 Okay, very good. Okay, thank you. Thank you very much. So I'll briefly go over the motivation for our talk and then introduce the governing equations for direct numerical simulations of fluid mechanical flows, discuss the role of dimensionless parameters and how the computational effort scales with them. That'll lead us to the need for turbulence modeling and I'll introduce two different approaches here and then I'll briefly touch on some other aspects of upscaling. So our interest is in performing high-resolution simulations in particular of turbidity currents, so of sediment transport flows from the continental shelf down the continental slope into the deep ocean. Just to remind everybody, turbidity currents are gravity-driven sediments flows down the continental slope. They represent an important part of the global sediment cycle. They're frequently triggered by storms or earthquakes or other natural reasons. They can also be triggered by man-made origins. They can be incredibly large. One turbidity current can transport many cubic kilometers of sediment. And just to give you an idea of their scale, they can travel over distances up to the order of 1,000 kilometers or more. They can have front velocities up to 10 meters a second and front heights up to 100 meters. So if we want to perform high-resolution simulations of such flows, we typically approach them as dilute flows, meaning we assume that the volume fraction of the sediments grains is on the order of 1% or less. We also assume that the grain radius is much less than the grain separation, so that collisions among sediment grains can be neglected. And we assume that the grains are relatively small so that they don't have any considerable inertia. Now, due to the small volume fraction of the grains, we can neglect their effect in the conservation of mass or in the conservation of volume equation, so in the continuity equation. Rather, the coupling between the fluid motion and the sediment motion occurs primarily through the momentum equation. So the sediment essentially varies the density, the local density field of the suspension, and then gravity acts on those density variations in order to trigger the flow. The sediment is also then assumed to follow the fluid motion and have a superimposed settling velocity. So these are then our governing equations. We have the fluid continuity equation, so conservation of mass. Here we have the conservation of momentum, where C is the sediment concentration, and so we see this is the effective density term here, and gravity then acts on those density variations in order to drive the flow. And the sediment is being tracked by means of convection diffusion equation, where we assume that the sediment moves with the fluid velocity, plus it has a superimposed settling velocity. So in making these equations dimensionless, we obtain three dimensionless parameters. There's the Reynolds number, which is the combination of characteristic velocity, such as the front velocity of the current, characteristic length scale, such as the height of the current, and the kinematic viscosity of water. There's also the Schmidt number, which is the ratio of the diffusion coefficients in these two equations, and the dimensionless settling velocity. Now the most important parameter here is the Reynolds number, and let's just see what kind of order of magnitude we have for the Reynolds number when we're looking at real field scale currents. So as I mentioned, they can have a front velocity on the order of 10 meters a second. They can be up to 100 meters tall, and this is the kinematic viscosity of water. So that gives us a field Reynolds number up to 10 to the ninth. Please keep that in mind. So Reynolds numbers in the field can be up to 10 to the ninth. Now if we want to do DNS simulation, so direct numerical simulations, we take a model configuration, which we call the lock exchange configuration. So this is essentially an apparatus that you initially divide into two compartments. So we have here one compartment into which we fill initially water and we stir sand into it, and then we have here clear water initially, and then we have here a little Gaussian bump just to introduce an element of complex bottom topography. At time t equals zero, we then remove the dividing wall between these two compartments, and now this combination of water and sediment is denser than the clear water here, so it slumps to the bottom and forms a current that propagates along the bottom of the container. So we have here the turbidity current propagating, and then the clear water has to get out of the way, and forms a counter-flowing current along the top of the container. So just for those people who are interested, a few facts on the numerical method that Mohammed developed in order to do DNS simulations of such flows. We're using second-order central differencing for the viscous terms, so-called ENO schemes for the convective terms, TBD, Runge-Kutta time stepping, so explicit time stepping. We use a projection method to enforce and compressibility and the domain decomposition approach in order to parallelize our code, and we heavily rely on the PETC software package that was developed at Argon in order to solve our systems of algebraic equations. We can also use non-uniform grids in order to resolve the regions near the bottom of the flow more finely because that's where more of the action occurs, and we have a very nice immersed boundary method that allows us to deal with complex bottom topographies in a very accurate way. So this is a DNS simulation, a movie of a DNS simulation that I want to show you that Mohammed made. So up here we will see the flow. In the plane below, we plot the values of the shear stress, and in the lowest plane, you see information on the deposit height. So here the current is developing. We see initially the development of so-called loben cleft instabilities. Then the current interacts with this three-dimensional bottom topography. It becomes fully turbulent. It's strongly three-dimensional. We