Adding buoyancy to 3D integration of diffusion equation

[dangerousdave]

I have managed to create a simple CFD model to integrate in 3D the heat/diffusion equation, namely dT/dt = k d2T/dx2.

This works fine, but I would now like to add the effect of 'less dense rising' assuming the gas is operating in air -- I think I'm right in saying that's called buoyancy (I'm not a physicist).

SO my question is, can someone post a (hopefully) simple extension to this nice little equation that includes the buoyancy effect. I would assume it involves divT dot g, (where g is a vector of gravity), but that's as far as I can fathom.

Any help much appreciated,
Dave.

 

[lippyka]

The equation for the buoyancy effect is: dT/dt = k d2T/dx2 + div(T*g) where T is the temperature and g is the gravity vector. This equation takes into account the fact that warm air rises due to its lower density.

 

[quicknote]

I would recommend approaching this problem by first understanding the underlying principles of buoyancy and its effects on fluid flow. Buoyancy is a force that arises due to the difference in density between a fluid and its surrounding medium, and it causes objects to float or sink in the fluid. In the case of your CFD model, adding buoyancy would involve considering the density variations in the fluid due to temperature changes and how it affects the flow.

One possible approach to incorporating buoyancy in your model could be by adding a term that accounts for the density variations in the diffusion equation. This could be achieved by using the Boussinesq approximation, which assumes that the density variations in the fluid are small and can be neglected except for the buoyancy force. This would result in an additional term in the diffusion equation that includes the density of the fluid and the gravitational acceleration.

Another approach could be to use the Navier-Stokes equations, which are more comprehensive equations that take into account the effects of both buoyancy and viscosity on fluid flow. These equations would involve additional terms that account for the buoyancy force and its interactions with the velocity and pressure fields.

Ultimately, the specific approach you choose would depend on the complexity and accuracy desired for your model. I would recommend consulting with a fluid dynamics expert or referring to relevant literature for guidance on incorporating buoyancy into your 3D integration of the diffusion equation.