Computational Fluid Dynamics | An Introduction

Computational Fluid Dynamics is a tool widely used in design and analysis of equipments related to fluid flow and heat transfer. There are wide range of commercial softwares available in market for this purpose, which are user-friendly and using this you will be able to generate lot of colourful results. But in mean time it is also important to understand what is behind this CFD packages. This lecture is for that, to give you a basic understanding on CFD.

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Purpose of CFD

In the figure below you can see lot of colorful figures, which are output of some CFD simulations.

Fig.1 Some flow problems solved using commercial CFD softwares

The million dollar question is what exactly CFD is searching for?. Assume the volume shown below is the domain where fluid flow occurs. Or this is the control volume.

Fig.2 Control volume of a flow problem

In a CFD problem what I have to solve is velocity field and pressure field inside the control volume. More precisely we want to find out u, v, w and p throughout the domain. So it will be a function of x, y and z plus time, if the problem is unsteady in nature.

Equations to solve the unknowns – Navier-Stokes Equations

If we can solve the unknowns u,v,w and p we are done with CFD. To solve 4 unknowns you have to have 4 equations. So next step is to derive these 4 equations. To formulate these equations we will use conservation principles on a small control volume. First principle is conservation of mass this will lead to one equation.

Fig.3 Differential control volume considerred for derivation of conservative equations

Where you will say rate of increase of mass at a given point is mass flux in minus mass flux out. It can be represented in differential form like this.

Remaining 3 equations are derived from conservation of momentum, which is same as Newton’s 2nd law of motion. Since momentum is a vector quantity, there will be 3 components for it. It will generate 3 independent equations. It can be represented in differential form like this.

For each direction, there will be one equation. So in total you have got 4 equations. All these equations together known as the famous Navier-Stokes equations.

How to Solve Navier-Stokes Equations

If you can solve N-S equations together you can find out the 4 unknowns. But there exists no analytical solution for Navier-Stokes equations. Because these are highly non linear coupled partial differential equations. The only way left is numerical method. Where instead of solving for general case, we will solve the problem for particular case at discrete points.

Fig.4 Numerical Vs Actual solution

There are various mathematical techniques like FEM, FDM or FVM available for numerical solutions.

Direct Numerical Simulation

If you solve N-S equations numerically that method is known as DNS. A highly accurate method, results of DNS are even more accurate than experimental results. But if you try to solve a practical flow problem using DNS even the most powerful computer will take years. So DNS is used only as a research method. We can apply this method only when flow is very basic. We will see how to solve N-S equations for practical flow problems in a separate article.