 Our first speaker this afternoon is Jim McElwain who's going to be telling us about the dynamics of granular flows. Great, thank you for inviting me. It's a great pleasure to be here. I've changed my talk a little bit after some discussions this morning so I was told that I should put as many equations in as possible and you didn't want any pictures or movies so my original background was as a mathematician so I'm very happy to go along with that for you. So, what is granular matter? So this is a little bit different from what a lot of you do but of course you see in the environment all the time that there are grains everywhere. So, how do I get the pointers away? The middle one, okay. Great, so this is not a natural granular flow but this is a ping-pong ball avalanche in Japan. 650,000 ping-pong balls designed to test the interaction of grains with a fluid. The next picture over we see what happens to a building if it gets hit by an avalanche. So an avalanche is of course a granular flow, it's grains of snow and it may behave like a dry granular flow so just dry grains but the big ones, the powder snow avalanches, it's a two-phase mixture. Now many industrial materials go through a granular processing stage, about 40% by value apparently so there's a huge amount of money. What you can see here is a sort of storage silo that's collapsed. Now these things have been built, there are millions of them all over the world. You'd think that the engineers would know how to build them but actually it's a continuous problem because the underlying equations that really describe accurately the stress distribution in the granular material and how it flows are not known. So there's huge problems scaling these things. So people build these large silos or mixing drums and then they don't operate and they have to go and hit them with hammers or rebuild them. So I showed you the powder snow avalanche at the bottom and then there's a picture here of a concrete pylon which is broken and here concrete is of course a very poorly sorted mixture of grains all held together in a matrix and every few weeks in the news somewhere around the world some poorly built concrete building collapses and often what the problem is there's not enough cement but it can also be due down due to the fact that the materials have segregated when you're mixing your concrete. If you have all small particles and one part of the structure and big particles somewhere else then the thing is going to break. So what we want to do is try to develop continuum models which will describe these grain flows in nature and in the industrial setting. And the problem at the moment is you know for fluids we have Navier-Stokes equations and as we've heard today you know that's incredibly hard to solve directly in many sensible cases or the practical cases in the environment so we have to do LES modelling or RANDs or things like that but with granular flow we don't know an equation of state and a constitutive relation that will have widespread applicability. There's a lot of empirical models and theories which may be good or not so good in different situations but it's not even clear when they're going to work. So one of the things to try to do is to directly simulate these systems, compare these simulations with experiments and then try to figure out how we can develop continuum models and upscale these things to the situations that we're really interested in. Now this is particularly important when we start looking on other planets. These are all pictures taken by the high-rise instrument on Mars. This is an amazing camera. It has a pixel size of about 70 centimetres. The bottom left-hand picture you can see some small powder snow avalanches on Mars which are a mixture of calm dioxide snow and dust. You can see some strange zigzag gullies which have happened on a very shallow slope, probably too low for granular flow. One of the things I'm going to talk about later is that some people have argued that friction is different on Mars because gravity is lower. The paper that claimed that actually got a lot of publicity and it's had a lot of citations and that's one of the problems is that until we really have a theory arguing about how these things should scale on other planets is a problematic and controversial question. Now if you are probably not so much in Colorado but suddenly a lot in Arizona and Utah you get Washboard Road which you see at the top. So this is a small-scale granular instability. There's lots of other type of patterns you can get whether it's the wind-blowing sand around making ripples in dunes or rivers leaving all sorts of bed forms at the bottom. And again by trying to combine a mixture of lab experiments and simulations and field observations we're trying to make progress in these areas. Now I was originally going to go through the details of some of these simulation methods so these next slides I'm just going to talk about one or two of them and then skip through to some more results. But what's nice about granular mechanics is we can do these direct simulations where we simulate every particle. And these were first done for granular systems in the late 70s but really the techniques go much earlier to when people were doing molecular simulations of liquids. So people were trying to calculate heat transport and viscosity of liquids from first principles by simulating all the interactions. And granular materials are actually easier to simulate because they just have local interactions. So the bottom picture is a schematic of one particle crashing in to some others. So what is the basic of this simulation technique? You detect when the particles interact with each other. So when they're very close then you make some sort of force law in between them which will push them apart and deal with the contact and you can put in friction and cohesion and it's very easy to make complicated force laws but most of the time the simulation results only depend very weakly on actually how you do this. And then you just integrate the differential equation of motion for each particle and you calculate and store the interesting data. Now what's nice about this is that the system is order n so if you've got n particles that's how it scales so if you have twice as many particles it only takes you half as much time. So that's the basics. If anyone's interested in more of the details how you do the contact detection, what are the type of force laws, all of this type of thing, what are the appropriate time integration methods I'd be very happy to go through this. Now this was the paper I was alluding to earlier where people were saying well you know maybe friction is different on Mars though anyone with a physics background would object to this on dimensional grounds