Scientific Method / Science & Exploration

For turbulent mixes, a good model beats an exact solution

A new study shows how a simplified answer can provide just as much insight …

Rayleigh-Taylor instability between two liquids

It is said that the enemy of great is good. In engineering, however, good can often be great—when you can develop a solution that gets you 90 percent of the way to perfection in 20 percent of the time, then you have to ask how much that last 10 percent is worth to you. For example, knowing ? to six digits, one can draw circle around the Earth that deviates from a perfect circle by less than 850?m per kilometer. When applying such an analysis to a physical system a bit more complicated than circles, good can actually be better than great.

An example chosen for some recent research is a system where a cool, dense fluid is at rest on top of a light, warm fluid. This setup exhibits a Rayleigh-Taylor instability, which results in the fluid switching places, mixing and demixing in the process, and producing complex patterns like the one shown above. Rayleigh-Taylor instability is encountered in systems such as cloud formation, supernovae explosions, and geological formations, so it has lots of real-world relevance.

For those who wish to study this type of system, solving the full set of coupled Navier-Stokes/heat transfer equations is usually going to be overkill. Here is where good can be better than great: if a simpler method can describe the system adequately using only a fraction of the mathematical overhead, then it would be much preferred. In this case, the ultimate details of the mixing interface are often not as important as understanding how big the interface region is at a given time.

Given the thickness of the mixing region as a function of time as a goal, the team sought to create a model that was able to describe the system with a subset of the relevant physics. An upcoming edition of Physical Review Letters contains the results, obtained by physicists from the Universita di Torino and the CNRS Laboratory in Nice, France.

Their solution involved a simplified transport model. Since the fluid motion is coupled to the temperature distribution, they first tried to see if a simple constant thermal diffusion model could describe the system. While giving good quantitative results throughout the bulk of the mixing region, it was not able to adequately describe the temperature at its edges, where it meets the bodies of cool or warm fluid. 

Adding a layer of complexity back to the simple diffusion approach, they used a thermal diffusion coefficient that was a function of the temperature difference across a given region. This model produced very good agreement with the results of detailed numerical simulations, but it requires only a single free parameter, which is a measure of the efficiency of the conversion of potential energy into the turbulent flow.

In order to fully describe this system, one needs to use advanced mathematical techniques for solving non-linear partial differential equations. However, the new work shows that it's possible to model it pretty well—and pretty well here is actually very good—with far less mathematical effort.

Physical Review Letters, 2010. DOI: Upcoming

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