Lecture 1: Fluid Dynamics

Key Takeaways

So, let's let's start with unidirectional flow. Oh, here I should tell you my plan of action, my tentative plan of action. We have 4 hours, four three to four classes. And I have four I've made up four PowerPoints. I'm not quite finished with the fourth, but it'll come along. And so, the first one is on fluid dynamics that are relevant to sediment transport. And then there's one on sediment transport, one on bed forms and bed configurations, and finally one on sedimentary structures created by by flows. And I'm hoping to get one and maybe most of two today done because the third and fourth are heavy. They're long and heavy, and I think we need more than just 1 hour to do it. So, that's that's my point. It's been so long since I've given classes that I I can't really figure out how long presentation's going to take me cuz I might go over tangents and things like that. All right. Osborne Reynolds, famous name in in in fluid dynamics. And I'm mentioning him because he did a classic experiment way back in the late 1800s. A tank with a tube coming out of it and a valve at the bottom so that he could he could make a flow in the tube. And he had a little dye injection system as well cuz he wanted to see what happened to the dye in the flow. These are directly from his his sketches. So, in A, at low flows, the dye streams along in the tube without anything happening to it. Well, come on in. Or it might expand very slowly, but nothing much happens to it. Uh but when he increases the velocity, the dye wavers and then breaks into turbulence. And to visualize the turbulence, this is a sketch. He shows all these fluid eddies moving along. And um this is a classic experiment. So, I think this is a pretty good definition of turbulence. Um and a little bit more about turbulence. If you had a flow through a duct or a channel and you you released little tiny buoyant markers, which would float along, neutrally buoyant markers, everyone would take a different path and a very irregular path. So, that's one element of turbulence. Every motion downstream is very irregular. And if you had your little magic velocity meter to put in the flow and you measure the time velocity, instantaneous velocity, and plotted it. This is a bad plot. I drew this cartoon. They shouldn't be leaning backward. They should be pretty much even. But anyway, you get the idea. So, the fluctuations in velocity in a turbulent flow would be something like 5 to 10% of the of the mean flow. So, it's substantial, but but only a small part. You could also if you had I love magic things. So, if you had a magic powder and you you sprinkled it into the flow, you would see swirls of integrating swirls of masses of fluid called eddies going down the flow. If you've ever seen a river in flood, a muddy river in flood, and you look at it at the surface, you see all these eddies at the surface. It's very impressive. I love to see that sometimes. But this is the kind of thing you can visualize. >> [clears throat] >> So, this is my definition of turbulent eddies. Now, you know, if you have a question, I didn't say but if you have a question or comment, flag me down, right? I'm glad to stop and deal with things. Keep that in mind. Be glad to do that. Okay. So, so let's let's think about this flow through tube. It's a good example of what we have to deal with in in fluid flows. So, you need you need four variables to account for this flow. Obviously, you need the flow velocity and the tube diameter, but also the fluid properties, fluid density and fluid viscosity. Those are four variables you need to deal with. Um so, if you tried to graph a a four-dimensional diagram, well, that's impossible, right? You can't possibly do that. So, you'd think that we're stuck in dealing with this or in representing this. But Oh, this is an example. This is just an example I threw in of existence fields. You probably in some P chem or thermal course you've had to deal with the phase diagram for water, ice, vapor, water. And and that's the kind of thing, but you can't do that in four dimensions very easily, right? So, there's a technique in mechanics called dimensional analysis. And you can you can get really useful information about something. You don't have a solution, but you have some idea about the physical effects. So, here's what you do. Buckingham's theorem. Edgar Buckingham, I don't know much about him. He was an American physicist, and he was one of a few people who developed this idea of dimensional analysis. You can look him up on YouTube on um on uh you can Google him and look him up on the on the internet. So, here's here's the theorem. Now, I went through the proof of this back maybe 60 years ago, and it's not difficult, but you're going to have to take my word for it that it works. The number of dimensionless variables corresponding to the number n of original variables that describe the physical problem is equal to n minus m. n is the number, and m is the number of fundamental dimensions, mass, length, and time. This this is this is a a time-honored and very useful thing to do in fluid mechanics and other mechanics. So, um the way it comes out, rho u d over mu, density, velocity, duct size, and viscosity. And it's called in his number in his in his honor, Reynolds number. I say a Reynolds