 Hello everybody and welcome to video number 31 of the online version of the fusion research lecture We are in chapter 6 turbulent transport and you might remember that in the last video We started to talk about turbulence and neutral fluids and introduced the Navier-Stokes equation This video we will look at the dimensionless form of the Navier-Stokes equation because this can be very helpful for our purposes so we will look at the Dimension dimension less form of the Navier Stokes equation and To do that we will first start with a simple sketch made to make a drawing We have some obstacle Maybe we introduce another obstacle So these two obstacles here This one as well So this is just this is just an obstacle and the size of the obstacle is Corresponds to a characteristic size in the system capital L Then we have the resulting streamlines So we have a flow streaming around the obstacles and that might look for example If it's some kind of water so the streamlines might look like this like this then this Here in between them. It's more or less straight This one then goes more like this and This one more like this for example Then we have also a typical velocity Which I use green Not sure if you can see this So this kind of thick green line here This should correspond. This is a capital U corresponding to a typical Characteristic velocity of the system. So we have L being a character characteristic size of the system capital U being a characteristic velocity of the fluid From these two quantities, we can of course also define capital T a characteristic time using the characteristic length capital L divided by capital U Which then corresponds to a characteristic time now Producing the dimensionless units then the following dimensionless units dimensionless units T prime Being T divided by capital T then U prime being U divided by capital L over capital T and capital L prime being R over L And using these dimensionless units, we can then get the dimensionless form of the Navier-Stokes equation which reads the derivative of With respect to T of the Characteristic of the dimensionless velocity minus the gradient of the pressure plus one over R E I will introduce that quantity in a minute Laplacian into the of the characteristic velocity in principle also for the Gradient and for the Laplacian we have to introduce this prime so that we also hear the normalized Coordinate system so let's highlight that We have here as I said Introduced a dimensionless an additional dimensionless units and that unit and that is R e This is the Ray-Noel's number you might have heard of that This is the Ray-Noel's number R e which is equal as the coefficient of The mass of the fluid density sorry Yeah divided by eta Viscosity the characteristic velocity and length or Being the same as a characteristic velocity times the length over mu Where mu is the so-called kinematic viscosity the kinematic Viscosity and our is the Ray-Noel's number an important Number an important quantity in the real hydrodynamic or on turbulence in general Because the Ray-Noel's number tells us something about the state the fluid is in so for a fixed geometry The Ray-Noel's number basically determines the state of the fluid So for Fixed Geometry the Ray-Noel's number R e Determines the state of the fluid What do I mean by that? Let's have a look at the three examples that are included here first of all in the left-hand side We have basically the same Example as in the drawing on the previous slide. We have three obstacles one two three and around the obstacles There is a flow and it is a laminar flow so you can see we have nice streamlines and We have a low Ray-Noel's number for this case if you see the the big the long error on the bottom here This means that the Ray-Noel's number increase is the further we go to the right Then on the middle plot we have obviously a slightly increased Ray-Noel's number and We see that we have some kind of vortices here. You see there's one basically One row of vortices after each obstacle. This is why these Rows are called Vortex Streets Then if we further increase the Ray-Noel's number We basically have a lot of vortices a lot of eddies here until we finally end up here with the state of fully developed turbulence So that's right that down and when increasing the Ray-Noel's number, so If we have again a similar drawing where the Ray-Noel's number increases into this direction Then we start with a laminar state Then from the laminar state we go into a state where we have these vortex streets and these are also called Karman vortex Streets and Those these are basically rows of vortices with opposite rotational directions. So these are such the Karman vortex street is a row of Vortices With Opposite Rotation Directions Now if we further increase the Ray-Noel's number we get first an irregular Irregular appearance of vortices So we get this kind of irregular appearance of Vortices Until we finally get a state of fully fully developed Turbulence fully developed Turbulence this is what we've seen in the previous in the drawing Now let's have a look at some actual experiment Illustrated here. We have three different states in the top left one Let's start with the top left one where we see again an obstacle around which a flow is a fluid is flowing and Here the stream lights Stream lines in the water have been made visible by aluminum powder in the water and Here we have a relatively low Ray-Noel's number of 1.54 and you see these nice closed stream lines if you go to the next picture We have you already see that you might see that here are two vortices behind that Obstacle so the Ray-Noel's number has been increased and it is here approximately 26 And then if we increase the Ray-Noel's number further you can see that in this area We basically in this region we have fully developed turbulence and here we have a Ray-Noel's number on the order of 2000 which Resides in such a state of fully developed turbulence and You can interpret the Ray-Noel's number as follows so one interpretation and oops, sorry Interpretation is that the Ray-Noel's number basically is the the quotient to the coefficient of the non linearity of the non linearity Over the dissipation over the