Mixed convection in a vertical channel

Project to use in DynamFluid to perform the simulation: project.

This example shows an scenario where both forced and natural convection happen, a situation that occurs in low Reynolds numbers and when the buoyancy contribution is significant. This example shows the flow pattern established in a vertical channel where a fluid travels upwards, the gravitational force acts downwards and both walls are at different temperature (The right wall is a higher temperature than the left wall). The fluid is entering at the same temperature as the cold (left) wall. Non-dimensional variables are used. The non-dimensional boundary condition values for the velocity, pressure and temperature are the following:

• Left vertical (cold) wall: Vx=Vy=T=0

• Right vertical (hot) wall: Vx=Vy=0, T=1

• Bottom horizontal wall: Vx=T=0, Vy=1 (Uniform non-dimensional velocity equal to 1)

• Top horizontal wall: P=0

The gravitational force acts downwards.

In this project, the Reynolds number is 100 and the Grashoff number is 25000.

The width of the channel has been set to non-dimensional value of 1m. The length of the channel has been a non-dimensional value of 20m, long enough to allow the flow developing alongside the channel.

In this type of simulation (Open flow at the exit = Exit boundary condition) is important to set the pressure equal to zero at the exit, as it has been done in the project for this simulation. If not set, the simulation won't converge.

The analytical solution for a fully developed mixed convection profile is given by Aung and Worku G (1986), Developing flow and flow reversal in a vertical channel with asymemtric wall temperatures, ASME Journal of Heat Transfer, 108, 299-304:

Vy= Gr/Re * (1- Rt)*(-x1^3/6+x1^2/4-x1/12)-6x1^2+6x1

Being,

Gr = Grashof number

Re = Reynolds number

Tc = Temperature of the cold wall

Th = Temperature of the hot wall

Tf = Temperature of the fluid entering the domain (upwards).

Rt = (Tc-Tf)/(Th-Tf)

At very high Richardson numbers, the flow may reverse (move downwards) due to the buoyancy effect created by the density difference caused by the temperature difference between the hot and the cold walls.

The following drawing shows the comparison between the predicted analytical solution (Aung and Worku) and the profile obtained by the DynamFluid software for the fully developed flow, at Rt = 0. As it can be seen, the obtained profile matches very well the analytical solution.

The following drawing shows the comparison between the predicted analytical solution (Aung and Worku) and the profile obtained by the DynamFluid software for the fully developed flow, at Rt = 0.5. As it can be seen, the obtained profile matches very well the analytical solution.