number cuz there are other Reynolds numbers coming along of the same kind of thing. So, we've collapsed four dimensional variables into just one dimensional variable. All you need to specify to describe this flow through the tube is the Reynolds number. Nothing else, just the Reynolds number. It's a very powerful tool, and we're going to be using it. Um so, let's look at another problem, which is relevant to sedimentary geology, flow past a sphere. What's what's the drag force acting? What's the drag force which moving in a vis- viscous fluid? Again, we have four variables. Same ones, rho u d over mu, but u is now the approaching flow, d is the diameter of the sphere. So, the drag force is a function of the Reynolds number, another Reynolds number. Does that make sense to you? But it's not quite as simple as it looks. Why? Why is that not too simple? Here Here you have a sphere held there and a fluid flow fluid flows by it. And there's a drag force. But there's another way it could be done. You could drop a sphere into a body of still water. And there's still a force on it. It's a gravity force, and it's moving relative to the fluid. But what's the difference? Tell me the difference between those two cases. I'll give you a hint. The t- The t-word. In the case of holding the sphere steady and letting the flow flow by it, the flow is likely to be turbulent. And the the the sphere will will have to react to certain differences in the speed of the flow because of the turbulent eddies going by. Whereas if you drop the sphere, a a negatively buoyant sphere, into a body of water, there's no turbulence in the water. It's still. And that makes a a small but non-negligible difference in how it works. Does that make sense to you? Incidentally, we're going to get to this. I'll tell you more about this later. Now, let's look at open channel flows. That's clearly a an important thing to think about when we're dealing with with sediment transport, by unidirectional flows, and that's what this what this hour is about. Say you're standing out there next to a river, the sun is shining, and you and you try to look sideways at the flow downstream. What's it like? So, I have to introduce two things here. A steady flow, this is definitions. A steady flow means no change with time. And a uniform flow means same at all cross sections downstream. All right. So, so here are the forces involved. What what pulls the flow down slope? Well, it's it's the down slope component of down slope component of the weight of the fluid. You you all know about doing vectors and figuring out components of vectors, right? But there has to be a resisting force if it's a uniform flow, and that's the the the friction force at the base. So, these two are balanced when you have uniform flow. Makes sense? So, what what variables do we need? Now, we're going to do the same thing in dimensional analysis. What are the variables we need? Mean flow velocity and depth, obviously, density and viscosity sim- but there's one more. The acceleration due to gravity. And why is that important? This is not That's That's not an easy question. Pretty to think about. Think about it. Why do you need gravity here? Now, you don't need it because of the down slope force cuz that's all taken care of by the by the by the by the weight of the fluid. Surface waves. Wind blows over the surface and it can create surface waves. So, that's that's an additional thing about the flow. It turns out that waves develop on the surface not because of wind, but because of an instability that develops at high flows high flow speeds. Uh and I'll say a little bit more about that cuz there are things called antidunes. Have you ever heard of How many people have heard of antidunes? All right, good. Anyway, uh because those are waves on the water surface. They can be stationary or moving upstream and they affect the sediment transport. So, we need five. Well, that tells us by dimensional analysis how many how many dimensionless variables do we have? We had four variables in the in the previous and we got one dimensionless variable, right? Three less than the than the number of variables. So, how many would we have here? You have two of them. Right, okay. So, [clears throat] we have another Reynolds number. This one would be based on flow depth and flow velocity. There are lots of Reynolds numbers around. Um based on depth and but we need another one. It's called the Froude number and it has G in it, U squared over GD. Sometimes people take the square root of that and call that the Froude number. Anyway, because of certain possible surface gravity waves. Here he is, William Froude. Engineer, fluid dynamicist, naval architect, did a lot of stuff. Uh and I've always wondered how he pronounced his name, Froude or Froud. And I finally recently learned that it's Froude. So, it's the Froude number. I've also heard Freud number. Say again? I've also heard Freud number. And Freud as in Sigmund Freud? Yes, yeah. Pronounced that way. Really? Yeah. They probably mean this. They almost certainly mean the Froude number. Yes, yeah. Freud number. I hesitate to think about what that would describe. >> [laughter] >> Anyway, so so let's think about the velocity structure of the flow. And we have our little magic Here's another magic. I love magic things. We have a