CFD - Computational Fluid Dynamics [Fluid Mechanics #17]

Summary

In this informative video lecture, Prof. Van Buren introduces the fundamentals of CFD (Computational Fluid Dynamics) within the realm of fluid mechanics. The lecture transitions from theory and aerodynamics to understanding the processes and purpose of flow simulations. CFD enables the use of computational techniques to solve flow equations and analyze fluid systems at a high resolution without physical experimentation. The discussion covers various types of simulations, highlighting their advantages and constraints, including the challenges of turbulence modeling and sources of error in computational simulations. Tools like DNS, RANS, and LES are explored for their roles in handling complex fluid dynamics scenarios.

Highlights

• The video shifts focus from theoretical aerodynamics to practical flow simulation using CFD. 🚀
• Various types of simulations, like DNS, RANS, and LES, are used to tackle turbulence. 🌪️
• DNS provides high accuracy but is highly resource-intensive, requiring substantial computing power. 💪
• Mesh size and structure significantly impact the accuracy of CFD simulations. 🔍
• Understanding errors from simulations, such as physical and numerical errors, is essential to approach results critically. 🔎

Key Takeaways

• CFD allows us to simulate complex fluid dynamics on computers, providing detailed insights into flow fields without experiments. 💻
• Direct Numerical Simulation (DNS) is the most accurate but computationally expensive method. ⚙️
• Large Eddy Simulations (LES) and Reynolds Averaged Navier-Stokes (RANS) methods offer alternatives to DNS with varying trade-offs. 🔄
• Simulations require careful definition of the domain, geometry, and boundary conditions. Precision in meshing is crucial! 📏
• Each simulation introduces errors due to physical approximations, rounding, and discretization, but still remains invaluable for engineering applications. 📐

Overview

In this lecture, Prof. Van Buren delves into Computational Fluid Dynamics (CFD), a powerful tool used in fluid mechanics for analyzing flow systems through computational means. Moving beyond traditional experiments, CFD empowers researchers and engineers to dissect complex fluid interactions using simulations, handling flows ranging from simple laminar cases to the chaotic nature of turbulence, all on a virtual platform.

Throughout the video, the process of setting up a CFD simulation is thoroughly detailed. This setup includes defining the computational domain and geometry, applying appropriate boundary conditions, and carefully considering the discretization of the flow field through meshing. These steps are foundational in capturing the intricate details of fluid motion, especially in turbulent regimes that demand rigorous resolution and accuracy, often requiring supercomputers.

Prof. Van Buren emphasizes the various flavors of CFD, such as DNS, RANS, and LES, each with unique benefits and drawbacks. While DNS is lauded for its precision, the high computational demand makes it less practical for everyday use. Conversely, RANS and LES offer methods to balance accuracy with computational feasibility. Despite inherent errors in simulations, CFD remains an invaluable facet of fluid mechanics, significantly advancing the field's capabilities and applications.