dissipation Where non linearity is the part which creates which drives the turbulence and dissipation the part which Reduces it and this is why when you have a low Ray-Noel's number We have a low non linearity term a small non linearity term or large dissipation We do not have a turbulence if you have a large Ray-Noel's number the non linearity driving forces large and we get a state of fully developed turbulence and and Oops, ah there was sorry slight missing So here we see again two Photographies and these are flowing soap films So these are flowing soap Films vertically flowing from the top to the bottom so they're basically falling down and You can see on the left Photography we have these nice Stream lines. So these are where very well separated, right? There's a bit of The non linearity driving term here. So the Ray-Noel's number increases if you go further down here You can see these nice vortices which we have for here for example But interesting thing happens if we increase is a Ray-Noel's number as you can see on this Photography here because for a large Ray-Noel's number The behavior is quite different and you get a lot of enhanced mid-saying mixing properties, right? There's a lot of Exchange going on into the perpendicular direction. This is something which you can also see nicely here in this photography Well, these are smoke lines so smoke is injected into a system and You can clearly see these kind of enhanced mixing properties due to increased cross-transport and Increased perpendicular transport. So what we can see here is so first of all, this is smoke what we see here Nice photography is beautiful. Photographies of smoke taken from this book which is Excited here. So definitely worth a look and we can see that turbulence leads to enhanced Mixing properties Or if you are maybe more importantly to say it is it leads to increased cross Well by cross I mean perpendicular increased cross transport as nicely illustrated by this photography a Way to study the turbulence the Navi Stokes equation is to by studying the Borger's equation which is very similar to the 1d Navi Stokes equation and also a very famous equation so the Borger's equation Which you might know from your mathematical physics lecture also from your hydrodynamics lecture So this equation is similar To the 1d Navi Stokes equation Assuming that the fluid density is 1 and that there is no pressure then it reads the derivative with respect to t of a velocity plus the velocity times the spatial derivative of the velocity Equal to the to mu times the second spatial derivative of the velocity This can be solved for example numerically as you might have done it in your numerical plasma physics lecture One could use finer differences for that. So these are shown here are numerical Solutions for that. So what is done here? This was solved numerically One could use finer differences and just involve it in time then using the Condition that u so velocity as a function of x is equal to u knot Times sorry for writing it like this sine times the wave number times x Then the nice thing about numerics as you know, you can simply switch off The separate terms in the equation by studying then the other terms for example on the left hand In picture so here What we have looked at here or what we look at here is by just having an equation Which reads the time derivative of the velocity is equal to minus the spatial derivative of the velocity Such that only this term in the equation has been used now And this one we have cancelled out and just omitted it and neglected it here Then the solution shown here oops shows us two plots the dashed line here is the initial The starting point t equals zero and then the other line the solid line is the Solution after a characteristic time step and we can see that we just get the propagating wave by that Yeah, so this is just a propagating perturbation. This is just a propagating Perturbation as you might have expected from that now Just looking at the time evolution of the equation of the spatial derivative of u Being equal to mu and then the second derivative of u So only the part on the right hand side here just looking at this equation at this part of the equation We can then see that again in this drawing the dashed line is the initial The starting condition t equals zero and then after a characteristic time step We have the solid line and we see that the amplitude has been reduced. So this term corresponds to damping. So this is the viscosity which we have here the viscosity and this results in damping and This is from the microscopic level. This is basically inner friction which we have here Now if we finally look at the non linearity So meaning the partial derivative of u being equal to minus u times the special derivative of u So looking at this term here Then we see that the non linearity resulting that results in steep gradients so that the non linearity Results in steep gradients As you can see in the plot. So again the dashed line is the initial value The initial solution the starting point and then here you can see after the time has evolved Here you can see these kind of steep gradients which we have here Yeah, so basically we have something like if you could also say we have higher harmonics here And this can lead then to eddies and now these two Parts so the viscosity and the non linearity These two are always in competition. So these two are always in Competition competition and One way to express that is as we said the Reynolds number. So these are in competitions to either Which either results in a state where we have some kind of turbulence going on or if you have more laminar system Okay, that's it for this video where we looked into the dimensionless form of the Navier-Stokes equation and introduced the very important Reynolds number which determines the state of a fluid You have looked at a few examples introduced the expression Kaman vortex street Looked at again a few more examples then here finally in this equation look at the Borgers equation Which is similar to the one D Navier-Stokes equation in order to have a look at the role of the different terms in the equation Okay, that's it for this video. See you in the next video