Mixed convection in a partially and differentially heated cavity − a finite volume complete flux analysis
Authors: Rathish Kumar B. V. , Pandey Chitranjan 
Published: January 2025

Purpose
The purpose of this study is to derive a physics based complete-flux approximation scheme by solving suitable nonlinear boundary value problems (BVP) for finite volume method for mixed convection problems, to study the mixed convection phenomenon inside partially and differentially heated cavity for various sets of flow parameters. And, to study the impact of source terms on the cell-face fluxes for various sets of flow parameters for mixed convection problems.

Design/methodology/approach
The governing equations have been discretized by finite volume method on a staggered grid, and the cell-face fluxes have been approximated by local nonlinear BVP. The cell-face flux is represented as a sum of homogeneous and an inhomogeneous flux term. The proposed flux approximation is fully physics based as it considers the pressure gradient term, thermal buoyancy term and the other source terms in the cell-face flux calculation. The scheme comes out to be second order accurate in space tested with known solution. Also, the scheme has been implemented to study the mixed convection problems in a partially and differentially heated cavity.

Findings
The numerical order of convergence study shows that the proposed scheme is of second order in space. The scheme is first validated with existing benchmark literature for the mixed convection problem. As the proposed cell-face flux approximation scheme is a homogeneous part and an inhomogeneous part, this study quantifies the influence of the several source terms on the cell-face flux with the help of the inhomogeneous flux term. Then, the mixed convection problems in a partially and differentially heated cavity has been studied. Also, the effect of heat transfer rate at the hot wall is studied for different height of the heat source with different directions of wall movement. The numerical findings show that the local Nusselt number at the left wall is higher when the top and bottom walls move in opposite directions compared to when they move in the same direction, regardless of the Richardson number. In addition, the heat transfer rate at the hot portion of the left wall increases uniformly as the Richardson number decreases when the walls move in opposite directions. However, when the top and bottom walls move in the same direction, the increase in heat transfer rate is not uniform due to the formation of secondary re-circulation of the fluid near the bottom wall.

Originality/value
In this work, the flux approximation is conducted through local nonlinear BVPs, an approach that, to the authors’ knowledge, has not been previously applied to mixed convection problems. One of the strong advantages of the proposed scheme is that it can quantify the influence of source terms, namely, pressure gradient, cross-flux and the thermal buoyancy force, on the cell face fluxes required in the finite volume methods. Furthermore, the study explores mixed convection in a partially and differentially heated cavity, which is also novel within the current literature. These factors contribute to the originality and scientific value of the research.