Chapters

• 00:00 - 00:30: Introduction to Fluid Mechanics This chapter introduces the principles of fluid mechanics, focusing on the fundamental conservation equations. These include the conservation of mass and momentum, and are applicable to both enclosed flows, such as in channels and pipes, and external flows, such as boundary layers. Additionally, these principles can be applied to analyze and understand global forces like lift and drag.
• 00:30 - 01:00: CFD and Flow Simulations This chapter pivots from aerodynamics and theory to focus on flow simulations, specifically Computational Fluid Dynamics (CFD). It emphasizes solving problems using computational power and complements previous discussions on fluid measurement, framing experiments as the counterpart to simulations. The goal is to enhance understanding of fluid dynamics through simulations.
• 01:00 - 01:30: Understanding CFD This chapter delves into the concept of Computational Fluid Dynamics (CFD). It examines how flow simulations operate, including the various types of simulations and the potential errors that can arise. CFD is highlighted as a method for solving conservation equations within a discretized domain, enabling the exploration of flow physics through computational methods without physical experimentation.
• 01:30 - 02:00: Example of Flow Problem: Cylinder The chapter examines a flow problem over a cylinder, illustrating various simulations from simple personal computers to supercomputers. The example focuses on incompressible, uniform, and steady flow that begins as laminar and transitions to turbulence along the cylinder's surface.
• 02:00 - 02:30: Solving Conservation Equations In this chapter titled 'Solving Conservation Equations', the focus is on the analysis and resolution of conservation equations in fluid dynamics, particularly around a cylinder. The complexities of flow are examined, including the separation at the back of the cylinder, leading to a chaotic and turbulent boundary layer in the wake. The flow is highly three-dimensional and unsteady, providing a practical context to re-examine and write out the conservation equations for mass and momentum in the x-direction. This provides a basis for understanding the dynamics involved in such flow scenarios.
• 02:30 - 03:00: Analytical Methods This chapter deals with Analytical Methods, specifically focusing on developing a deep understanding of unsteady, three-dimensional flows. These types of flows are challenging because they change in the x-direction and involve all velocity and pressure terms, which cannot be neglected in the analysis. The goal of this chapter is to determine the complete velocity and pressure fields as functions of the three spatial dimensions and time. Once these fields are known, they allow for various calculations, such as the global forces of lift and drag on objects within the flow.
• 03:00 - 03:30: Experimental Methods and Limitations In the chapter titled 'Experimental Methods and Limitations,' various techniques for solving partial differential equations (PDEs) in fields such as aerodynamics and hydrodynamics are discussed. The chapter covers the challenges of solving these equations without assumptions and suggests traditional analytical solving methods like pen and paper. Techniques mentioned include separation of variables and the shooting method, among others, commonly utilized by mathematicians.
• 03:30 - 04:00: Computer Simulations and Benefits The chapter discusses the use of computer simulations in solving complex equations, specifically highlighting the limitations of traditional strategies. These strategies, while useful, are severely limited and applicable only in the simplest forms of equations. The chapter emphasizes the challenges associated with solving the Navier-Stokes equations, which are notoriously difficult to solve directly. As an alternative to computational simulations, the chapter mentions the potential for conducting physical experiments, such as using a wind tunnel to perform flow measurements on a cylinder with the aid of hot wire instruments.
• 04:00 - 05:00: Steps for Running a Simulation The chapter titled 'Steps for Running a Simulation' discusses the challenges faced in conducting physical experiments, such as those involving wire or particle image velocimetry. It emphasizes the difficulty in capturing comprehensive data like velocity and pressure fields in a single experiment due to limitations in spatial and temporal resolution of measurement devices. Additionally, the high cost of equipment and facilities, along with space constraints, are mentioned as significant hurdles that can restrict the feasibility of conducting such experiments.
• 05:00 - 06:00: Defining Geometry and Computational Domain In the chapter titled 'Defining Geometry and Computational Domain,' the focus is on the benefits and limitations of using computer simulations in comparison to physical experiments. It highlights that simulations provide high resolution and accurate flow field data, including the entire uvw flow field and pressure, which is challenging to achieve in experimental settings. However, it also notes that simulations can be time-consuming.