see how these flow structures have a strong influence of the local shear stress behavior, which of course then has an effect on the erosional behavior. Maybe I can show this movie once again. And we can also see how the deposit height locally is very much affected by this topographical feature here. So, yeah, just to see that movie once again. So initially it's nearly two-dimensional. It develops some frontal instability, and then becomes strongly three-dimensional when it interacts with this Gaussian bump. And you can see here how that affects the local deposit height and how it affects the local shear stress along the bottom wall. So this is a DNS simulation, as I mentioned. It was carried out not for Reynolds number of 10 to the 9, so for the Reynolds number corresponding to a real field scalar current, but for a smaller Reynolds number of 2,000. Now if we think about what that means, that corresponds more to a front velocity of two centimeters a second to a length scale perhaps of 10 centimeters, and here again the kinematic viscosity of water. So this very much corresponds to a laboratory scale current, not to a field scale current. So we can do the DNS simulation for the laboratory scale, but not for the field scale, and that's of course something that we would like to overcome because eventually we would like to have simulations for field scale currents. Here's another simulation, another DNS simulation. This is one of a turbidity current that propagates down a submarine channel, and there are some, there are levies on both sides of the channel as well. And people have speculated that when a turbidity current like this encounters a bend in the submarine channel, then it might partially overflow, it's called flow stripping, and so we wanted to see if we see that in the numerical simulations as well. So here's our turbidity current propagating down the submarine channel, and indeed there's this flow stripping occurring in the bends of the submarine channel. So we see that these things are being reproduced in quite some detail. So to summarize, for DNS simulations we see they have quite a few advantages. They accurately reproduce the physics because we resolved all of the scales. They provide very detailed information on the flow field, on shear stresses, on deposit profiles and so on, and they require only a minimum of empirical modeling assumptions. The disadvantages are that they are computationally very expensive and they are limited to low Reynolds numbers. So the question then is why can we not do a DNS simulation at Reynolds number 10 to the 9? So why can we not simply say we want to resolve all the scales for such a high Reynolds number flow? Well, in order to understand that, it helps to think of the Reynolds number as effectively a measure of the ratio of the largest length scales in the flow to the smallest length scales in the flow. So the largest length scales in the flow, the so-called integral length scales, those might be something like the front height of the current, whereas the smallest length scales of the flow, the so-called comagorov length scales, those are the very tiny vortices that we have in the flow where viscosity is strong enough in order to convert kinetic energy into heat. So that's where dissipation occurs. And turbulence theory shows us that this ratio of the largest length scales to the smallest length scales scales with the Reynolds number to the 3 fourth. So DNS simulations, which of course need to resolve these smallest length scales, and at the same time have a control domain that's large enough to capture the largest length scales, they then require this kind of scaling, Reynolds number to the 3 fourth in all three directions, so that means they require Reynolds number to the 9 fourth grid points. In addition, the time step also has to scale with the grid spacing, so that tells us that in the end the computational effort is on the order of Reynolds number to the third power. So when you compare now that we did our DNS simulation for Reynolds number of about 10 to the 3, but the field scale current was for Reynolds number of 10 to the 9, then we see that the field scale simulation would require on the order of 10 to the 18 times the effort of the laboratory scale current. And the laboratory scale current of course already took several days on maybe about 100 processors, so even with Janice I'm sorry to say we're not going to overcome this factor of 10 to the 18 anytime soon. So the question then is how can we perform simulations at the field scale? So here the key idea is that the large scale flow features are unique for every flow, but perhaps we can assume that the very smallest length scales of the flow, the ones at which dissipation occurs, may be somewhat universal. So in other words the smallest length scale, the smallest vortices of turbulent flow fields may be very universal in nature and very similar in all kinds of flows. And so as a result it might be able to develop a turbulence model that allows us to model the effect of the smallest length scales without having actually to resolve them explicitly. And the most important effect of the smallest length scales that we need to capture is the energy that they extract from the largest scale. So we need to make sure that we can capture this effect of the small length scales to extract energy from the large length scales and if we do not explicitly resolve the small length scales then we need to have a model, a turbulence model that extracts the energy for us. And so there are essentially two established approaches in order to accomplish this. One is based on the temporal averaging of the governing equations so spatial filtering and that leads us to large eddy simulations. So let me just give you the key ideas of how these approaches work. So first on ranch simulations so here we want to do temporal averaging so we take all of the variables of the flow fields such as velocity, pressure, sediment concentration and so on split them into one time average value and into