little current meter that we can place at various positions above the base of the flow and we measure the time average velocity. Just time it for a minute or 2 minutes or 5 minutes and you get a profile. Let's talk about the profile. What's characteristic about it is um it's fairly evenly distributed in the upper part of the flow, but it rapidly goes down to zero at the bed. And um we need to think about why that is. There's something called the no slip condition. The fluid at the boundary has the same velocity as the boundary. It's not like tilting up the table, putting a brick on it, having brick slide down the table. Cuz there's actual there's a jump discontinuity in velocity, but that's not the way fluid flows are. The the fluid motion is always zero in contact with the boundary. So, now you can think about why is the profile much steeper near the base than it is higher up? Think about it. Somebody come up with an answer here. Why would why would Again, again, I'll I'll give you the I'll give you the hint, the T the T word. How about the T word? Is the thickness roughness the thickness roughness means? I'm sorry, I couldn't hear Does the thickness roughness near the bottom the structure means? Well, that's that's relevant, but that's there's more to the story and I'm going to get to that pretty soon. This is this is this is a simpler answer. Um It's a turbulent flow. And the turbulences are doing their thing, but the closer you get to the bottom, the less possibility there is for the for the turbulences to exchange momentum vertically. And that goes to zero at the boundary because the flow is always parallel to the boundary at the boundary, right? So, so over most of the flow, there's very effective cuz I use my hands. It's very effective mixing back and forth eddies exchanging positions and evening out the the flow structure, the velocity. Whereas you go out there near the bottom and the turbulence becomes imperceptible. And so, um the the flow has a very steep gradient right near the boundary. And it's called the viscosity dominated layer. Now, you know about viscosity. I'm going to have a slide on it pretty soon. It's basically a measure of the resistance to shearing in a in a in a continuous medium. And so, so here here's that profile we had and it's turbulent up in here, but down near the base, the turbulence is almost stamped out and any horizontal motions are very weak. And so, the flow here is dominated by viscous effects, not turbulence effects. That's that's a a basic characteristic of channel flow that you need to know about. So, some terminology. It's not too important for our purposes, but so it's a turbulence dominated region and a viscosity dominated region and sort of what they call a buffer layer in between. It's just It's just terminology and it's not to scale, obviously. All right, so now um let's see. Um Oh, I can't remember your name. Curran? Kozulin. Ko Zulin. Okay. You Um We have to think about not just smooth planar boundaries, but in our case as sedimentologists, we have to think about rough boundaries. And it seems simple, but it's not really as simple as it seems. This is a tough slide tough tough slide. It's in the notes. So, we have something called dynamically smooth flow and dynamically rough flow. And a dynamically smooth flow is like what you showed what I showed you. This is a viscosity dominated layer and the and the surface grains or particles are immersed in this viscosity dominated layer and it's as if it was a planar boundary. As far as the flow is concerned, it's as if it's over a planar boundary. But um for stronger flows, the particles stick up above the viscosity dominated layer. There basically is no viscosity dominated layer and and the the particles are shedding turbulence and the turbulence dissipates right down near the grains. So, that's that's worth keeping in mind. So, this this may help you. We'll see. A boundary can be physically smooth and with all viscous drag and no roughness, it's dynamically smooth. But it can be physically rough and when it's physically rough, it can be dynamically smooth because it's all viscous drag and pressure drag, which I'm going to talk about soon. Um and um on the other hand, at higher velocities, it can be it can be dynamically rough. So, that's the difference between them. Um I haven't told you what RE star is, but I'm going to get there eventually. Boundary Reynolds number. There's another Reynolds number. Boundary Reynolds number, roughness Reynolds number. It's a kind of dimensionless boundary shear stress and it's useful. Rho to the one half tau zero is the boundary shear stress to the one half D divided by mu. So, this this is useful useful little Reynolds number that you have to deal with. Okay, so now to change the pace here, there's something called the Navier-Stokes equation. It's a it's a long partial difference equation that accounts for all of fluid flow. The trouble is it's very hard to solve. You can't solve it analytically for most for most uh problems. And um I'm not going to tell you about the Navier-Stokes equation except to show you the two people who developed it. Claude Claude Louis Navier, who was a French physicist and civil engineer, and Sir George Gabriel Stokes, mathematician, physicist. Um Fellow of the Royal Society, big deal, right? So, there's