MATLAB: Stationary Convection-Diffusion Equation 2-D

convectiondiffusionfemgalerkinpetrov

I need to solve Convection-Diffusion Equation for a square domain using triangular mesh. The equation is:

Where V is the velocity vector, T is temperature, K is the diffusion coefficient and Q is power.

The study is about finding the air temperature in a closed space with forced convection and a constant power; the velocity goes in just one direction and ist constant too. I need to create a square of 11×11 centimeters because that is the size of the machine I'm working with, and I want to create a triangular mesh but I don't know how to.

Is it possible to solve the equation using the Petrov-Galerkin method? If the answer is yes, how can I do it?

If the answer is no, what other method can I use?

The idea is to create a program that allows you to assign a constant value to the power and speed and gives back as a result the temperature values, but my concern is that in this Finite Element Method you can not use the regular Weight Functions.

Please excuse my english.

Best Answer

Hi Andrea,

I suggest using programmatic workflow.

Re: "In the case of the temperature study," : Dirichlet BC will specify the value of temperature on the boundary. Neumann BC would specify the heat flux.

For your original advection problem. You need to define the Q as f-coefficient, a V*gradT as a-coefficient. Both of these could be spatially varying, you can user functional form of coefficients to do so. Having said that there is no stabilization mechanism in PDE Toolbox, so you might encounter numerical instabilities depending of your problem is advection dominated or diffusion dominated.

Regards,

Ravi

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Radial heat equation with lateral heat diffusion would be a BVP(Boundary Value Problem) in multiple dimensions, but PDEPE function solves boundary value problems in one dimension.

PDEPE is therefore not suited to this type of problem, as there are too many edge and boundary conditions to convert the PDE to one-dimensional. Approximating this PDE to be one-dimensional would force discontinuities at the boundary edges.

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I think pdepe function will work for your problem. Please see the first example of the documentation page. https://jp.mathworks.com/help/matlab/ref/pdepe.html

It uses finite difference for spacial derivative and ode15s for time integration.

If you want to explicitly code the finite difference part by yourself, the following File Exchange entry might be of use.

https://jp.mathworks.com/matlabcentral/fileexchange/59916-simple-heat-equation-solver

The above code solves 2D case with the Neumann boundary conditions. You need to drop one dimension and modify the boundary condition of one end where you need Dirichlet boundary condition. And of course, please modify the initial condition too.

getRHS.m has the following line of code that implements Neumann boundary condition.

ubig(end,:) = ubig(end-1,:);

The idea is that the finite difference derivative at the boundary should be zero. In order to use the Dirichlet boundary condition, you can change the above to

ubig(end,:) = 2*BC - ubig(end-1,:);

where BC is the boundary value that you want to specify. The idea here is to fix ubig(end,:) value so that the interpolated value at the boundary to be BC.

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