but anyway these people wanted to try this out so they got this plane and they had all these drums which you see in the bottom left here and the idea is that they're going to rotate them and they're going to measure the angle of the surface and say this is the friction and if you rotate at a particular slow enough speed you get solid body rotation and then intimacy and avalanches so the idea was they'd measure these so they put these up in the plane and lo and behold the mean angles were lower when the plane was on one of these low G things Now the problem is they can only maintain the low gravity for about 30 seconds so they would have poor statistics, the gravity varies there's a lot of vibrations and we were really unconvinced that this actually made any physical sense so we decided to look at this so let me... where is the mouse gone? I need to click on one of these things and I can't... okay so this is the avalanching state if you have a drum that you rotate it very slowly now when you're doing these simulations your time step has to be a small fraction of the collision time between grains grains collide very quickly this is a very slow process so this is partly why I've used so many CPU hours and I'm very pleased to have been able to get all of these that takes a lot of time now if you go a little bit faster you have a steady flow and here this is pretty much a constant angle so this is a fairly constant dynamic friction angle that you can see in this state again this is an experiment obviously now when you start getting... when you start going faster other types of things can happen before in this steady case there's a local balance of driving force and friction here there's inertia taking place in the flowing layer so you get a curved effect and if you go faster it starts cascading and looking a bit like a waterfall so let's go back to all screen click on that again view view okay thank you it should be right where you are after you're done okay and now we have the cursor so the question is can we get these with simulations so believe it or not this simulation is actually moving but when you have... this is a simulation with perfectly round particles it turns out and there's no vibrations of course in the simulation so this is what's really nice you can eliminate all these extra effects even in any experiment there's always some vibration you have end walls this is a periodic simulation but you have to go incredibly slowly to really get these this is actually playing isn't it? you have to go... so you can see a few small rearrangements you have to go incredibly slowly to get this avalanching state it's really really sensitive you see one happening and it will stop and then it's going to stay for a long time but actually at first when we started doing these simulations we were a bit concerned because we just couldn't find this avalanching state because we had to go so slow now you can also do these simulations with angular particles or you can glue the spheres together to make much more effectively rough shapes and then this happens much easier but we were quite relieved when eventually we did manage to get so we can get the steady state again and this is sort of this S-shaped state where inertia starts becoming important and if we go really fast we get a cascading state where they're just about to start centrifuging around the edge I can also look at what happens when we start making the thing really long so of course one of the things you might want to know about is what happens with the effects of the end walls so what this is telling you about is the lateral transmission of stresses in the system so again what you see here with this slow rotation rate is that these actual avalanching behaviors are correlated over quite a long width so if a small amount of the material starts moving at some spot it starts in moving at another spot so there's long range order going on and physicists would expect this because effectively we're looking at a sort of a phase transition it's somewhere between the flowing state and the steady state and what the correlation is between this motion tells us a lot about what the underlying physics is of these flows so the basic idea with these things is there's a few non-dimensional groups about the radius of the drum divided by the particle size so how big the drum is in terms of particles both diameter and length and then the only way that gravity enters in is just through the rotation rate so the idea is if you change the rotation rate it's the same as changing gravity and of course if you do simulations you're going to non-dimensionalize them mostly so you're going to set gravity or something to one anyway so you really... there's no way that gravity can actually affect the results if all you have is a frictional law if you put in cohesion and other forces like this then you expect a difference but what we wanted to do is reproduce or try to explain the results from this LOG experiment so this is just a trace of what one of our experiments looks like and in fact it could just as well be a simulation or an experiment what's going in the top is the trace of the surface angle so it's moving up and then it's dropping down and these triangles we identify the maximum and minimum and then we draw these histograms and what you'll see is there's a huge variation in these so to actually get any kind of reasonable values even for an experiment we have to run it for hours and hours and not 30 seconds and then we get a distribution of starting and stopping angles so what we can see is there's no well-defined dynamic friction angle but there's a whole range of these now why is this particular transition so interesting? well this is because this is the natural state for most granular material if you have a desert sand dune the material is deposited till it gets onto this critical state a bit more sand and it might fall down if it's much flatter it will just build up so the middle frame these are let's just talk about this one in the middle left these are histograms of the starting and stopping angles for the avalanche so that's unfortunate the colours are that way round they're swapped sorry about that, well spotted you're definitely awake in the front row I won't go over the details of what these other things are for now but there's another complication so if you look to this so these are all the different experiments we did so we vary the food number essentially of the centripetal acceleration to gravity and we look at the mean of these distributions now you might think we must have varied something else here these are different sized particles but in fact these are the same particles just over running the experiment for a couple of weeks so what of course happens with natural