• 06:00 - 07:00: Defining Boundary Conditions The chapter titled 'Defining Boundary Conditions' delves into the intricacies of setting up simulations for complex flow problems. It emphasizes the challenges in achieving timely and accurate results due to the complexity of flow dynamics and the computational power required. The discussion highlights the necessity for programming skills if one intends to develop their own simulation software. The chapter sets the stage for understanding the process of starting simulations, preparing the reader to explore the methodologies involved in the workflow of simulation development.
• 07:00 - 08:00: Discretization and Meshing The chapter titled 'Discretization and Meshing' provides a basic introduction to running a simulation. It begins by noting the author's limited expertise in simulations, emphasizing their background as an experimentalist. The author conveys that their guidance, though not expert-level, aims to be useful. The initial step in any simulation process is highlighted as simplifying the equations involved. This simplification often mirrors initial explorations conducted to determine if the flow problem could be tackled manually, using pen and paper methods.
• 08:00 - 09:00: Importance of Proper Mesh Density This chapter discusses the importance of proper mesh density in simulations, particularly in cases of turbulent or separated flow where many traditional assumptions (e.g., incompressibility, two-dimensionality, steadiness) do not apply. It emphasizes the need for accurate simulation settings and geometry definition in computational domains, noting that real-world geometries might often be simplified for the purpose of simulations.
• 09:00 - 10:00: Solving the Flow Field The chapter discusses the importance of domain size in solving flow field simulations. It explains that the domain should be large enough to ensure that the flow at the edges is not influenced by the flow around a cylinder, which is crucial for setting appropriate boundary conditions. However, it also warns against making the domain unnecessarily large, as this requires more computational resources and time, which may be limited.
• 10:00 - 11:00: Direct Numerical Simulation (DNS) The chapter titled 'Direct Numerical Simulation (DNS)' delves into the precision required for accurate simulations, particularly in the context of fluid dynamics. It emphasizes the necessity to carefully choose the dimensions of the simulation domain: a few body diameters in the vertical direction and a longer extension in the predominant flow direction (x direction). This consideration is crucial, especially when studying flow over an object, as the domain must be extended further downstream than upstream.
• 11:00 - 12:00: Turbulence and Grid Density The chapter discusses the importance of boundary conditions in computational domains, specifically focusing on both the boundaries of the object (a cylinder) and the overall computational domain. It emphasizes that every boundary and wall must have an appropriate boundary condition. The chapter illustrates a cylinder and domain, highlighting common boundary conditions, particularly the left boundary where the inflow condition is applied for free stream velocity, maintaining a parallel steady flow.
• 12:00 - 13:00: CFD Techniques for Estimating Turbulent Flows The chapter discusses Computational Fluid Dynamics (CFD) techniques for estimating turbulent flows. It highlights the application of velocity boundary conditions, such as allowing fluid to move parallel to the domain walls, while using no penetration conditions to ensure there is no flow leaving the domain at the top or bottom. The chapter also elaborates on challenges in defining outflow conditions due to variable velocity fields. Overall, the focus is on designing domains large enough to minimize vertical velocity effects at boundaries.
• 13:00 - 15:00: Error Sources in Simulations The chapter discusses error sources in simulations, particularly focusing on flow around a cylinder. It emphasizes the influence of boundary conditions, such as atmospheric pressure on outflows and no-slip conditions on cylinder surfaces. The effects on open versus enclosed flows, like those in pipes, are also considered.
• 15:00 - 16:00: Review and Conclusion In the 'Review and Conclusion' chapter, the focus is on how to approach solving flow problems in computational domains. It involves applying boundary conditions such as no slip and no penetration at the surfaces enclosing the domain. The text explains the shift from continuous domain solutions typical of analytical approaches (ODE and PDE solutions) to discretized domains necessary for numerical methods. Here, the flow must be divided into sections rather than solved as functions over a continuous range.

