Jf t S dt Atf tn Sn649

(a) t0 t0+At (b) t0 to+At (c) t0 t0+At (d) t0 t0+At FIGURE 6.10 Approximation of time integral. (a) Explicit, (b) Implicit, (c) Mid-point, (d) Trapezoid.

(b) Implicit Euler: Integral is evaluated using the value of $ available at the next node:

Integral is evaluated using the value of $ available at the mid-

(d) Trapezoid rule: Integral is evaluated using linear interpolation:

J f (t, $ )dt = At1 [f (tn, fa) + f (tn+1, <f>n+l)] (6.52)

The first method is an explicit method while the remaining three are implicit methods (to varying degree). The Euler explicit and implicit methods are first-order accurate (errors are proportional to At) while the remaining two methods are second-order accurate (errors are proportional to At2). Explicit methods have minimum requirements for memory and computations but are unstable at larger time steps. Implicit methods may require an iterative solution (and more memory) to obtain the values at the new time step but are much more stable. Apart from the two-level methods discussed here, there are multi-level methods such as the Runge-Kutta methods and Adams methods. Detailed discussion of these methods can be found in Press et al. (1992). For computational flow modeling, if the spatial discretization is second-order accurate, two-level methods for integration with respect to time will generally be sufficient, and are widely used. For special purposes, when higher order spatial discretization is used (for example, in large eddy simulations), higher order schemes can be used. Here we discuss application of two-level methods to solve the generic unsteady transport equation (Eq. (6.1)).

Integration of first term of Eq. (6.1) over a computational cell and over a time interval can be written as:

The procedure for evaluating integrals of the remaining terms of Eq. (6.1) over a control volume remain the same as discussed earlier. To evaluate integration with respect to time, it will be necessary to employ one of the two-level methods discussed above. As mentioned earlier, generally all the terms appearing in Eq. (6.1) are linearized when carrying out discretization. Linearization simplifies the task of time integration. Integration of with respect to time can then be written (considering the example of a term containing E):

where 0 is a parameter controlling the degree of implicitness. Zero implies an explicit scheme, and one implies a fully implicit scheme (0.5 corresponds to the Crank-Nicholson scheme). Carrying out such a procedure for all terms of the governing transport equation, a discretized equation similar to Eq. (6.11), is obtained for the unsteady simulations:

+ |ap - Ç (1 - 0) anb + (1 - 0) Sp j 0P + Set (6.55)

where v^ n n AV aP = 0 > anb + ap - 0SP$ ap = p--(6.56)

nb At

For physically realistic and bounded results, it is necessary to ensure that all the coefficients of the discretization equation are positive. This requirement imposes restrictions on the time step that can be used with different values of 0. It can be seen that a fully implicit method with 0 equal to unity is unconditionally stable. Detailed stability analysis is rather complex when both convection and diffusion are present. In general, simplified criteria may be used when an explicit method is used in practical simulations:

These criteria can be interpreted as no fluid particle (information) can propagate more than one grid length in a single time step. If the details of development from the initial guess to the final steady state are not important and only the final steady state is of interest, such a restriction on the time step may limit the rate of convergence. In such a case, implicit methods are advantageous. Since implicit methods are unconditionally stable, large time steps can be used and it might suffice to do a single iteration per time step. Such a pseudo-time-marching approach can be conveniently used to obtain steady state solutions to complex flow problems. Pseudo-time-marching is analogous to employing an under-relaxation. Pseudo-time-marching uses the same time step for all CVs, which is equivalent to using a different under-relaxation factor for each CV; use of a constant under-relaxation factor for all CVs is equivalent to applying a different time step for each CV. Typical steps in applying SIMPLE-like algorithms to solve unsteady flow problems are summarized in Fig. 6.11. Within a single time step it may be necessary to carry out several iterations to obtain an adequately converged solution of the governing equations. Within each such iteration, there may be internal iterations to solve algebraic equations. Some issues related to the overall performance of such a solution procedure are discussed in the next chapter.

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