one fluctuating value. And then we take our governing equations that I had written down earlier we take the time average of all the terms in the governing equations in order to derive a set of equations for these time average quantities here. And then the hope is that if we can solve only for the time average values then we don't have to have as fine a grid. But the problem that appears is that we have non-linear terms in our equations both in the sediment transport equation and also in the momentum equation and these non-linear terms when you take the time average of them, of these products of the time average value and the fluctuating value then you get this term here the time average of the product of the two fluctuations that is not equal to zero and it's not negligible and so this is a term that we cannot get directly out of calculating just the time average values of the individual quantities. So this term here is essentially the crux of the problem because it doesn't disappear and because it means we cannot entirely neglect the fluctuating quantities and so we have a closure problem because we have more unknowns now than we have equations. So in some sense then this is the term that we need to model when we perform turbulence modeling and so many such models have been developed to capture the effects of these terms and you may have heard of a few of them mixing length models, K epsilon models Reynolds stress models and so on all of these problems they are of course at various degrees of sophistication but all of these problems all of these models involve several empirical constants and these constants again depend on the flow physics on the flow geometry on all kinds of things and so it's very difficult to determine reliable values for these empirical constants especially when we're looking at flows involving complex physics such as sediment transport, erosion, deposition and so on over complex topography so in complex geometries and so that means as a result there's a large amount of uncertainty associated with these empirical constants and so there's a large amount of uncertainty associated with the results of Rand simulations. So Rand simulations do offer some results but at the same time they have these drawbacks. While there's this alternative approach of large eddy simulations and so remember for Rand simulations we used a time averaging approach for large eddy simulation we essentially use a spatially averaging approach or a spatial filtering approach so we take our original velocity field and we then do this filtering process here and there's a spatial length scale associated with the filter and so in that way we get a filtered velocity which now contains only the large scales and not the small scales and so again the hope is that we can derive a set of equations just for these large scale quantities and so in that way we may not have to resolve the small scale quantities but just like for the Rand's approach we find that again here we do get a closure problem because we cannot entirely neglect the effect of these unresolved quantities and so again we need empirical models in order to capture the so called sub grid scale effect so the effect of stresses and transport that occurs at the smaller scales than our grid resolution tells us so we need then LES turbulence models and again a variety of models have been developed the most well known one perhaps is the Smogorinski model but again these models involve empirical constants now people have actually developed very sophisticated procedures so called dynamic models that automatically determine these constants within the course of a simulation so you do not have to prescribe these models all these constants before the simulation and then they stay constant but through a double filtering procedure actually these constants can be determined automatically during the simulation so that has brought some progress as a result LES generally can be considered to be more accurate than Rand's but it's also more expensive computationally as an approach so we can say that there's still some uncertainty associated even with LES modeling and so clearly there's more research needed on this whole issue of turbulence modeling but let me just show you a couple of results so these are simulations LES simulations that Centiel who's in the back of the room Paul Stock and my group has performed so here we see the low Reynolds number simulation that Mohammed did for Reynolds number of a thousand and then by comparison an LES simulation of the same flow for a Reynolds number of 200,000 you can see that the turbidity current in the LES simulation has moved much farther at the same time so it has a larger front velocity and also it shows much more small scale structure than the DNS simulation and that's exactly what we would expect for a large Reynolds number that we see a lot more small scale deformations of these concentration contours and small scale vortices and so on this is at a later time and again we see the same difference the high Reynolds number current has moved faster and it exhibits a much more small scale structure so we see there's certainly some promise in these LES simulations in order to help us to get closer to these field scale Reynolds numbers and we're currently working on further refining these LES models so that indeed we can go to Reynolds numbers up to 10 to the 7 10 to the 8 and 10 to the 9 so this turbulence modelling is one aspect of upscaling results so going from arbitrary scale currents to large scale currents there are a couple of other aspects of upscaling when we think of upscaling again is going from very small scale phenomena to large scale phenomena and I want to mention those as well so this is the work of Zach Borden who had a very nice experience of winning the poster award last night and so we can look at his work also as one aspect of upscaling in that what he does is he wants to use particle based simulations so simulations that resolve each grain of the sediment in order to gain a better