something called creeping flow. What you do is forget about turbulence and just look at the at the viscous fluid flow at very low Reynolds numbers, which I told you about. And it's called creeping flow. This is what it looks like past the sphere. The the streamlines diverge, go around the particle, converge again, but it even I didn't draw quite right. I didn't emphasize enough. They they they come together slower than they went apart. Which means there's a force on the particle. Okay. So, um this is useful for small particles and slow flows. Stokes solved the Navier-Stokes equations by ignoring viscous effects. You throw out viscosity and you have a solvable equation. And no details here. Um but uh the thing about it is that uh turbulence is not is not a not an issue here. Fluid density is not in that because there's no fluid accelerations or very very weak accelerations, very slow change in dimension in the in the direction of the flow. And um this has a practical application. You can measure settling You can measure the settling velocities of particles so long as the Reynolds number is low. You have a tank of water and you have a little s- sort of fine sand silt sized particles and you drop them in the flow, it settles according to Stokes' law. And you measure the settling time, the rate of settling. And you can figure out the size back back out the size from the that R is is a radius. If you put diameter in you change the coefficient. And this is done. This is this is common thing in studying sediments is fine sediments is to measure the the settling velocity and figuring out the uh the size. It's very useful. There are devices that do it. Okay, now let's look let's look at higher Reynolds numbers. So here's a surface a solid surface that sort of breaks away from the main flow. And what happens is that the flow has so so much inertia that it keeps on going. It separates from the from the um solid surface where it uh where it breaks away and it falls away from the from the flow. And so you get very strong shear and that shear develops turbulence on it. And you have an expanding turbulent shear layer. And you got a lot of mixing a lot of entrainment. And down here you get a kind of a counter rotating irregular turbulent [snorts] vortex. And as we're going to see a lot of this when we talk about bed forms, ripples, dunes, etc. because this is this is classic kind of thing that goes on. Um >> [clears throat] >> And although I didn't mention in the slide, it takes a long way down past the point of separation for the flow to adjust itself to conditions that are as as if there had been no direction. That's a very complicated sentence, but you see what I mean. You have to go a long way before the flow comes back to what it was before it approached this breaking away point. And that's that's sort of painful for people who are studying uh flows around things in open channels because uh channel has to be very long before the effect of the thing that's there disappears. But you have to live with that. Um so this is what it would look like around a sphere. Now this this separation is not a line. It's a it's a circle that goes around the the flow transverse cross section. Pardon me. And that's where flow separates. There's a ring of flow separation ring that goes away goes downstream of the flow with the kind of thing I described, you know, a turbulent shear layer developing and and all around as as an it is with this with this uh blobby mass wake in the in right behind the sphere. I want to introduce two terms here, skin friction and form drag. I used I used the term form drag in earlier figure and I didn't think to to define it for you. Skin friction is a viscous forces on a solid boundary and I've told you about viscosity. Form drag is the front and back pressure forces because of flow separation. It turns out that I should go back to this slide. It turns out that the pressure is high is I'm sorry. It's high at the front. And then in the back the low pressure above and all around that circle um means low pressure. And so that pre pre creates a drag force just because of this the structure of the flow. And I have another slide here to show you. Wait a minute. I have to go. Okay. So things like the sphere or a sediment particle can be called a bluff body or a roughness element. Bluff bodies. Your house is a bluff body in the wind. Uh a semi trailer. You come up behind a semi trailer, it's a bluff body and you're you're you're you're tailing him in the wake and you're feeling it like this. Then you go to pass him and you're hit with a with the flow, right? And we we are bluff bodies. We are rough elements. Did you ever think of yourself as a bluff body and a rough element? But we are. So for low Reynolds number and not bluff body smooth boundary skin friction, that is the the friction right at the boundary between the flow and the solid surface and is much greater than this form drag that I mentioned on the last slide about the flow around a sphere. Big pressure difference and and you get a lot of a lot of pressure a lot of uh a lot of force on the particle. But in relatively high Reynolds number around bluff bodies form drag is far greater than skin friction. So just keep that in mind. Now there's something called an inviscid fluid. And there's no such thing as an