Module 1: Introduction to Finite Difference Method and Fundamentals of CFD Lecture 5:

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1 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture5/5_1.htm 1 of 1 6/20/ :22 PM The Lecture deals with: Explicit and Implicit Methods

2 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture5/5_2.htm 1 of 1 6/20/ :22 PM Let us now tempt a different discretizion of the original partial differential equion given by Eq. (3.3). Here we express the spial difference on the right-hand side in terms of averages between and time level (5.1) The differencing shown in Eq. (5.1) is known as the Crank-Nicolson implicit scheme. The unknown is not only expressed in terms of the known quantities time level but also in terms of unknown quantities time level. Hence Eq. (5.1) a given grid point, cannot itself result in a solution of.

3 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture5/5_3.htm 1 of 2 6/20/ :23 PM Eq. (5.1) has to be written all grid points, resulting in a system of algebraic equions from which the unknowns for all can be solved simultaneously. This is a typical example of an implicit finite-difference solution (Fig. 5.1). Figure 5.1: Crank Nicolson Implicit Scheme Since they deal with the solution of large system of simultaneous linear algebraic equions, implicit methods usually require the handling of large mrices. Generally, the following steps are followed in order to obtain a solution. Eq (5.1) can be rewritten as (5.2) where or or (5.3)

4 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture5/5_3.htm 2 of 2 6/20/ :23 PM Eq. (5.3) has to be applied all grid points, i.e., from to A system of algebraic will result (refer to Fig 5.1). Finally the equion will be of the form: Here, we express the system of equion in the form of A, where, C: right-hand side column vector (known), A: tridiagonal coefficient mrix (known) and : the solution vector (to be determined). Note th the boundary values and are transferred to the known right-hand side. For such a tridiagonal system, different solution procedures are available. In order to derive advantage of the zeros in the coefficient-mrix, the well known Thomas algorithm (1949) can be used.

5 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture5/5_4.htm 1 of 1 6/20/ :24 PM Explicit and Implicit Methods for Two-Dimensional He Conduction Equion The two-dimensional conduction is given by (5.5) Here, the dependent variable, (temperure) is a function of space and ( )and ( ) is the thermal diffusivity. If we apply the simple explicit method to he conduction equion, the following algorithm results (5.6) When we apply the crank-nicolson to the two-dimensional he conduction equion, we obtain (5.7) where the central difference operors and in two different spial directions are defined by (5.8)

6 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture5/5_5.htm 1 of 2 6/20/ :24 PM The resulting system of linear algebraic equions is not tridiagonal because of the five unknowns and In order to examine this further, let us rewrite Eq. (5.7) as (5.9) where Figure 5.2: Two-dimensional grid on the ( ) plane. Eq. (5.9) can be applied to the two-dimensional (6 6) computional grid shown in Fig A system of 16 linear algebraic equions have to be solved time level, in order to get the temperure distribution inside the domain. The mrix equion will be as the following:

7 file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture5/5_5.htm 2 of 2 6/20/ :24 PM (5.9) where The system of equions, described by Eq. (5.9) requires substantially more computer time as compared to a tridiagonal system. The equions of this type are usually solved by iterive methods. These methods will be described in a subsequent lecture. The quantity is the boundary value. Congrulions! You have finished Lecture 5. To view the next lecture select it from the left hand side menu of the page or click the next button.