GUTS OF CFD: Navier Stokes Equations

let's dig into the guts of cfd part one navier's stokes equations hello everybody i'm nick the naval architect so first off why does this even matter why do you need to understand all of the CFD techno mumbo jumbo stuff well to create in an analogy a lot of software vendors would like you to think of CFD as a calculator just type in the numbers and get the same answer and you're done in reality CFD is much more like an advanced Excel spreadsheet where you as the operator are typing in your own formulas and the CFD software will very happily execute those formulas and it's up to you as the operator to understand what is actually being executed it's up to you as the operator to actually understand how that mathematics is translating into real-world numbers because all of the CFD machine knows is mathematics it doesn't actually understand if that mathematics translates into accurate real-world predictions so now abandoning our analogy CFD you're not necessarily typing in actual formulas but you are actually controlling the setup and a lot of the accuracy of that setup has to do with the grid sizing and a lot of the physics settings that you input you input that all of those settings what is a good value for those settings depends on the physics and on the equations you determine those settings which means you know you need to understand how those settings feed into the physics so you need to understand the physics and how they're interpreted by the CFD solver let's start with the most fundamental of all CFD physics navier's stokes equations Napier Stokes equations describe all of the forces in all of the motions that can act on a fluid they are the equations of motion for a fluid now that sounds very big and scary but I'm actually going to show you the basis of how they get derived and I want you to understand that they're actually based on a pretty simple concept that simple concept it's basically Newton's laws of motion translated into fluid terms so if we just take a simple six-sided box this is a six-sided cube of fluid some tiny little chunk of fluid and we're going to say what are all the possible forces that can act on the surface of this fluid well we've got direct pressure that can act on all six sides but then we also have shear stresses that can act on all six sides okay let's add up all the forces on all six sides and to avoid making this too complicated we're going to start by just adding up stresses in the X direction only so first we have the X faces that's the minus and positive X face those are direct pressures only so we've got the stress terms those direct pressures times the areas of those faces dy DZ next we add in the shear stresses from the y sides so we've got the minus y and the plus y those are shear stresses so we've got shear YX on the minus side and then share YX on the plus side and then multiply that by the area of that face which is DX DZ now we have to add up the other area which is going to be the shear stress again but this time for the top and bottom faces going to get multiplied by the area of those faces which is going to be d DX dy and you can see by adding up all three of these terms we now have all six of our faces on this little infinitely small cube of fluid here if we group some similar terms together here do a little bit of algebra to clear clean up our equation and simplify it down a bit we get this simplified equation this right here is adding up all of the stresses on our infant s amol element of fluid in the X direction only now we have to add in one extra force our infinitesimal fluid we were just looking at forces on the surface of that fluid there is one other thing that can act on that fluid which is body forces these are things that not don't act on the surface but actually act throughout the entire volume of the fluid these would be things like grab or magnetism okay we'll add that in and now from here on out I'm going to skip a few steps because it's more important for you to understand the essential idea behind navier's stokes rather than the full complicated mathematics of it and the essential idea behind navier's stokes is that it is force equals mass times acceleration so the equation of motion in the x-direction is mass times acceleration in the x-direction equals force all right let's turn that around a little bit what's our mass well our mass is our density times our three dimensions of our cube dxdydz our acceleration in the X direction I'm going to start slipping in some calculus terms here that turns into big d u-- big dt which if you've ever taken differential equations you'll know that the big d is significant it stands for the full derivative across all the variables and that's probably bringing back some nightmares of differential equations right now but understand that basically that means acceleration and on the right-hand side of that equation that's the forces well what are our forces we've got our surfaces our surface forces and our body forces that's the f1 X and our f2 X so there they are remember we had those before and if I just take off that shading you can see there they are mmm this equation is starting to look a little complicated but okay we remember at the core of it it's still just mass times acceleration equals force I'm starting to see some common terms there that we can cancel out on both sides of the equation and there well that's not looking too bad that's a fairly simple looking equation but we're definitely now into the realm of calculus that's okay we kind of knew we would have to get there anyways and at this point I'm going to skip about five steps jump ahead to the end and show you what the full Navy or Stokes looks like we're going to get rid of those stress terms and make some substitutions to turn them into some of the more fundamental properties of the actual physical fluid which is viscosity that's what we really can measure once we do those substitutions we get something that looks like that now I know that looks a lot different from the previous equation I've also switched to vector notation which is pretty fancy so yeah it looks very different but this is the Navy or Stokes equations for full fluid motion just in the x-axis direction in vector notation it looks different but the core thing you have to remember is this is just mass times acceleration equals force that's still what we're dealing with here add up all the forces and that's what the acceleration is that's the fundamental idea behind Navy are stokes equations so Navy or Stokes is one of four critical things that we need once you get these four pieces of information you own the fluid you know everything there is to know about that fluid first thing is navier's stokes equation if you can solve that equation you're halfway there then you need the equation of continuity this is a very fancy way of saying if fluid cannot simply cease to exist right in the middle of space you can't have sudden vacuums and voids and in the fluid it can't happen it can change in density but it can't just suddenly cease to exist there might be thermodynamic relationships so yeah fluids can expand they can contract they can heat up that's all possible so that might happen and the boundary conditions which is to say things like if your fluid is sitting inside a bowl that bowl is a boundary it has a basic mathematical relationship of saying fluid cannot leave the bowl that those sorts of things you get these four pieces of information you know everything there is to know about that fluid to summarize up the main thing that you need to remember from this today is navier's stokes equations you're going to see this written in many different forms it gets expanded it gets simplified people like to write it in different formats they have different notations for writing it but at the core of it all you really have to remember is that navier's stokes equations is force equals mass times acceleration it's the momentum equation it's the momentum equation and that is the basic essence of navier's stokes equations I hope this helps you thanks very much i'm nick the naval architect thanks for watching don't forget to click that like button and subscribe for more videos and did you know that we produce more than just videos at DMS check out our website to find more articles free downloads and other help with ship design we offer a host of engineering services for budgets large and small so check us out to see if we can make your next project easier

Channel: DMS | Marine Consultant

Views: 16,901

Rating: 4.9747901 out of 5

Keywords: naval architect, ship design, engineer, marine, ship science, marine engineer, marine engineering, CFD, Computational Fluid Dynamics, Computational, Fluid Dynamics, Fluid Mechanics, Navier Stokes, Mesh Generation, CFD engineer, ANSYS, ANSYS CFX, ANSYS Fluent, OpenFoam, StarCCM+, CD-Adapco, CD-Adapco StarCCM+, fluid motion, newtons law of motion, law of motion, fluid law of motion, equation of continuity, surface forces, fluid body forces, fluid surface forces, fluid stress tensor

Id: YIY-lWftuzY

Length: 9min 42sec (582 seconds)

Published: Mon Jun 24 2019