In this video we will learn how to solve the complex Navier-Stokes equations, within power bound of your PC. Here concepts of RANS and FVM are introduced in a logical manner.
Detailed description of video lecture is given below.
Actual Vs Averaged Solution
As we discussed in previous article, exact solution of N-S equations is too much of accuracy.It captures every minute details of turbulent flow. But an engineer is not interested in such a solution, what he needs is an averaged solution as shown below.This is in fact averaged solution of actual solution.
Fig.1 Actual Vs Averaged solution
Any variable in a turbulent flow can be represented as sum of mean value and fluctuating value. An example is given below for variable u.
Where averaged quantity is found out using following operation.
Here time interval of integration should be carefully selected. It should be small enough to capture any unsteadiness in flow, at the same time it should be big enough to smooth out fluctuations due to turbulence. A pictorial representation of averaging operation is shown below.
|Fig.2 Averaging operation done on a turbulent flow|
How to obtain averaged solutions ?
To obtain this averaged values, instead of solving actual N-S equations we can solve something called, averaged N-S equations. Navier Stokes equations generated after averaging operation is known as, Reynolds averaged Navier Stokes equations.
Fig.3 Conversion of Navier-Stokes equations to Reynods Averaged N-S equations using averaging operation
The averaged equations are represented here in index notation form. Here F represents external force acting on fluid.
Reynolds Stress – Turbulence Modeling
But RANS is not purely in terms of mean values. The last term of RANS is in terms of fluctuating components. This term is known as Reynolds stress.
There are various turbulence models available in order to represent, Reynolds stress in terms of mean quantities. Selection of proper turbulence model is one of the most important step in a CFD project.
Solving RANS numerically
RANS is not so difficult to solve numerically. Current CFD packages use mainly 3 numerical methods. They are FEM, FDM and FVM. Some commercial packages using these methods are listed here.
Fig.4 Different numerical schemes used in CFD and some commercial CFD packages using them
It is clear that FVM (Finite Volume Method) is the most common method of all. So in this article we will concentrate only on FVM.
Finite Volume Method
The fundamental flow equations are derived in FVM, using integral approach. Here instead of considering a differential volume we will consider a finite volume of arbitrary shape. And we will say that when flow passes through it, rate of increase of quantity inside the volume will be same as flux in minus flux out, plus generation of quantity.
Fig.5 Basic philosophy of FVM : Conservation of quantity
You can apply this to any quantity. Quantity can be either mass or momentum component. Such a concept will lead to conservative equations in integral form. It will have general form like this.
Where U represents quantity. So we can apply this general form to mass, so U will be density in to velocity. It will lead to conservation of mass equation. Similarly we can apply it for 3 components of momentum. We will get 3 more equations, which are conservation of momentum equations. So again we have 4 equations in total. This time in integral form.
Now the challenge is to solve these equations throughout the control volume numerically. Here there are surface and volume integrals. We will execute numerical integration of these equations on small non overlapping cell volumes, of arbitrary shape. So before executing numerical integration, actual control volume is split into small cells, as shown below.
Fig.6 Conversion of Control volume in to small non-overlapping cells : Meshing
This process is known as meshing. There are various meshing schemes available in CFD.
Now we need to execute surface and volume integrals on these cells. Such a cell is shown here. The method of surface and volume integrals are also shown pictorially.
Fig.7 Numerical approximation of volume and surface integrals in FVM
In FVM volume integrals are approximated as volume of cell multiplied by average value of quantity at centroid of cell. Similarly surface integrals are approximated as midpoint averaged values. So for this cell we have to do same operation on all four surfaces. Now we can apply same operation throughout the cells. This will generate series of equations in terms of averaged values. We can solve them together, with help of some boundary condition. An appropriate mathematical solver will do this task for you. Usually CFD solvers use iterative method to get the solution.
CFD at a Glance
The steps we have explained so far are summarized here in step by step manner.
Fig.8 Step by step CFD procedure
Before winding up the lecture, we will also see a sample CFD problem.
Fig.9 A sample CFD problem
How to build a career in CFD?
Hope you got a good introduction to CFD from here. Now its time to get hands on experience on CFD packages. Most used CFD packages in industry and its relevance are given below.
- ICEM CFD – A perfect software to do meshing
- Gambit – Another meshing software, but has become obsolete nowadays
- Fluent – Most preferred CFD solver in industry
- CFX – Another good solver
- Icepak – Electronics thermal management
Now it is clear from range of CFD packages available in market that, it is not possible to learn all of them at a stretch. So first thing you have to do is to find out your area of interest and select a good CFD package which will suit your need. You will be able to produce colorful results from CFD, but the real challenge is in understanding physics behind the problem ( make sure you are good in fluid mechanics and mathematics ). Please keep in mind that most of the time CFD produces rubbish results. Reasons might be poor quality of mesh, wrong physics used in CFD solver, wrong solver selected or wrong boundary condition applied. So it is imperative to spend lot of hours running simulation and analyzing results of simulation in order to become a good CFD engineer.