inviscid fluid. But um it's also called ideal fluid. No viscosity. Only pressure and velocity, that's all that's involved. And uh it turns out that for large Reynolds numbers outside that thin viscosity dominated layer the nature of the flow is pretty much like it is in an inviscid fluid, a good approximation. And that helps us because um there's a a relationship between pressure and velocity. That V is a velocity there. As you as flow moves around an object um so there's an inverse relationship between velocity and pressure. And I'll show you why that could be useful by looking at an airfoil of an airplane wing. You you're probably familiar with this. The shape is such that when the flow goes around because of the because of the stronger arching on the on the upper surface the the streamlines are crowded, velocity is higher, pressure is lower, so you get a lift. May maybe you knew that about how airplane wings work. Uh the Wright brothers discovered this without knowing why back in 1900 or something like that. This is a complicated slide and I'm going to show it in two parts. Fluid forces on a sphere at low Reynolds numbers and at high Reynolds numbers, A and B. And just so you can read it even more easily, I'll show you first the one at small uh Reynolds numbers. This is the one I defined for you, the boundary Reynolds number. And so um there's some lift. But there's a lot of drag and so the total fluid force is at a moderate sort of shallow shallow angle. And the the the line of action is uh is fairly high up on the particle. Those pluses and little arrows are the the pluses and minuses are pressure differences and the and the little single barbed arrows are skin friction. But then I'll show you this the other one for large large Reynolds numbers. This would be anything coarser than sort of coarse sand up into the gravel range. And um you can see there's difference. It's rough flow. There's flow separation and and a big pressure difference from front to back. Still some skin friction. So it's mainly dominated by form drag. And and the lift lift is substantial. I mean it's almost equal to the to the to the horizontal component of the the flow. So very different depending on the Reynolds number. And And this this is relevant to we're looking at a bed of silt or fine sand at relatively slow flows, that would be the first case. If we're looking at coarser sediment like medium to coarse sand up in the gravel range, we we and then strong and large Reynolds numbers, strong flows, we we behave more like this. You get a lot of lift. Which is important, we'll see that as time goes on. All right, that's my pitch on oscillatory flow. Now I have to stop for a minute and uh you in the back will have to readjust cuz I want to take my sweater off. This is the other main topic we have to deal with. We've looked at unidirectional flows down a channel. It could be a river, it could be a tidal current, tidal flow, a longshore current in the shallow ocean. Uh but but there's also oscillatory flow, back and forth flow. And it's another important thing to deal with. And we did we the approach is going to be a little different from what I just gave you, but bear with me and see what you think. All right, so you probably know that that's a wave with a crest and a trough, a wave height and a wavelength. That's a typical wave that we would deal with. And and this this problem with this slide here. Um this is supposed to be a circle. And I know why it happened. I was going to I was resizing it trying to make it bigger and go up and down and I don't know how on PowerPoint you can you can expand something without changing the dimension. There must be a way of doing it, but I had to relearn PowerPoint. It's been 15 years since I made a PowerPoint. Literally, yes. So I had to relearn. But anyway, so so that's that's what we see when the wave goes by. Um but there's a difference, an important difference between the deep water waves and shallow water waves. In a deep water wave the the the bed is seabed is so far down that the orbits die out slowly with depth to get to where it disappears before you reach the bottom. And that's not relevant to our business here. But um in the shallow water waves the the orbits are ellipses. And I can't tell you why, but they're ellipses. And ellipses die back. But there's still back and forth motion right at the boundary even at even at the solid boundary at the seabed. And so that's relevant to us. We're going to think about how sediment is moved by oscillatory flows. It's going to wait. Boy, it's going to wait. So what is the flow like at the bed? Well, again we'll think about variables, wave period, max orbital velocity. The wave period, you know it. And orbital velocity, it moves this way and this way and that's an orbital diameter, how far the move moves back and forth. And the maximum orbital velocity in the middle of the orbit. That's the maximum velocity. So, there's three variables. And actually only two are needed. Now, I'm not going to prove this, but it's well known that that the velocity equals pi times the orbital diameter divided by the period. Just have to accept that. So, there's only two of these are are are So, but it's UM and T that that I and we as sedimentologists are mostly interested in. So, we have four variables, UM T, rho and mu. And