?About the author: A Matlab simulation developer who loves scientific research. He cultivates his mind and improves his technology simultaneously. For cooperation on MATLAB projects, please send a private message.
Personal homepage: Matlab Research Studio
Personal credo: Investigate things to gain knowledge.
For more Matlab complete code and simulation customization content, click
Intelligent optimization algorithm Neural network prediction Radar communication Wireless sensor Power system
Signal processing Image processing Path planning Cellular automaton Drone
Content introduction
In physics, convection-diffusion is an important phenomenon that is widely used in many engineering fields. It describes the transport and dispersion process of substances in fluids and is critical to understanding and optimizing many practical problems. This article will introduce a numerical method based on the finite volume method (FVM) for simulating transient convection-diffusion processes.
The convection-diffusion equation is a mathematical model describing the material transport and dispersion process. It combines the two main mechanisms of convection and diffusion and can be used to explain many practical problems, such as air pollution transport, heat conduction, and chemical reactions. The general form of the convection-diffusion equation is:
?C/?t = ? · (D?C) – ? · (vC)
where C is the concentration of the substance, t is time, D is the diffusion coefficient, and v is the fluid velocity. This equation describes the variation of concentration with time and space, taking into account the effects of diffusion and convection.
To solve the convection-diffusion equation numerically, we can use the finite volume method (FVM). FVM is a numerical method widely used in fluid mechanics and heat transfer, which discretizes continuous equations into algebraic equations by dividing the computational domain into discrete volume elements. In FVM, we divide the computational domain into grids and perform calculations on each grid cell. For the convection-diffusion equation, we need to calculate the change in concentration at each grid cell.
In FVM, we use control volumes to calculate changes in physical quantities. The control volume is a virtual volume corresponding to the grid cell and enclosed on the boundary of the grid cell. For the convection-diffusion equation, we need to calculate the rate of mass change within the control volume. This can be achieved by calculating the mass flux into and out of the control volume.
In the convection-diffusion equation, the convection and diffusion terms are calculated slightly differently. For the convection term, we use the product of fluid velocity and concentration to calculate the mass flux. And for the diffusion term, we use the product of the concentration gradient and the diffusion coefficient to calculate the mass flux. By adding these two mass fluxes, we can obtain the rate of mass change within the control volume.
In FVM, we also need to consider boundary conditions and initial conditions. Boundary conditions describe the behavior of matter at the boundaries of the computational domain. Common boundary conditions include fixed concentration, fixed flux, and reflection boundaries. Initial conditions describe the distribution of matter at the beginning of the calculation. By applying boundary conditions and initial conditions to the control volume, we can obtain a complete numerical solution.
By using FVM, we can simulate the transient convection-diffusion process and obtain the changes in substance concentration over time and space. This is useful in many engineering fields such as environmental engineering, chemical engineering, and biomedical engineering. Through numerical simulation we can optimize system design, predict contamination transport and improve process efficiency.
In summary, FVM-based transient convection-diffusion simulation is an important physics application. It realizes the numerical solution of the convection-diffusion process by discretizing the continuity equation, using a control volume to calculate the mass change rate, and considering boundary conditions and initial conditions. Through this method, we can better understand and optimize many practical problems and provide strong support for engineering practice.
Part of the code
function FVToolStartUp()</code><code>%</code><code>% SYNOPSIS:</code><code>% FVToolStartUp()</code><code>%</code><code>% PARAMETERS:</code><code>% No perameter</code><code>%</code><code>% RETURNS:</code><code>% None</code><code>%</code><code>% EXAMPLE:</code><code>% n.a.</code><code>%</code><code>% SEE ALSO:</code><code>% PVTinitialize, FVTdemo</code><code>?</code><code>%{<!-- --></code><code>Copyright (c) 2012-2021 Ali Akbar Eftekhari</code><code>All rights reserved. </code><code>?</code><code>Redistribution and use in source and binary forms, with or</code><code>without modification, are permitted provided that the following</code><code>conditions are met:</code><code>?</code><code> * Redistributions of source code must retain the above copyright notice,</code><code> this list of conditions and the following disclaimer.</code> <code> * Redistributions in binary form must reproduce the above</code><code> copyright notice, this list of conditions and the following</code><code> disclaimer in the documentation and/or other materials provided</code> <code> with the distribution.</code><code>?</code><code>THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"</code><code>AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO,</code><code>THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR</code><code>PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR</code><code>CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,</code><code>EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,</code><code>PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA,</code><code>OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF</code><code>LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING</code> <code>NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS</code><code>SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.</code><code>%}</code><code>try</code><code>p = mfilename('fullpath');</code><code>file_name = mfilename;</code><code>current_path = p(1:end-1-length (file_name));</code><code>addpath([current_path '/Boundary']);</code><code>addpath([current_path '/Calculus']);</code><code>addpath([current_path '/Classes']);</code><code>addpath([current_path '/Discretization']);</code><code>addpath([current_path '/MeshGeneration ']);</code><code>addpath([current_path '/Solvers']);</code><code>addpath([current_path '/Utilities']);</code><code>addpath([current_path '/Visualization']);</code><code>addpath([current_path '/Examples']);</code><code>addpath([current_path '/Physics ']);</code><code>addpath([current_path '/Tests']);</code><code>?</code><code>try</code><code> addpath( [current_path '/PhysicalProperties']);</code><code> addpath([current_path '/FieldGeology']);</code><code>catch</code><code> disp([\ 'Some of the physical functions are not available in this copy.' ...</code><code> ' It does not affect the functionality of the FVMtool']);</code><code>end</code><code>?</code><code>% Check for other solvers</code><code> % check for AGMG availability</code><code> cd(current_path);</code><code> cd('Solvers');</code><code> if exist('AGMG_3.2', 'dir') == 7</code><code> addpath([pwd '/ AGMG_3.2']);</code><code> disp('AGMG 3.2 linear solver is available.');</code><code> elseif exist('AGMG_3.0', ' dir') == 7</code><code> addpath([pwd '/AGMG_3.0']);</code><code> disp('AGMG 3.0 linear solver is available.') ;</code><code> else</code><code> disp('AGMG 3.x linear solver is NOT available (Not necessary).');</code><code> end</code> <code> % check for Factorize availability</code><code> if exist('Factorize', 'dir') == 7</code><code> addpath([pwd '/Factorize' ]);</code><code> disp('Factorize is available.');</code><code> end</code><code> cd(current_path);</code><code>% end of check for other solvers</code><code>?</code><code>% Check for the PVT package</code><code>if exist('PVTtoolbox', 'dir') = = 7</code><code> addpath([current_path '/PVTtoolbox']);</code><code> if exist('PVTinitialize', 'file') == 2</code><code> cd('PVTtoolbox');</code><code> PVTinitialize();</code><code> cd(current_path);</code><code> disp('PVTtoolbox has started successfully.');</code><code> else</code><code> disp('PVTtoolbox is found but cannot be initialized.');</code><code> end</code><code>else</code><code> disp('PVTtoolbox is NOT available (Not necessary).');</code><code>end</code><code>disp('FiniteVolumeToolbox has started successfully .');</code><code>catch err</code><code> error('An error occurred while tryng to start the FiniteVolumeToolbox.')</code><code>end</code> <code>?
Operation results
