Comparative Analysis of Natural Convection Flows Simulated by both the Conservation and Incompressible Forms of the Navier-Stokes Equations in a Differentially-Heated Square Cavity

Abstract

This report illustrates a comparative study to analyze the physical differences between numerical simulations obtained with both the conservation and incompressible forms of the Navier-Stokes equations for natural convection flows in simple geometries. The purpose of this study is to quantify how the incompressible flow assumption (which is based upon constant density advection, divergence-free flow, and the Boussinesq gravitational body force approximation) differs from the conservation form (which only assumes that the fluid is a continuum) when solving flows driven by gravity acting upon density variations resulting from local temperature gradients. Driving this study is the common use of the incompressible flow assumption in fluid flow simulations for nuclear power applications in natural convection flows subjected to a high heat flux (large temperature differences). A series of simulations were conducted on two-dimensional, differentially-heated rectangular geometries and modeled with both hydrodynamic formulations. From these simulations, the selected characterization parameters of maximum Nusselt number, average Nusselt number, and normalized pressure reduction were calculated. Comparisons of these parameters were made with available benchmark solutions for air with the ideal gas assumption at both low and high heat fluxes. Additionally, we generated body force, velocity, and divergence of velocity distributions to provide a basis for further analysis. The simulations and analysis were then extended to include helium at the Very High Temperature gas-cooled Reactor (VHTR) normal operating conditions. Our results show that the consequences of incorporating the incompressible flow assumption in high heat flux situations may lead to unrepresentative results. The results question the use of the incompressible flow assumption for simulating fluid flow in an operating nuclear reactor, where large temperature variations are present. The results show that the use of the incompressible flow assumption with the Boussinesq gravitational body force approximation should be restricted to flows where the density change of a fluid particle along a pathline is negligible.4

A Novel Multivariate Spectral Local Quasilinearization Method (MV-SLQLM) for Modelling Flow, Moisture, Heat, and Solute Transport in Soil

“…where u and w are velocity components, and P is the pressure of the fluid; ρ and μ are density and viscosity of the fluid, respectively; and F and k are, respectively, the inertia resistance coefficient and the permeability coefficient. Following the Darcy-Boussinesq approximation assumption [79] on the vertical flow, g is the force due to gravity, and β and β * are, respectively, volumetric coefficients of expansion of heat and contaminants. The terms T 0 and C 0 in equation ( 3) represent medium surface temperature and concentration, respectively.…”