of course we get a a dimensionless variable. But it turns out that this doesn't help us much. I just sort of mentioned that you can do that. It holds for a smooth bed with no sediment, but we're dealing with the mobile bed case pretty soon, you know, for sediment transport. So, and here's an important point. For shallow water waves with very small amplitude, the time history of velocity is is symmetrical. The the the forward velocity is equal to the back velocity. But for large amplitude waves, and that's pretty typical of large waves, when the troughs are broad and the crests are sharp. It turns out that um that uh when the wave crests are sharper and the troughs are broader, um it's not symmetrical. It's asymmetrical. We just did that. Okay. So, the thing is if you look at the maximum velocity when the when the crest is passing over you, forward, um the the velocity is higher, but the time of passage is smaller. And it's the opposite for when the when the trough is over you and it's moving backward. So, there's no there's no net movement of sediment, but it goes this way fast and this way back slow. And and so because, and I'm I'm not going to tell you too much about it, but the sediment transport rate, how much sediment gets moved, is a very steeply increasing function of the flow velocity. And so, the consequence is that there's some net sediment movement under a purely oscillatory flow, which seems counterintuitive, but that's the way it works. And as you know, the waves come from the deep ocean and they show, and um uh when the when the depth gets to be less than about a half a wavelength of the of of the of the wave, uh it begins the waves scrunch up. It's a technical term. Scrunch up and finally break, of course. Um and but but in this range from here to here, uh even in purely oscillatory flow, there is some net transport toward the toward the toward the toward the shore. And this is little a business about shoaling waves um that I just mentioned. We're talking about a single oscillation, a single wave. Goes in, water moves back and forth. But you can have more than one oscillatory component. There can be Get rid of this. There can be waves running this way and waves running this way or maybe this way. And they superimpose on one another. They they add to the effect. But um they act at the same time and they make for very complicated water movements at the bottom, as you can imagine. So, when the sea is fully developed under a storm, under a wind, uh how big the waves get depends on both the wind wind strength, but also how how far what greater distance that the wind has a chance to operate on the on the on the surface. And in a storm, you if you've ever been out on a ship during a storm, there's pretty good size waves running. They're very complicated. So, there's a spectrum waves running in different directions and with different characteristics and they all add together. Um but they're self-sorting because when the waves when the waves move out away from the storm, they sort themselves out by direction, obviously, but they also there's a low-pass filter effect. The bigger waves die out slower than the smaller waves. And so, when you get well away from the storm, it gets simpler because there's only usually one dominant um oscil- oscil- oscillation component. Unless unless there were a couple of sub-equal ones in different directions, which can happen in major storms. Um I just talked about this. But we have to think about the water motion the bottom water motions because when we talk about the sediment movement by these by these oscillatory flows, that's an important consideration. So, they can be at right angles, which is a fairly simple kind of thing. The ellipses would be the orbits would be in ellipses rather than straight lines. And for more if you had two oscillations not at right angles, it gets a very complicated pattern, which is infinitely repeating. Have you ever heard of Lissajous figures? How many people have heard of Lissajous figures? Go to do a There's a nice Wikipedia entry on Lissajous figures. And just go and look at it sometime. It's clever, okay? Some guy named Lissajous developed this and I don't know who he is or when he lived or anything like that. Anyway. Um Now, we have to deal with combined flows. We've talked about unidirectional flows, various things about the forces and motions. And we've talked about oscillatory flows. But it's it's natural to think that there can be oscillatory flow with a super with a superimposed unidirectional flow. They're combined flows. And and it's in terms of sediment transport, it's a jungle. It's amazingly difficult to deal with. John Grotsinger, my long-term colleague, would call it it's a dog's breakfast. He He loved that expression. It's a dog's breakfast. Anyway, it's really complicated. And we're going to get to that when we talk about sediment transport and bed forms. But you have to live with that. Um the simplest case is they're collinear. And that's easy to do in the lab because you have a channel with a flow in it. At the same time you're making waves. And uh and you can uh you can look at the effects when they're collinear. And in this case, you can see that this is a steady this is a spectrum. Steady unidirectional flow. Time time and invariant velocity. But it can be pulsing