Finite volume method for unsteady flows

To compute unsteady flows, the time derivative terms in the governing equations need to be discretized. The major difference in the space and time co-ordinates lies in the direction of influence. In unsteady flows, there is no backward influence. The governing equations for unsteady flows are, therefore, parabolic in time. Therefore, essentially all the numerical methods advance in time, in a step-by-step or marching approach. These methods are very similar to those applied for initial value problems (IVPs) of ordinary differential equations. In this section, some of the methods widely used in the context of the finite volume method are discussed. 

For unsteady flows, discretization schemes need to be devised to evaluate the integrals with respect to time (refer to Eq. (6.2)). The control volume integration is similar to that in steady flows discussed earlier. The most widely used methods for discretization of time derivatives are two-level methods. In order to facilitate further discussion, let us rewrite the basic governing equation as an ordinary differential equation with respect to time by employing the spatial discretization schemes discussed earlier  

By integrating with respect to time between two grid points, one obtains  

Since the variation of 0 with time is not known, some approximations are necessary to evaluate the integration of the function. Four commonly used approximations are detailed below (shown schematically in Fig. 6.10). 

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

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