CFD - Computational Fluid Dynamics [Fluid Mechanics #17] Transcription

• 00:00 - 00:30 hi and welcome to another lecture in fluid mechanics to this point we have covered our conservation equations which dictates how a fluid moves including conservation of mass and conservation of momentum we did this for both enclosed flows like the channel and the pipe and external flows like the boundary layer and if you zoom out you can use the solving of these flows to analyze global forces like lift and drag
• 00:30 - 01:00 today we're going to pivot away from aerodynamics and theory and start to think about flow simulations specifically we're going to be talking about cfd a way to solve problems using only computational power this video complements the fluid measurement video from not too long ago where fluid experiments are often the other side of the coin to fluid simulations our goal is to better understand how
• 01:00 - 01:30 flow simulations work how simulations generally solve a flow system the types of simulations there are and the errors associated with it so let's jump right in computational fluid dynamics commonly referred to as cfd is how we make use of governing equations using computers and exploring flow physics without ever leaving our keyboard it is a method for solving the conservation equations in a discretized domain to get the detailed flow field
• 01:30 - 02:00 this can span from doing simple simulations on a tablet or personal computer to using the world's largest supercomputers to solve a single flow problem let's consider an example flow problem that we'll consistently refer to throughout the video here we have incompressible flow over a cylinder the incoming flow is uniform and steady flow starts laminar and transitions to turbulence along the cylinder surface
• 02:00 - 02:30 and at the back of the cylinder we have separation that leads to a chaotic week in the turbulent boundary layer and the wake we have highly three-dimensional unsteady flow it's been a while but let's write out our conservation of mass and momentum in the x-direction for this flow you may notice that there are regions of
• 02:30 - 03:00 this flow that are unsteady developing meaning changing in the x direction and three-dimensional these all mean you cannot remove any of the velocity or pressure terms in these equations we're stuck with them in their daunting entirety our goal is to eventually arrive to the entire velocity field and the pressure field which is a function of the three spatial dimensions and time using the known flow field we could do things like calculate the global forces of lift and drag on the
• 03:00 - 03:30 body for something like aerodynamics and hydrodynamics so without being able to make assumptions how do we solve these equations to get these values there are a number of ways to approach the problem some of which we've seen already first we could try to solve it with a traditional pen and paper this is analytical solving there are a number of techniques to solve pdes you could use separation of variables shooting method along with others that mathy people will know much more about
• 03:30 - 04:00 than i do however these strategies are severely limited they can only be used in the simplest forms of the equations before they can be solved outright after all the navier-stokes equations are famous for being so difficult to solve alternatively if we wanted to do something with our hands we could build an experiment and take physical measurements in this example we would stick that cylinder in a wind tunnel and we would do some flow measurements with a hot
• 04:00 - 04:30 wire or maybe particle image velocimetry with this we would be able to piece together the velocity field and we might be able to say something about the pressure field but we would have difficulty obtaining all these variables in one single experiment additionally the spatial and temporal resolution would be limited to our measurement devices and the equipment and facilities cost a lot of money and also sometimes space is a limiting factor not everyone has a warehouse with a wind
• 04:30 - 05:00 tunnel some people just have a desk and a computer in these conditions we can make use of the computer to run a simulation here we would take a model and directly simulate the flow around the model with great resolution and accuracy also we would get the entire uv and w flow field along with the pressure field all at the same time something very limiting in experiments however the simulations can take a long time
• 05:00 - 05:30 literally years depending on the flow complexity and often we need results faster than that they also can takes quite sophisticated computers depending on how detailed and complicated your flow problem is and if you want to build your own code to do the solving you're going to need some programming capability and this brings us to the focus of today simulations you might be thinking how do we even start with a simulation what's the process let's go through a
• 05:30 - 06:00 very cursory overview of how you might run a simulation i want to note before we start that i'm an experimentalist and have very limited experience with simulations i have never written my own simulation code and am by no means an expert hopefully you still find this useful step one of any simulation is to try and simplify our equations to the best of our ability you likely already did this when you first explored if your flow could be solved on pen and paper
• 06:00 - 06:30 here we would say things like incompressible two-dimensional steady or all of those other useful assumptions we've been using throughout the course however if you have turbulent or separated flow we can't use many of these assumptions which is why we have to run these simulations in the first place the second step is to define our geometry in our computational domain here sometimes geometries are simplified maybe the cylinder has a bump or a divot
• 06:30 - 07:00 in it that you don't think is important no need to include it in your simulation then the domain is everywhere you want to try to solve for the velocity generally you want your domain to be big enough so that flow at the edges is not impacted by the flow of the cylinder this helps for the boundary conditions in the next step however you don't want to go any bigger than that bigger domains means larger computational resources and our time is limited so you really want your domain as small
• 07:00 - 07:30 as possible without missing anything important or making any accurate simulations typically in the vertical direction you would choose something like a few body diameters in each direction and in the x direction which is the predominant float direction you would use a little bit longer and also if you're studying flow over an object like we are here you have a longer domain downstream than upstream of the body in the next step we need to define our
