**1. Introduction**

Werner Heisenberg: "*When I meet God, I am going to ask him two questions. Why relativity and why turbulence? I really believe he will have an answer for the first*."

Richard Feynman: "*There is a physical problem that is common to many fields, that is very old, and that has not been solved. It is not the there is a problem of finding new fundamental particles, but something left over from a long time ago … over a hundred years. Nobody in physics has been able to analyze it satisfactorily in spite of its importance to the sister sciences. It is the analysis of circulating or turbulent fluids.*"

These quotes, both from Nobel prize winning scientists, may be apocryphal. But they nevertheless help provide context for our current understanding of fluid flow. And while these statements were reportedly made many decades ago, there is still truth to them. But at a risk of being brazen, we may now have the critical tools necessary to solve complex fluid flow problems with acceptable accuracy and fidelity. Those tools of numerical simulation are the focus of this chapter and this book.

It is not an overstatement to call numerical methods in general, and numerical simulation of fluid flow, in particular, a critical development in science in the past 100 years. In this chapter, we intend to briefly discuss the historical development of this science before quickly moving into the essential aspect of its practice. In our view, perhaps the most important aspect of numerical methods is that they provide solutions to problems that would otherwise require extensive experimentation or would otherwise be intractable. Simply put, numerical simulation opens the door for solutions to many academic and real-world problems that could otherwise not be solved.

One reason for the importance of numerical simulation for thermal-fluid problems is that the governing equations of motion are highly non-linear and coupled. That is, solutions require the simultaneous consideration of momentum (in all three coordinate directions), conservation of mass, conservation of energy (particularly for problems that involve heat transfer), and the potential for additional turbulence equations and species conservation/reaction equations. Such problems are not capable of being solved analytically. Furthermore, experimentation is often prohibitively expensive, time consuming, or impossible.

As we will see, numerical simulations of flows are important at many spatial and temporal scales. From the nanoscale to astronomical scales, from microseconds to millennia. At the large scales, climate and weather simulations are nearly ubiquitously used to make predictions. Small-scale examples include flows of fluid through microchannels or around micron-scale (or smaller objects) are

representative. The breadth of scales is indicative of the wide applications this technique has been applied to.
