## Numerical Simulation of Natural Turbulent Convection with Vorticity Vector Formulation

##### Résumé

Turbulent natural convection in an enclosure plays a big part in heat transmission and the building
environment. Sophisticated buildings around the world are outfitted with costly heaters and coolers
to maintain comfortable temperatures for human existence, manufacturing, and sophisticated farming
methods, a scenario that many people cannot afford. Over a time researchers have consistently developed
a number of numerical study models to simulate the natural turbulent flow in these rectangular
enclosures to solve complex problems associated with turbulent flows. In spite of several experimental
studies and model simulations on the structure of natural turbulence convection, the fundamental
mechanism in turbulent phenomena is still incomplete. Significant variations in experimental data and
model simulation data in previous studies have been noted. This is because the unknown turbulent
correlation coefficients resulting from the nonlinear terms of the turbulent flow control equations make
it difficult to accurately determine fluid flow variables such as mean velocity distribution, temperature
distribution and kinetic energy in a model simulation. Thus an accurate numerical prediction of natural
turbulence convection is crucial to solving the nonlinear equations for subsequent practical applications.
The performance of a numerical turbulence model k-ε in estimating the amount of heat transfer
that occurs as a result of the naturally occurring turbulent convection that takes place within an air-filled
rectangular enclosure is investigated in this work using vorticity vector formulation. The workflow of
simulating the heat transfer which results from the action of natural convection within an enclosed rectangular
cavity takes into account the effect of turbulence for the Rayleigh numbers Ra = 1.552×1010,
Ra = 9.934×1011, Ra = 1.552×1013 and Ra = 2.425×1014. The Low-Reynolds-number turbulence
k-ε model was employed in this numerical study to model the non linear relations ∇· ρu′
iu′
j and ∂CpT′u′
i
∂ xi
in the averaged Navier Stokes equation and energy equation respectively to complete the governing
equations. Apart from the hot and cold walls, which are maintained at 308K and 288K, respectively,
all of the walls of the enclosure are adiabatic. The vorticity vector formulation allowed the pressure
term to be removed from the momentum equation. Finite difference approximations were used in the
FLUENT program to solve the vorticity, energy, vector potential, and two resultant equations for each
model together with their boundary conditions. The outcomes of the study for the distribution of the
velocity and temperature components are presented, demonstrating that the number of contours and
vortices increases proportionally with the Rayleigh Number. In addition, a higher Rayleigh number indicates
more turbulence, which in turn implies a higher absolute value of the velocity hence increased
Kinetic energy.