[MUSIC] Hi, welcome again to the fifth week of our class, Simulation and Modeling of Natural Processes. This module will be about the equations and challenges in computational fluid dynamics. The very fundamental equations when you do computational fluid dynamics are the so called Navier-Stokes equations. Today we will work with one version of these equations called the Navier-Strokes equations for incompressible flow. They express the conservation of momentum in the fluid and the incompressibility of the flow. They are very simple, and they neglect many effects, many physics, which you can observe in actual fair fluids. They don't account for the compressibility of gases, for example. They don't account for temperature or thermal transition effects. They don't account for chemical reactions and many more phenomena, yet despite their simplicity, they are very much used on the most common versions of Navier-Stokes equations in computational fluid dynamics. In particular, they are very frequently used to simulate both gases and liquids. Although gasses are compressible, neglecting compressibility, in many cases, is not so important. Therefore, the incompressible version of Navier-Stokes equation will be the baseline of our study today. This is the mathematical shape of the incompressible Navier-Stokes equation. The first line expresses the conservation of momentum. It is really three equations, because it consists of vector quantities like the velocity u. The velocity u is the central quantity for which we are solving. Once we know the value of the velocity in every point in space and at every moment in time, we consider the Navier-Stokes equations solved. The velocity is a three-component vector in 3D space or a two-component vector in 2D space. Therefore the Navier-Stokes equations consist of three scalar equations. The pressure which also occurs in this equation is not an important variable. It is defined throughout the volume of a fluid and it is responsible for ensuring the incompressability of the fluid one if you apply some external forces on it. The viscosity is a measure of the resistance of the fluid to shearing forces. The viscosity corresponds to what in everyday life we call the thickness of a fluid. For example, honey is thicker than water, which means that honey has a higher viscosity. The first term in the Navier-Stokes equations expressed the acceleration of velocity fluid. The second term is the convective acceleration, it has some importance if the velocity changes over space. For example, if a fluid speeds up when it enters a narrow channel, it is the only term of Navier-Stokes equations which is non-linear. And it is responsible for all non-linear effects in the fluid for example, fluid turbulence. The pressure gradient is a term which is responsible to maintain the incompressibility of the fluid. The viscous term is responsible for the answer for the response of the fluid to shooting forces. The viscous term takes the shape of a vector Laplacian, which means it is a diffusion equation for the momentum. Momentum in the Navier-Stokes equations are diffused in a similar way as heat is diffused in a heat equation. Finally, the force on the final equation of the Navier-Stokes equations are the continuity equation. They simply state that the fluid incompressible, and we use them as a tool to calculate the pressure turn. I will not talk in more details about the mathematics or the detailed significations of the mathematical operators in the Navier-Stokes equations. Because today, when we derive a numerical method, we will not derive it from the Navier-Stokes equations. Instead, we will take the lattice gas automata and develop them further into numerical scheme to solve Navier-Stokes equation. Navier-Stokes equations today are more theoretical background framework, which gives us some help to understand the physics of fluid flow. One very interesting properties of a Navier-Stokes equations, becomes obvious if you make the variables dimensionless. To do this, choose some characteristic quantities of your simulation. Characteristic velocity capital U and the characteristic length capital L. If you've modeled an airplane, then the length could be the diameter of the airplane and u could be the cruising speed of the airplane. Both of which are characteristic properties of your system. Then we can make the velocity in the Navier-Stokes equations dimensionless by dividing them by the characteristic velocity capital U. We can get the dimensionless velocity, U star. In the same wa, the pressure is made dimensionless by dividing it by the characteristic velocity squared. We also make the differential operators dimensionless. The time derivative is made dimensionless by multiplying it by the length divided by the velocity. And the gradient is made dimensionless by multiplying it by the characteristic length. Once we have done that, we get on the bottom of the slide a dimensionless version of the Navier-Stokes equation, which means it has no length and time scales. The only parameter which is left in this model which you can still choose is the constant in front of the viscous term. It is conventional to take the inverse of this constant and call it the Reynolds number. The Reynolds number is equal to the characteristic velocity times length divided by the viscosity. The fact that we were able to reduce the Navier-Stocks equations to a dimensionless form governed by a single parameter means that these equations are scale invariant. You can simulate two systems, one of which is big and one of which is small, and have the same physics if they are scaled properly. This is a property which is exploited, for example, when you explore the aerodynamics of an airplane on a model airplane which is placed in a wind tunnel. The model airplane is smaller than the real airplane. But you can get the same Reynolds number if you increase the velocity of the wind around the airplane, or if you replace the air by another gas which has a lower viscosity. The dimensionless formulation of the Navier-Stokes equations is also exploited when you do numerical simulation. Because the units of your actual physical system is most often not so convenient a numerical simulation where we will use a different system of units. I would just make sure that we have the same Reynolds number in the simulation as we have in the experiment, so you can match the two together. But differential equations are not the full story in computational fluid dynamics, as the boundary conditions are just as important. I show you here an example of a possible numerical simulation, which is a two-dimensional flow around an obstacle. There's a bounding box around the system, which defines the extent of the simulation, of which we will solve the Navier-Stokes equations. There are different types of boundaries in the system. Around the obstacle, where here is a circle obstacle, we have boundary condition which you can consider to be a physical boundary condition. It is conditions by the physics by the surface properties of the obstacle which is placed inside the flow. But we have also virtual boundary conditions which have no physical mean which are just placed there to satisfy some condition, some constraint of our numerical system. For example, at the inlet, well we'll just cut the domain, the real physical domain is bigger but we have to cut it somewhere so we need to find the boundary condition. Which is there to pretend that the system was really bigger, to minimize the influence of the reduced domain size on the solution of the flow equation. The same we do on the outflow to have the fluid exiting the system into nothing in an appropriate way, and also on the lower and on the upper boundary. This ends our module Equation and Challenges in Computational Fluid Dynamics. Stay tuned for the next module. [MUSIC]