CFD 12 : Equilibrium and Marching Problems

In principle, all fluid flow and heat transfer processes evolve with time. From the practical viewpoint, it is, however, desirable to classify them into two groups: equilibrium (time-independent) and transient (time-evolving)

The classification is determined by the nature of the process and by the purpose of the analysis. The equilibrium problems appear when our interest is in a steady state of the system, where the properties do not significantly change with time. For example, such a situation arises when we want to know the air resistance coefficient of an airplane cruising with constant speed and latitude or the temperature distribution within a bioreactor operating in a steady-state mode. As an approximation, we assume that the distributed properties are functions of space but not time and replace the time derivatives in the governing equations by zeros.

In other cases, the evolution toward the equilibrium state is of interest or the equilibrium state does not exist even as an approximation. Returning to our examples, this would correspond to an airplane taking off or landing, or to a change in the regime of operation of a bioreactor. Full transient equations should be solved in such cases.

There are special situations in which the PDE problem is formulated as transient even though the underlying physical process is steady-state. This is done when the solution is known to evolve in one particular direction in a timelike manner.

The coordinate in this direction assumes the role of a new time line. A prominent example of such behavior is the solution of the approximate equations derived for steady-state boundary layer flows. Among our model equations, the Laplace and Poisson equations correspond to equilibrium problems, while the heat equation, wave equation, linear convection equation, Burgers equation and generic transport equation describe transient evolution of time-dependent systems. CFD can be applied to both kinds of processes, but different numerical approaches are required. For equilibrium problems, the equation has to be solved numerically only once to determine an approximation to the time-independent solution u(x) in Ω,
For transient processes, the so-called marching problems must be solved. Starting with initial conditions, which give the state at t = t(0), the solution is advanced or marched forward in time to determine approximations to u(x, t ) at t >t(0).