unidirectional with a minor oscillatory component, the flow pulses. Slow, fast, slow, fast, slow. Right? And then there's a middle thing, stop-start. It flows and stops. It flows and stops. And but and but in terms of sediment transport, it's not much different from this the case of steady unidirectional is not much different from pulsing unidirectional stop-start. But if the oscillatory component dominates over the unidirectional component, you get an asymmetrical oscillatory flow until finally you get to a symmetrical oscillatory flow. And and any any one of these unidirectional components and oscillatory components can be combined with each other just cuz we have combined flow. Now, if they're at right angles to each other, that's a little more complicated. And I don't know of anybody who's done this in the laboratory, but I had an idea long ago and I thought about doing a research proposal on it. Um have a channel with flow flowing down it. But on the sides of the channel, all along the sides of the channel, there are reservoirs with entrances into the channel. So, you tilt the channel back and forth, slosh the water sideways while it's flowing down that way. I never did it. You know, maybe I should have done it sometime, but I I never never got into actually writing a proposal for it. So, that's that's pretty that's pretty easy to deal with. The most difficult case is when there currents plus multi-directional waves. And the only thing I can think of if you want to do experiments, you'd have to build a very large wave tank with wave oscil- wave wave makers at various angles, plus you pump water in at one end and you pull it out of the other end. And I don't think anybody's done that. But an experimentalist like me thinks in terms of doing something like this. Um and you could instrument the seabed. People do that and look at sediment transport and bed configurations, ripples on the seabed. But it's difficult and expensive. And it's especially difficult in major storms where much of the action takes place. So, it's it's just difficult. And that's it. So, now I have to go to Let's deal with sediment transport. Now that you know all about fluid flow, let's deal with sediment transport. And the obvious place to start is initiation of motion. How strong does the flow have to be before the sediment starts to move, right? It's an important thing to deal with. It's a classic problem. Much has been written about it over the many many decades. So, here's a thought experiment for you. An imaginary field trip. You can shrink yourself down to microscopic size. You're a micro person. And if you're doing it in water, you have to have scuba gear, obviously. And you and and you have you have shoes that are equipped so that you can grab onto any surface down there. We're dealing with sediment particles on a sediment bed. You want to be able to get your feet onto. And so, you go down among the particles sitting there on the bed and below the bed as the flow is flowing. What would you see? What would you feel? Well, clearly there'd be flow around the surface particles, but there'd also be slower flow around some of the particles fairly down well down in the sediment. And it would be it would be viscosity dominated flow, but there would be irregularities because overhead underneath overhead above the the the sedimentary bed, there'd be turbulent flow. And and so, there'd be velocity fluctuations that would be transmitted down into the subsurface. But that's my field trip. But there's more to it than that because uh if you Now I'll turn into a fluvial geomorphologist. You know about effluent streams and influent streams? Maybe not. Okay. So, look at the stream and the the the the groundwater table comes down to the surface of the stream. And so, if the groundwater table slopes up from the channel, water is coming in from the subsurface up into the channel. And that it's it's a not a small it's a small effect, but it's not negligible cuz that would tend to move the particles upward and entrain them. An effluent stream, the groundwater table slopes away from the channel and you're losing water. And so, there's a downward flow and that would tend to inhibit the particle transport. I don't think it's a big deal. I should talk with with Taylor Perron about that sometime. So, um I told you about the boundary shear stress. Average over time and space. Skin friction, form drag, and lift. We We did this in the in the in the previous part. But you can look at things a little more closely and look at a sediment bed as a flow over the sediment bed. And here particles, they have contact points, probably usually three contact points. They're sitting there and they have a weight and there's a fluid force of the kind I just talked about. And so, and so, that's what we have to think about if we're thinking about making the move start happening. This is the kind of picture we have to think about. And in more detail, we could say there's this particle maybe ready to move. And so, there are skin friction forces, there are front to back pressure forces. Um and that's what actually moves the particle. And uh which are more important, the viscous forces or the pressure forces, depend on a Reynolds number as you might guess because that's what Reynolds number is all about. So, uh this is from what