## Introduction

Fluid mechanics is a challenging science; the great Richard Feynman famously said “Turbulence is the most important unsolved problem of classical physics”. Given the importance of fluid mechanics, why is there so much we still don’t know?

We’ve actually known the equations that govern the motion of fluids for over two hundred years. These equations are called the Navier-Stokes equations, after G.G. Stokes and M. Navier, who derived them independently. Unfortunately, knowing what the equations are doesn’t mean we know how to actually solve them. In fact, a general proof of the Navier-Stokes equations is considered so important (and difficult) that it is one of the seven Millenium Prize Problems. A solution to one of these problems will earn the mathematician \$1M USD, so if you’re a talented mathematician, get to it!

If we can’t analytically solve the equations, we’re left with essentially two options. We can try to solve the equations numerically, which is what we call Computational Fluid Dynamics (CFD). This is not a simple process, and it gets more difficult as the fluid moves faster or on larger scales. The other option is to measure a flow in the laboratory, using a range of experimental techniques.

A key parameter that we will need to discuss is the Reynolds number. The Reynolds number represents the ratio of momentum to viscous forces in a flow. Pouring honey out of a bottle is an example of a very low Reynolds number flow: the momentum is very low, and the viscosity very high. The flow of air around a supersonic aircraft, or water around a ship, are examples of very high Reynolds number flows: lots of momentum, not much viscosity. The Reynolds number governs the behaviour of the fluid in a whole range of ways, determining whether or not turbulence is present, and the nature of that turbulence.

## Computational Fluid Dynamics:

The common perception these days seems to be that we can just build a computer model for everything. To a certain extent, this is true, we can certainly build a model. Whether we’ve built an accurate model is an entirely separate question. Building accurate models for fluid flows is incredibly challenging, and is certainly a hot topic in the fluid mechanics community.

Broadly speaking, there are three approaches to CFD: Direct Numerical Simulation, Reynolds Averaged Navier Stokes models, and Large Eddy Simulation.

### Direct Numerical Simulation (DNS):

Direct Numerical Simulation (DNS) is the most “pure” approach. It simply takes the full Navier-Stokes equations, discretizes them onto a numerical grid, and then solves them. No approximations are required; the full equations of motion are solved. In this sense DNS is a “numerical experiment”, it truly is a real fluid flow being solved. Unfortunately, everything comes with a cost, and this high degree of fidelity to the true physics has a very steep price tag. The number of computational operations required to produce a DNS solution scales as the Reynolds number to the third power. This means if you want to model a flow that is twice as large, or twice as fast, it is eight times more expensive.

This means DNS is mostly restricted to studying small scale, fundamental turbulence problems. Calculating the flow of air over a Boeing Dreamliner would take a computer larger than the known universe. The role of DNS is thus very much at the pure physics end of the spectrum. This shouldn’t be interpreted to mean it is not useful in a practical sense though; only by understanding the fundamental physics can we build models to tackle larger, more applied problems. Direct Numerical Simulations are almost exclusively performed on high performance computing facilities.

### Reynolds-Averaged Navier-Stokes (RANS):

Direct Numerical Simulation is so expensive in part because every scale of the flow must be resolved. If the cost is too prohibitive for the problem being considered, a common approach is to solve the Reynolds-Averaged Navier-Stokes equations rather than the full Navier-Stokes equations. Rather than figuring out exactly what is happening at each moment in time, the RANS equations instead describe a time averaged flow field, with a statistical representation of how much things deviate from that time average. The turbulent fluctuations (ie the deviations from the time average) are dealt with using an approximation, rather than solving the equations. Most of the important physics are lost, however careful RANS simulations are often still accurate on the larger scales, making it a powerful engineering design tool. A RANS simulation can often be produced on a desktop computer, meaning they are relatively very cheap (though this of course depends on the complexity of the problem). These days RANS mostly finds its use in design, research is more typically done with DNS, or our next topic, LES.

### Large Eddy Simulation (LES):

Sitting between the two extremes of DNS (solve everything) and RANS (model everything), is Large Eddy Simulation (LES). LES can be considered something of a hybrid between the two approaches: directly solving the larger scales in the flow ala DNS, and then modelling the fine scales ala RANS. The more computationally expensive you make your LES, the closer it becomes to DNS. Vastly more information about the flow physics is available than through a RANS simulation, but LES can be extended to high-Reynolds number flows. The flexibility of LES means it is used on everything from small scale workstations up to massive high performance computing clusters.

While I have tried to give a brief overview, I want to make clear that I am not a numericist and have done only very basic computational fluid dynamics (RANS models). However, Prof. Julio Soria works extensively in numerical modelling, as do a number of our students and postdoctoral researchers. Some examples can be seen on the Laboratory Webpage.

## Experimental Techniques:

Our second approach is just to take a flow in the laboratory and measure it directly. We don’t need to worry about solving the equations now, nature does it for us! The challenge instead becomes that of accurately measuring the flow, which is harder than you might think. We’re interested in measuring the parameters that we see in the Navier-Stokes equations: fluid velocity, density, pressure, temperature, viscosity, and so on. Getting a measurement of velocity might seem quite simple; we measure wind-speed all the time for instance. However measuring the velocity without changing the flow we’re measuring is actually very difficult. If you stick a measurement probe into your flow, the presence of the probe will change the flow significantly: you have changed the flow by observing it. As an example of this, see this video by Jeremy Veltin & Dennis McLaughlin from Penn. State

You can see that as they move the pitot probe array (intrusive devices used to measure pressure) through the jet, the flow changes dramatically.

To try to get around this problem, in our Laboratory we use a range of optical measurement techniques. Our gold-standard technique is Particle Image Velocimetry, which we use to measure the velocity of liquid and gas flows. We also use schlieren photography to obtain estimates of fluid density in gases. Other techniques that allow us to measure temperature or pressure non-intrusively include Planar Laser Induced Fluorescence (PLIF). Check out the links to see these discussed in more detail.

## Challenging Fluid Flows:

All fluid motion is difficult to study. Things get even trickier though when our flows become more complex, i.e. when they start moving really fast, or changing state between liquid and gas. When flows move very quickly, we start to have to consider effects, which complicates both numerical simulation and experimental measurement. It’s this type of flow that I am most interested in, and almost all my research involves a consideration of compressibility effects. When two or more states of matter are present in a flow simultaneously (i.e. liquid and gas), then we are dealing with multi-phase flow, which also complicates matters. Check out the pages for compressible and multi-phase flow to learn about them in more detail.