03.17.06
Why Simulating Free-Surface Flow is Difficult, Part 2
(This is a continuation of Why Simulating Free-Surface Flow is Difficult, Part 1.)
In Part 1 of this series, I talked about why it’s difficult to do an accurate computer simulation of a single drop dripping from a faucet. That’s a very simple example of a free-surface flow, however, and many of the flows that are relevant to other scientific or engineering questions are far more complicated.
One general class of more complicated free-surface flows are “sprays”, also referred to as “spray flows” or “atomization” – a liquid is forced at relatively high pressure through a small nozzle (or set of nozzles), and when it comes out it breaks up into tiny droplets. Sprays are nearly ubiquitous in any handling of liquids; for instance, in my day-to-day life, I might take a shower in the morning (under a spray of water from the showerhead), wash the shower walls with a cleaner in a spray-bottle, use an aerosol spray can of cooking oil to oil the pan to cook my eggs in, and wash the pan using the spray attachment on my sink. My car has a fuel-injected engine, which means that there are nozzles in it that spray the fuel into the air intake. If I were to paint the (sadly rusting) hood on it, I’d use a spray can. The train that I take to work has a diesel engine; the term “diesel” means that in it the fuel is sprayed directly into the engine’s cylinder, where it burns immediately as it’s being sprayed in.
In every one of those cases, it’s important to get the spray “right”. The shower-cleaner, cooking-oil, and paint sprays need to provide an even coating on the walls without spraying any off to the sides. The shower head needs to produce rather large drops that act like rain rather than fog. The fuel injectors in my car need to produce tiny droplets that will evaporate quickly, while the ones on the diesel locomotive need to produce droplets of the right size to get to the middle of the cylinder before they burn up. Thus, it would be very useful to be able to simulate the spray that comes out of a new nozzle design, so that it can be tested and improved without needing to do costly experiments.
As an example to talk about, here’s a fairly typical spray that’s easy to photograph – water being squirted through the nozzle from a Windex® bottle. I should credit my wife for patiently helping me take the picture; it took quite a few tries before I managed to get the flash timed correctly with the spray!
(click on the photo for a larger version)
Dilute Parts of a Spray: Lots of Little Droplets
One of the most obvious complications in simulating a spray like this is the number of drops involved. There are something on the order of 10,000 shown in this picture, and this is a fairly “tame” spray; a diesel-fuel-injector spray will produce millions at a time. If calculating the behavior of a single drop dripping from a faucet is compilated, this is obviously far, far worse.
This isn’t quite as complicated as it might seem from that estimation, however. For at least the left two thirds of the picture, the individual drops aren’t really doing anything interesting. Very few of them are breaking apart, and they’re mostly far enough apart that they’re not interacting with each other (except indirectly because the overall spray pulls along the air it’s moving through), and, though it’s not visible in the picture, they’ve pretty much all settled down to being nearly spherical. This part of the spray, where the drops are widely spaced apart so that they aren’t interacting with each other, is known as the “dilute region”.
That means that, for most of the dilute region, we don’t need very much information about a drop to know what it’s doing; if we know the size of the droplet and its velocity, it’s a reasonable approximation to say that it’s doing exactly the same thing as any other droplet of that size with the same airspeed. And that’s much easier to calculate than trying to track all of the individual details of where each point on the surface of each drop is.
Even when the drops are breaking apart – and sometimes they do, if they’re large enough and are going fast enough – they break apart in ways that don’t differ very much from drop to drop, and so it’s possible to approximate that by applying a statistical distribution that converts some fraction of the drops (depending on their velocity and size) into multiple smaller drops.
So, for the 10,000 drops of this spray, that reduces the calculation of the dilute region of the spray to something that could run on a desktop computer in a day or so. But even that’s a fairly long time, particularly if we want to simulate something with millions of drops. For those, there’s yet another simplification that we can make – with that many drops, we can start grouping them into “packets”; near a given spot in the middle of the spray, there are perhaps a hundred tiny drops that all have nearly the same diameter and nearly the same airspeed, and so there’s no reason to do a calculation for each one of them individually; we could just do the calculation once and say that they all do approximately the same thing.
There is, of course, the question of how good these approximations actually are in practice. This obviously depends on the details of how one does it and of the spray that one’s simulating, but it turns out that it’s possible to get quite good results for simulations of the dilute region of most sprays with this sort of method.
The Nozzle: The Other (Relatively) Easy Bit
The other part of the spray that’s relatively easy to simulate is the flow inside the nozzle and just outside it for the first half-millimeter or so. In this part, the surface is not doing anything especially interesting, and so it’s entirely possible to track all of the details of its motion in a simulation that could be run on a desktop computer in a few hours.
Dense Parts of a Spray: What About the Rest of It?
That, then, takes care of the spray in the left-hand two thirds of the image, and the first tiny bit coming out of the nozzle. The alert reader will have noticed that this leaves quite a significant portion of the spray undiscussed! This is where things get tricky. In this region of the spray, known as the “dense region” because the drops are sufficiently close together to affect each other directly, there are still too many drops and too much going on to simulate all of the details of the surface motion, but the process of the water breaking apart into droplets is far too complicated to accurately approximate as the motion of independent spheres.
This region is one of the biggest reasons why simulating free-surface flows that are of interest in engineering problems is difficult. At present, it’s an unsolved problem.
One common way to deal with the dense region is to simply take the independent-spherical-drops approximation from the dilute region, and continue it all the way up to the nozzle. The drops in the simulation are created at the nozzle in such a way that, when they get downstream to the edge of the dilute region, they match some experimental data for the distribution of real drops there. How good is this approximation? It’s not really clear; it works fairly well for cases where the dilute region is the only part that’s important (though it’s not completely clear how true that is in most interesting sprays), but of course it’s only applicable in cases where there’s good experimental data to start with.
Meanwhile, there are ways to do approximations working from the upstream end. If you look at the spray as it comes out of the nozzle, it’s (roughly) a conical sheet. To a rough approximation, this conical sheet first breaks up into rings, and the rings then break up into drops. (That’s a very rough approximation for this spray; it’s a bit clearer in some with more precise nozzles.) By assuming that these processes can be separated and calculated independently, it’s possible to get mathematical models for how quickly the sheet breaks apart and what size drops are produced, and these tend to be reasonable as a very rough guess for the actual drop distribution.
So, at present, there are two big remaining problems. One of the problems is that the existing sets of approximations don’t really meet in the middle; there is a part of the dense region after the calculations of the initial exit from the nozzle have done what they can accurately do, and before the independent-drop approximations of the dilute region are accurate. The second problem, even if we solve the first problem, is that the computational methods corresponding to these two sets of approximations are very different, and it’s not clear how to combine them in the same simulation in a way that’s mathematically consistent.
The research that I’m doing for my dissertation is something to address a tiny piece of both of those two problems. It won’t solve them – not by a long shot – but it’s a small step up the hill.