• 07:30 - 08:00 boundary conditions here we consider both the boundaries of the objects under consideration in this case the cylinder and the boundaries of our computational domain every boundary and wall should have some sort of boundary condition here we draw our cylinder and the domain with common boundary conditions first the left boundary of the domain is where we give our inflow condition or we set the the free stream velocity here we have a parallel steady flow with
• 08:00 - 08:30 velocity u infinity on the upper and lower domain walls we would allow fluid to be moving parallel to the wall but we could use the no penetration conditions to make sure that flow is not leaving our domain in the top or bottom wall ideally we've designed our domain big enough so that there would be no substantial vertical velocities out here anyway on the outflow it is a bit more difficult to define a condition because the velocity field is likely
• 08:30 - 09:00 still impacted by the presence of the cylinder typically on the outflow we use something like a constant pressure or atmospheric pressure assuming your flow is open to the atmosphere and on the surface of our cylinder we have the no slip and no penetration velocity condition that we've been using throughout the course when we've studied these flows analytically it's important to note that if we were doing an enclosed flow like a pipe
• 09:00 - 09:30 we would do all the same conditions but the walls of our domain would be our surface so we would give them the no slip and no penetration condition after our boundary conditions we need to take our domain and discretize it when solving flows analytically with ode and pde solution methods you can solve the flow in a continuous domain as functions but here we're not doing that for a numerical solving we have to divide the flow up into sections
• 09:30 - 10:00 called discretization where we solve for one single value in each section this is similar to taking a video in reality time and space are completely continuous and you could zoom in on your face until you saw molecules but if you take a video of yourself you now have discretized your visual information pixels limit your spatial resolution and the frame rate of your video discretizes time you do not know what happens at scales smaller than a pixel
• 10:00 - 10:30 or in between your frames we do the same here but with the conservation equations this can be as simple as just dividing our domain up into a grid which in cfd is commonly referred to as meshing each grid point is called a cell and each cell gives you a single value for u v w and p at a certain time much like the pixel of an image in reality meshing is an extremely
• 10:30 - 11:00 complex and sensitive aspect of running simulations typically the number of cells is huge like in the millions and although we can draw a very simple grit here they are often different shapes including rectangular triangular pentagons etc additionally the size of each cell varies throughout the domain either increasing or decreasing the cell density this depends on the anticipated local flow characteristics
• 11:00 - 11:30 like how big a structure is or a velocity gradient in that area to ensure you have a good mesh density and size what you do is called a convergent study you do a sample simulation and you make sure that if you increase or change the number of cells your answer doesn't change substantially if this is true your mess is generally good to go when choosing cell sizes we have to follow some general rules your cell can be big if you are far from
• 11:30 - 12:00 a boundary anticipate fairly uniform flow with small gradients and it's not turbulent however if the flow is turbulent you have high velocity gradients you're near a wall or there's separation locally your cell size likely has to be pretty small we can label in our diagram areas where we think we can get away with big cells and where we need cells to be smaller
• 12:00 - 12:30 to show why we need to make sure the cells are small enough consider a vortex if the vortex is big we draw a decent sized grid on it doing dns on this grid would give us a velocity vector at each cell and in the end result we would be able to see the vortex very clearly this is properly resolved and has proper cell sizes however if we have a group of small vortices on that same grid
• 12:30 - 13:00 and we do dns on that we would notice only sparse velocity vectors that barely even indicate locations of a vortex but no detailed information about the vortices at all this is under resolved our cells are too big so you want to be sure that your domain mesh is properly resolved before moving forward the next step is to go ahead and solve the flow field there are a ton of tools out there to solve discretized systems like finite
• 13:00 - 13:30 element methods finite difference methods but the one that's most predominant in fluid mechanics is finite control volume methods to best understand it let's bring up our domain again here we have the boundary conditions labeled and our cylinder is ready to go in the bottom corner let's consider a single cell of our mesh and zoom in on it here the corners of our box have known x
• 13:30 - 14:00 and y locations and we're looking to calculate the velocity vector and pressure in the center of this box in reality we can treat this the same way we treat control volumes which is how we derive the conservation equations in the first place because we picked a box in the corner we can say something about the velocity on the left wall and the bottom wall due to our boundary conditions referring back to our conservation of mass we can show how we would discretize
• 14:00 - 14:30 the equations because we are no longer concerning our continuous system similarly we could discretize the momentum equations but we won't show that here the computer takes the knowns linearizes the equations between points and iteratively solves them once you have the entire domain iteratively solved to a tolerance you're okay with you then move forward a time step and you do it all again and a given simulation could have
• 14:30 - 15:00 hundreds of thousands of time steps so you can start to see why you might need supercomputers to do these simulations we have millions and millions of cells each cell needs to be iteratively solved at each time step and then we need to do every time step of which there could be thousands in the end after all these calculations you get something like this a grid of velocity vectors and pressure info at each grid point these describe what a
