Solving Linear and Non-Linear Convection using MATLAB
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1) The project aims to solve differential equations for linear and nonlinear convection using numerical methods like Euler's integration scheme. 2) It studies the effect of time step size and number of nodes on the velocity profile, finding that more nodes better resemble the original profile while a larger time step can cause instability. 3) The CFL number is used to predict stability, where values above 1 will be unstable.
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An Algebraic Multigrid Solver For Transonic Flow Problems
Solving Linear and Non-Linear Convection using MATLAB
- 1. RESEARCH POSTER PRESENTATION DESIGN © 2012 www.PosterPresentations.co m COURANT – FRIEDRICHS – LEWY CONDITION The CFL condition ,denoted by CFL = u indicates the stability of the solution obtained . If the value of CFL number is greater than one , it implies that the solutions obtained will be unstable . REFERENCES 1. Advanced Engineering Mathematics – H K Dass 2.www.wikipedia.com ACKNOWLDEGEMENTS • Mechanical Engineering Department – St Joseph’s College of Engineering • Edxengine ,www.edxengine.com User group meeting presentations 2015 Solving Linear and Non-Linear Convection using MATLAB Cyril Mathew Samuel , S. Sai Ganesh , Suresh K Dept. of Mechanical Engineering, St. Joseph’s College Of Engineering INTRODUCTION Numerical methods are used to solve differential equations where an analytical solution does not exist .While taking this approach, several parameters have to be studied to gain confidence in the results. OBJECTIVE This project aims to solve the differential equations of Linear and Non Linear Convections through numerical methods, study the effect of time interval and nodes on the velocity profile and the importance of CFL number . CALCULATION OF VELOCITY PROFILE The solution for the differential equation +u = 0 ,gives the required velocity profile .Euler’s integration scheme is used with first order type spatial discritization. EFFECT OF NODES ON VELOCITY PROFILE The velocity profile of a higher noded element resembles the original profile, than that of a lower noded element . The shifting of velocity profile from a step profile to a bell curve is called as numerical diffusion . EFFECT OF TIME STEP ON VELOCITY PROFILE It can be seen that the velocity solution gets unstable as the time step increases beyond a critical value. CONCLUSIONS Of the several differential equations which define a fluid flow , the 1-D convection equation was studied in this project . Numerical diffusion was lesser for higher noded elements and conversely simulation time was higher for the same case. A larger time step resulted in unstable solutions . CFL number was used to predict if the solutions obtained would be stable or not. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.25 1.5 1.75 2 Length of the Element Velocity(m/s) 41 Nodes 81 Nodes 101 Nodes 201 Nodes Original velocity Profile 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.25 1.5 1.75 2 Length of the Element Velocity(m/s) Time step= 0.001 s Time step= 0.004 s Time step= 0.007 s Time step= 0.01 s Original Velocity Profile 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Length of the Element Velocity(m/s) CFL Number = 0.9 CFL Number = 1 CFL Number = 1.11 0 50 100 150 200 250 300 350 400 0 0.007 0.014 0.021 0.028 0.035 0.042 0.049 0.056 0.063 0.07 0.077 0.084 Number of Nodes SimulationTime(Seconds) EFFECT OF NUMBER OF NODES ON RUN TIME The run time increases with an increase in number of nodes , as spatial loop has to be executed for more times