Stream Function/Vorticity Formulation to Study the Cavity Flows by Unstructured Grid Method

A derived variables formulation, called the stream-vorticity formulation, of the two-dimensional incompressible flow is presented as an alternative to the velocity-pressure approach where the pressure needs to be calculated explicitly. The finite volume method for unstructured meshes, composed of quadrilaterals, is used to compute steady, 2D incompressible flows in square cavity. However, the proposed formulation is not restricted to the simple topology. The efficiency of the method is verified by comparing the present predictions with two steady-state benchmark solutions: lid-driven cavity flow, and natural convection inside a cavity with two isothermal walls.

To valid the method a square enclosure with sinusoidal protuberance is used as the model problem. Numerical solutions are obtained for a Prandtl number 0.71 and for Rayleigh number up to 10⁶. The effect of the sinusoidal protuberance located at the lower surface of the cavity on heat transfer and free convective motion is also analyzed.