• 15:00 - 15:30 cylinder does when you put it in a cross flow theoretically it would match exactly what would happen if you put this cylinder in a real cross flow because we know that the reality follows the conservation equations which we've solved for using the simulation and in a final step if you wanted to do some post-processing you could then trim your data set to save space for example typically you don't need all those time steps you've calculated
• 15:30 - 16:00 a lot of times you only save what you need because these data sets get very big after cleanup and post-processing you can calculate the global forces you need from the velocity and pressure distribution this method completely solves the conservation equations and is called direct numerical simulation or dns it's very accurate and is the gold standard of cfd in research however it's typically very
• 16:00 - 16:30 computationally expensive there are different techniques for making the simulations less expensive if you don't have the time or money to do full proper dns the reason we've developed different models for estimating flow is usually because of turbulence if you remember the smallest cell sizes are determined by the smallest structures you anticipate having in a flow field unfortunately turbulence creates very
• 16:30 - 17:00 small often microscopic flow features it is a defining feature of turbulence itself as an example let's consider a turbulent channel flow and all that chaos that comes with it to the naked eye some structures are big some are medium and some are small however if you zoom way in with a magnifying glass you would still see even smaller structures you have to zoom in to the point where the vortex is so small
• 17:00 - 17:30 that it would rather turn into heat than keep spinning these are the smallest structures that turbulence makes and these tiny flow structures are everywhere the flow is turbulent and mostly near the walls so if your flow is turbulent you need an astronomical grid density to estimate it accurately your grid density is generally a function of the reynolds number the higher the reynolds number the flow is the bigger the gap between your
• 17:30 - 18:00 biggest flow structures and your smallest flow structures here that would be the gap between the channel height which represent the biggest structures and the smallest flow structures that dissipate to heat so the number of grid points is a direct function of the reynolds number and unfortunately many flows we want to simulate like commercial aircraft are extremely high in reynold's number there are a ton of alternative methods to simulations that we don't have time to talk about
• 18:00 - 18:30 but we can lay out the major players that are used to estimate turbulent flows what we've already talked about is dns which fully solves the navier stokes equations they're great for accuracy but at high cost then we have the reynolds average navier stokes or rand's estimations which solve the flow field only in time average and utilize the reynolds decomposition method we learned about in a previous video
• 18:30 - 19:00 these are easier to solve and give reliable time average information but they require the difficult modeling of the reynold stretches which are very flow specific and tough to obtain accurately lastly you could do large eddy simulations or les here you would just ignore the small structures and keep your grid relatively large then after you solved out that flow you add back in the small scale information through models which are tuned to your
• 19:00 - 19:30 flow type it works decently well and is fast though you need to be careful with it near surfaces where those small structures are more important there are many more types of cfd techniques to study that we don't have time to cover and i don't know them well enough to talk about them lastly we consider the error simulations aren't perfect and we need to say something about the error associated with a simulation for measurements the error was fairly
• 19:30 - 20:00 straightforward at least in my opinion it's a bit easier to quantify and account for however for cfd there are many sources of error that can become significant if you're not careful let's go over the main ones first the error and physical approximation you do some sort of cad model to use in your cfd and often the surface is extremely perfect and can't match reality our actual cylinder might have roughness or imperfections that aren't accounted
• 20:00 - 20:30 for in our ideal cad model second there are rounding errors as you do a calculation you can only save a certain number of decimal places usually this is quite a lot but over trillions and trillions of calculations these small errors can add up third by definition we are solving the flow iteratively which means we are stepping again and again until we are within some tolerance that we've designed this
• 20:30 - 21:00 tolerance means we got close to the right answer but not necessarily the perfect right answer as a result the tolerance carries with an error and the more tolerant we are to error the more error we have fourth is the discretization error we have divided up a continuous domain into discrete parts and that isn't perfect this is usually a larger source of error and a computation and last we have the error due to
• 21:00 - 21:30 linearization when we estimate the functionality of our differences and how we do our computations and finite difference methods we usually expand things out in terms of the equations and ignore higher order terms moving forward by making it easier to solve dropping these higher order terms we get error associated with that despite these errors stimulations are still incredibly useful and a foundational component of fluid mechanics analysis in research and
• 21:30 - 22:00 industry without simulations like these we wouldn't be anywhere near our current capability with aircraft race cars sports or even fluids transport hopefully this brief overview helps you better understand the use and limitations of simulations and how it fits into experiments and theoretical analysis and that's it let's review we started by introducing the concept of solving a flow through a computer simulation
• 22:00 - 22:30 you start by defining a domain and geometry then apply boundary conditions to that domain next you discretize by creating a mesh ensuring that all your cells are small enough to capture the smallest structures in that area you then solve for the flow field at each cell by iteratively solving the linearized discretized conservation equations this builds your flow then we considered all the different types of simulations
• 22:30 - 23:00 and ways to model turbulence behavior without needing to simulate the smallest structures in the flow and lastly we consider the main sources of error that come with simulations and what to look out for when designing our simulations i hope you enjoyed the video and thanks for watching