I gave you before. You've seen this. And just remind you that um the nature of forces on a particle that might be ready to move is very strongly a function of this of this uh Reynolds number RE star, boundary Reynolds number. So, let's do more more to I will love her if the dimensional analysis I guess you can see you can see that. Um so, here's a flow over a granular bed, loose granular bed. And what what do we need to worry about? We need to we need the boundary shear stress, that's what moves the sediment. Um we need uh the particle size. You also need rho and mu, the fluid properties. But we also need the specific weight, the weight per unit volume of the sediment. Now, you might be saying to yourself, uh why do we have to have the fluid density rho and the uh uh specific weight? And I'm going to run through a little thing with you, this will probably be the last thing I can do. Um let's think about the threshold as a function of fluid properties, specific weight of the particles, size, and shear stress. And so, as as before, we have four of them, we have two dimensionless variables. This is a a kind of a uh dimensionless uh shear stress. And the other is another kind of Reynolds number. This This is called the Shields parameter. And I'm going to tell you about Albert F. Shields in a minute. And uh U star is a kind of a fake velocity. It's uh it's uh tau zero divided by rho to the one half. And if you figure it out, that has that has the dimensions of a velocity. So, it's not really velocity, it's a it's a uh it's a shear stress in disguise. But anyway, that's the basic thing. Now, um Suppose you um have a smooth granular bed and you slowly increase the flow velocity. Few grains start to move, rolling, sliding, and they find new pockets here and there. Um but you know what that is? That's called a a a printer's fist. Do you know that? It's a printer's fist. I love those printer's fists. It's a to emphasize something. This is important to me. Printer's fist. The cutoff between no movement and established movement, there's no well-defined point. It is first a few grains start to move and then some more grains start to move and more and more and pretty soon a lot of movement means, "Oh, well, it's past the threshold." But there's no way to actually put a value on the particular threshold. Um this is probably the last thing I can do. But But here's a nice graphic way of thinking about how it works. This distribution is uh a distribution of instantaneous shear stress on the bed and it ranges from small to large depending on the turbulent eddy. Over the the bed, that's that's the distribution size distribution of the particles in the bed. So, when there's no movement, clearly the even the strongest flow doesn't move the most easily moved sediments, right? Um but you finally get to a point as you increase the flow velocity, the bed size distribution stays the same and the flow distribution shifts to the right and it starts to overlap and that's what we would call incipient motion. And then eventually gets to the point where the flow overlaps the bed quite a bit and we have general motion. So, there's a a famous diagram called the Shields diagram. Um I don't have a slide about Shields, but there's a nice interesting Wikipedia entry for Albert F. Shields. I read it sometime. He had interesting and varied career and I won't spend time talking about it. But anyway, this is a plot basically dimensionless boundary shear stress versus the boundary Reynolds number, which I told you about before. And he he plotted a number of points, data points from the literature. Ignore these these diagonal lines. And and he fitted a curve through it. And it looks pretty reasonable. But he extended the curve up to very small boundary Reynolds, that is small particles. Uh and and it it climbs without any data in there. And and last thing I want to do is talk about that. Fine particles tend to be cohesive because large particles, any cohesive forces between them, electrostatic cohesive forces, are negligible. But when you get down to really fine sizes, they dominate the situation and the particles tend to to clump together. And so, that makes uh more difficult for flow to move. And and although there's no data here, uh that's a reasonable way of looking at what happens in fine sediments. You have to really have a strong flow to to overcome those those um interparticle forces. And I think Oh, one more thing. I mentioned this to you. Look him up. He's interesting interesting guy. And uh and uh I will mention on in the notes in in in case any You You all got copies of the notes, right? Just in case you want to look at them. And I'm I'm referring these things to a particular page. Um there's something called the Hjulström diagram. I think he was Swedish, I'm pretty sure. And Sundborg, or I know he was Swedish. Sort of variants or extrapolations of the Shields diagram. So, this is standard initiation of motion stuff. So, I think I'm going to quit because um Well, I'll return to this. I got to stop. It's 4:00. So, I'll stop. But I will pick that up uh to what we do uh to deal with the initiation motion better and I'll go from there. When we have established motion, we have a lot of things to talk about. So, I have to quit. And uh I'll see you next uh next week, same time, same place, right?