particles which doesn't happen with the simulations the surface wears, you get dust so actually it's very difficult even to do these experiments repeatably continually degrading and changing so this is a typical problem with granular material that you have to work very hard to get reproducible results your state preparation can be very important and the particles degrade and change over time anyway we did eventually manage to get some repeatable results and we could show the same effects as they got from their zero G flights but we just can explain this as a change in the food number and the effects of vibration and now we've each one of these data points here this is several months of computer stimulation time where we've varied the food number the radius of the drum and the diameter of the drum and Q here is the surface the mean surface slope angle and we've got a little theory here which will collapse all of these surface angles onto this bottom one and the one on the right is a different type of particle which is a rough particle so we've glued them together to make them rough and again we get the same type of result so we've now got a model that can describe this behaviour in the drum now you might say this drum this is very far from a geophysical situation we've also been doing experiments with sand and things like that but what's nice about this experiment is we now think we have a protocol where you can go and get any material from the field you can put it in your drum experiment for a few weeks and you will probe how the material wears and what its behaviour is in the continuous flowing regime the transitioning so it's a very effective way of measuring nearly everything that you might want to know about a natural granular material and we can look at these are the pictures of where the transition happens so this d-scu is a measure of a skewness in the velocity distribution so we get a very nice typical kind of bifurcation here and the difference between the left and the right is the effect of the sidewalls which is quite profound in this case and that's because it affects the correlations now something else that's very important and of course all natural materials pretty well, they're a mixture of different sizes and you know there's the Brazil nut effect if you like muesli for breakfast the big things end up on the top and the question is you know what effect does this have on the flow dynamics and the first thing if you want to know that you also want to know how quickly will materials segregate this is a very simple experiment with a single intruder and you see it rising up to the top and this is extremely generic behaviour that happens all the time this shows you a before and after shot big particles at the bottom moving to big particles on the top again this is something you can do in a drum experiment with different sizes of particles and in this case it's doing a slightly different type of intermittent behaviour where you've got a hydraulic shock jumping up but as the stuff flows down it segregates up to the top and that's why it leads to this beautiful pattern and in a minute the drum is about to speed up and then you're going to get a continuous flow and it's going to start looking a little bit different it's still segregating but now it no longer has this it will still be segregating but it will no longer have this discrete structure here we go so we're getting this the central core is basically not moving and all the small particles are ending up around the edge now you can do these type of experiments but it's very difficult to take good observations if you're looking through the edge what you see is dominated by the end walls people are trying and we've tried to do experiments to memorize things you can also use radioactive particles it's extremely difficult to get good data whereas with the simulation we can chuck the particles in with whatever size we want we can have end walls or periodic and we can measure the position of every particle and really try to develop theories about how these things are moving in time the first thing perhaps is that we've done is just look at these things a lot and try to see this is the result of one simulation big particles going to the top again and try to develop intuition and then some theoretical postulates to make a proper theory so there's a little bit of math here sigma ij is the stress now if you've got two different sorts of particles there's a stress tensor corresponding to particle 1 and particle 1 and a stress tensor corresponding to particle 2 but there's also a stress tensor in between the two and it turns out that if you hypothesize that the stress components there's a kinetic bit which is just like the temperature of the molecules you might think and then there's a contact term which is also quadratic and the question is what are these coefficients beta and gamma so this was our hypothesis and it's how the stress is divided up between the particles that determines what happens because the big particles take more of the stress they're in a pressure gradient because of gravity so they push up to the top so we have this ansatz and then we want to see is it going to be true from the simulations where we can measure these stress components exactly and if you do a bit of math you can get a burgers equation which you can solve so we can predict what should happen in shear flows so here how can I get it to click on the on some other mode I want it to be on a movie but it's um oh there we go yeah no well let's go try this one it's not giving me the right type of pointer because right so this is speeded up a lot so this is a whole lot of particles all started off mixed it's actually it's doubly periodic but it's shearing down a slope you can see all the big particles going up to the top turns out you know for dense for a fairly deep flow like this this is very time consuming it actually takes quite a long time for the particles to move up if you have a shallow flow like you were seeing in those drum experiments it happens much much faster so if you had a space time plot where the color here represents all red is all big particles all blue slash purple is all small particles this is the height from the bed starts off mixed and then over time we just we get these two shocks moving until they merge and we get all the small at the bottom and all big at the top and this is a comparison with the theory you can also try what's in here the solution has an expansion fan and two shocks so here we've started off with the small particles on the on the top and the big particles and now the particles are going to reverse so you know you see this type of inverse grading and deposits all the time when you look at turbidity currents and all sorts of natural flows