Cavity flow non-slip boundaries Buoyancy

The test case described in here meets the typical reference problem for validating cfd software in a confined domain (Files for this test case can be downloaded from here) when considering buoyancy effects and the Boussinesq approximation is considered. By default, DynamFluid uses the Boussinesq approximation.

Configuration on the model to use for the temperature variation can be done in menu Edit > Properties > Model > State Equation -> Boussinesq Approximation

The domain used in this test case is a square (the edge lengh being 1m) containing an incomressible flow.

The boundary conditions applicable to the domain are:

• Solid wall (Vx=0 m/s, Vy=0 m/s) on all surfaces (walls)

• Upper and lower wall: Adiabatic wall (zero heat flux)

• Left wall: Wall at high temperature

• Right wall: Wall at low temperature

Initial conditions:

• The velocity components at every point of the domain are zero

• The temperature at every point of the domain is equal to the low temperature (except to points located at the left wall, where the temperature is equal to the high temperature)

A picture of the domain used, can be found in the next drawing

The elements used in the mesh used for simulating this problem have been (a) linear TRIA elements and (b) Cuadratic TRIA elements. A picture of the mesh used in linear TRIA elements can be found below.

The project has been simulated for several Rayleigh numbers (10^3, 10^4 and 10^5). The results have been compared with the following references:

• Barakos, G., Mitsoulis, E. (1994): Natural convection ow in a square cavity revisited: Laminar and turbulent models with wall functions, International Journal for Numerical Methods in Fluids, 18, 695-719

• Massarotti, N., Nithiarasu, P. and Zienkiezicz, O.C. (1998): Characteristics-based-split (CBS) algorithm for incompressible flow problems with heat transfer, International Journal of Numerical Methods for Heat and Fluid Flow, 8, 8

• de Vahl Davis, D. (1983): Natural convection of air in a square cavity: a bench mark numerical solution, International Journal of Numerical Methods in Fluids, 3, 249-264

The following table shows the comparison between the results obtained with Tria Linear Elements, Tria Cuad Elements and the references (The Nusselt number provided is the average nusselt along the vertical left wall):

Vx = Horizontal velocity

Vy=Vertical velocity

y/H = Relative vertical position

x/W=Relative horizontal position

The table shows the good agreement got for both Linear and Cuadratic TRIA elements, for a incompressible fluid when buoyancy effects are taken into account in stationary regime.

Tria Linear Elements - Rayleigh 10^3

Tria Cuad Elements - Rayleigh 10^3

Tria Linear Elements - Rayleigh 10^4