equations to provide an approximation to the second derivative of T, so we require that the coefficient of the second derivative in the summed equation be equal to 1. We further require that the coefficients of the first, third, and fourth derivatives be zero. This gives us four equations that can be solved for values of A, B, C, and D. They are:


    Replacing these values in the summed equation and rearranging gives:


    What is the expression for the error, ?

    Cylindrical coordinate system (DEM)
     

    Example: Steady State Heat Conduction
     


    Assume and, are uniform

    In the r, direction, define

    Write the Taylor Series expansion for each of these variables




    In this instance we are free to either deal with all four expansions as a single sum, or group the radial and theta equations separately. I will follow the pattern in the previous higher order example. Multiply the equation for TE by the unknown coefficient "A", multiply the equation for TW by the unknown "B", multiply the equation for TN by the unknown "C", and multiply the equation for TS by the unknown "D". Sum these modified equations. We want the summed equations to provide an approximation to the radial and azimuthal derivatives in the conduction equations, so we require that the coefficient of the first derivative of T with respect to r in the summed equation be equal to 1/ro. The coefficient of the second radial derivative must be one. We require that the coefficient of the first derivative with respect to theta be zero. The coefficient of the second derivative must be 1/ro2. This again gives us four equations that can be solved for values of A, B, C, and D. The solution is:

    Thus the finite difference equation is

    What is the error?




    -
    Or
    Special treatment near origin

    When symmetry exists at the origin (Geometrical & Thermal)

        Indeterminate

    Thus

    When no symmetry at r = 0, use Cartesian coordinates for the node at the origin:

    Everywhere else use cylindrical coordinate system



    Application for Higher Order Boundary Conditions

    As part of the second homework, you will notice a small uniform difference between the calculated and analytic solutions for the two 1-D verification problems.  This can be removed by going to a second order evaluation of the surface heat flux


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    Created by  Frank Schmidt
    Maintained by John Mahaffy : jhm@psu.edu