so now you know we have this model and which we verified with these simulations so we can say how long does it take for this type of segregation to occur so if we have a field deposit that we can look at and understand maybe we can say something about what happened with the flow and some people in the past for example have argued that if there's only a small size difference you get no segregation here we've got particles which are only 1% difference you wouldn't notice this in experiment the results here they look the same but we can see the dent differences in the profiles and if we go up to particles which are 2.2, the big ones are 2.2 times the small ones we see that we get a perfectly segregated layer at the top but because there's still some diffusion going on even the steady profile still has some of these big ones towards the bottom but it's not symmetric there are none of the small ones at the top and again we can predict that now with our theory that we've developed and there's a neat thing you can do with simulations which you could never do with experiments is you could try to get a steady segregating flux so here when the small particles get to the bottom we take them out and we drop them back up at the top and what that means is we have a steady state we have a steady state flux rather than the steady state being zero flux and we get this prediction for the profile that has a maximum in the middle rather than having all the particles at the top or the bottom and again we have an analytic solution which agrees quite well now I said the key thing is we're making this hypothesis to develop the theory of the stress distribution and this shows the three stresses the one two and the two two and the gray lines are a simpler theory which is not quadratic and the dots are the simulations and then the solid blue, black and red lines are fit with this theory so we can see that we get extremely good fit understanding how the stress is distributed and it's these small differences that give rise to all of the segregation now I'm going to finish off with showing you another a more natural granular flow rather than these idealized situations and how we can try to describe this with simple models so this is a big avalanche from the test site in the west of Switzerland in Valais de la Sion this is us walking up the debris so what's nice about here is that we can blast these avalanches from a helicopter so we can control them and what we did before this one is we went up there with a helicopter with a laser scanner and a good GPS so we get a very detailed surface height measurement of the snow we blast the avalanche and then we can re-scan it afterwards and we can look at how is the surface height changed so the blue regions here are regions where the surface height has decreased so the snow has been eroded the red and the yellow and the black regions some of them more than 10 meters of snow have been deposited so can we try to explain this type of behavior this is my friend who is in the bunker under those 10 meters of snow at the bottom we have a concrete bunker with some other instruments had to dig him out from this one we can calculate we know something about the snow properties so because there's erosion and deposition happening at some places you have to be a bit careful to actually calculate the deposit but these are for two different avalanches where we did this and we're looking at what their deposit depth well that's actually let's go here the deposit depth is a function of the slope angle and what you'll see is that as the slope angle gets to about 35 degrees there's no snow deposited because it's well above the angle of repose even with cohesion so here we have a frictional cohesion granular material and once we get below about 20 degrees the material is below the angle of friction so it won't flow so it could be any depth it can deposit deep but in this intermediate regime we have a I won't have the equation here we have a simple theory that matches up friction with cohesion which gives these curves so there is a tremendously complicated thing you know and we're still trying to develop a full model for this we can actually from the deposit infer what the friction and the cohesion was and get good agreement so what I hope I've shown is the direct simulation of granular systems is a really powerful tool very really powerful tool you have to be quite patient and you have to have some friendly supercomputer owners who will let you do what you want to do but the nice thing is you can really help yourself understand experiments much better because experiments often your measurements are going to be invasive you have effects of sidewalls and vibrations and other things with the simulations you can put these things in or take them out and really understand exactly what's going on and these rather obtruse things which you'd never be able to measure directly such as you know individual components of the stress tensor we can measure everything in a granular simulation and then compare our theoretical assumptions to develop models and what I haven't talked about here is how the grains actually couple with the fluid but this is sort of the new frontier for these type of simulations that people are working on and it's going to be very helpful for understanding these complete flows in the future thank you. Fluid coupling what do you intend to explore there? Well actually it's people like Eckert who are mostly developing that so if you just try to loosely couple the grains with the fluid and you don't take into account the volume of the grains you miss a lot of the physics it's okay for dilute systems but to really get a lot of the effects the poor pressure effects for example you have to include that so to be able to so there's all these models for debris flows for example and I've worked a lot trying to develop models for debris flows but it's extremely hard to get the poor pressure right and that's because we don't quite it's very hard to get good measurements and one of the things I'm hoping is that with these type of direct simulations where we can figure out what's going on at the microscopic level we can then develop better theories for debris flows in particular. So I missed the answer about G little G it has it has no effect on friction it's impossible on dimensional grounds if you change gravity in one of these experiments it's exactly the same in a drum experiment it's the same as just changing the rotation rate in a shoot experiment it's the same as rescaling time so a granular flow on Mars down at dune or something would look exactly the same as a granular flow on on earth if you change time by you know about 50% or something if you played the movie back differently you'd see no difference cheers