understanding of erosion and the balance between erosion and deposition so that from these very detailed small scale simulations we can then get better correlations between for example wall shear stresses and effective erosion rates so the idea here is to carry out very small scale simulations that only resolve a few hundred or a few thousand sediment grains in order to obtain relationships which we can then use in much larger continuum based simulations so let me just show you this little movie that Zach made so here we have a flow propagating over a sediment bed and you see as soon as the flow starts it starts to erode sediment grains at the same time gravity wants to bring those sediment grains back down and so we have this balance of erosion and deposition and so we hope that these kinds of simulations will tell us more about this very important region where the turbidity current touches the sediment bed and allow us to develop better models for erosion and deposition so if we then can derive such continuum models then we can do simulations such as these where now erosion is just a function of the bottom wall shear stress and these are two simulations one is for a slope angle of three degrees and you see here erosion is weaker than deposition so here the flow dies whereas if we go to four degrees then erosion has become stronger than deposition and so in this case the amount of suspended sediment grows and we have an avalanche like effect the current accelerates and grows in in magnitude we can then take those kinds of calculations in order to try to get some first stratigraphic information so this shows a picture of a series of simulations the deposit profiles that come out of a series of simulations I think here we had about 40 different simulations they were polydispers so in each simulation we had both small particles which are given here by the blue color and large particles, large grains the red color and these currents all came from the left and then deposited their particles and we see initially of course more of the large red grains are being deposited the small blue grains are deposited further downstream but then we also see these interesting striations here so we see for example how the first current how the first current deposited the very large red particles then the somewhat smaller yellow particles on top then comes the next current depositing first the red particles then the yellow particles and so we do get some idea of what kind of deposit profile we may form here one other way in which we can look at upscaling is just sketched out here so we can again bridge the difference between small scales and large scales by saying okay let's use a course mesh for a very large domain but then let's zoom into a small region here and let's zoom further in so that we have a very small region here now which we can resolve with a much finer mesh so this is again another way in which scales can be bridged and this is what is called a nested grid approach so we have a fine grid here and a much coarser grid here and actually we're just starting a project that employs this kind of technique sponsored by BOEM in the Gulf of Mexico where we're trying to couple coarse grained ROMS model to a fine grained turbidity current model in order to study the interaction between the large scales and small scales in these Gulf of Mexico events so let me just summarize so if we do DNS simulations they have great advantages they give us all kinds of detailed information very accurate but their computational effort increases with the third power of the Reynolds number and so as a result for realistic field scale currents we cannot really perform DNS so that means there's a need for some turbulence modeling and we can do either RAND simulations or LES simulations however both of these approaches require some empirical constants, some empirical closure models and so as a result there are some uncertainties associated with them and then I briefly touched upon some other aspects of upscaling of bridging the divide between small scales and large scales such as going from microscopic particle based models to large scale continuum models or using nested grids okay thank you very much again everything crystal clear yes just one question is how do the particles affect the turbulence in your model excuse me how? how do the presence of the particles affect the turbulence you mean in the continuum flow simulations? in the large eddy simulations in the large eddy simulations well the the turbulence is essentially generated because the flow is driven by density differences and so the particles determine the local density of the suspension and so if we have high concentrations of particles in some regions, low concentration in other regions then gravity will pull this region down much more strongly than this region and so that sets the fluid in motion and that generates the turbulence but when you look at the erosion processes near the bed it's probably affected by the presence of particles the turbulence stress and so on yes that's right and that's exactly why we're doing these microscopic particle based simulations because we want to see how exactly the turbulence and the particles interact and there clearly we have particle interactions so collisions among particles and so there the situation becomes much more complicated that's why we're doing these very refined small scale simulations okay so you have interactions in the model yes, yes the particles interact with the fluid and among each other and so the turbulence can be damped by the particles which is something that people have also observed in laboratory experiments so we expect that we will capture all those effects, yes very interesting one of your last slides you mentioned Gulf of Mexico but I think you showed Monterey Canyon that's right, that's right, yes but I was wondering where in the Gulf of Mexico is this study intended to be, is it a particular site yes, James do you want to comment on that over or why we picked a certain site