Two-dimensional transient laminar natural convection in a square cavity containing a porous medium and inclined at an angle of 30∘ is investigated numerically. The vertical walls are differentially heated, and the horizontal walls are adiabatic. The effect of Rayleigh number on heat transfer and on the road to chaos is analyzed. The natural heat transfer and the Darcy Brinkman equations are solved by using a finite volume method and a Tri Diagonal Matrix Algorithm (TDMA). The results are obtained for a porosity equal to 0.45, a Darcy number and a Prandtl respectively equal to 10^{−3} and 0.71; they are analyzed in terms of streamlines, isotherms, phase portrait, attractors, and spectra amplitude as a function of the Rayleigh number. It is found that, as Rayleigh number increases, the natural convection changes from a steady state to a time-periodic state and finally to a chaotic condition.

This work is part of a recent branch of nonlinear physics devoted to dissipative dynamical systems. The heat transfer in porous medium is found in various applications, cooling of electronics components, oil recovery, applications related to geothermal energy extraction, thermal insulation and safety issues in the cores of nuclear reactors.

Scientists have long sought to geometrically represent order hidden in chaos by a trajectory the trajectory represented by an attractor in a phase space called (phase space) which describes the whole evolution of the system, this trajectory converges towards a particular region of phase space, it is called (attraction pool).

In 1987, Moya et al. [

In recent years, the appearance of more powerful computers has made it possible to simulate numerically the phenomena responsible for the appearance of unsteady natural convection and their interpretation in a concrete way.

Ragui et al. [

Aklouche-Benouaguef et al. [

Oztop et al. [

Subhash et al. [

An experimental study of laminar mixing in several types of two-dimensional cavity flows by means of material line and drop deformation in an experimental system composed of two sets of roller pairs connected by belts is presented by Chien et al. [

A two-dimensional numerical simulation is performed for natural convection in a V-shaped cavity heated from below and cooled from above using a finite volume method [^{3} and 7.6 10^{3} and the Hopf bifurcation occurs between Ra = 1.5 10^{7} and 1.6 10^{7}. Additionally, another periodic state bifurcation to chaotic occurs between Ra = 5 10^{7} and 6 10^{7}. The power spectral density, the phase space trajectory and the largest Lyapunov exponent of the unstable flows during the transition to a chaotic state have been described. In addition, the heat transfer in the cavity is calculated and the corresponding dependence on the Rayleigh number is discussed and quantified.

Finally, Adjal et al. [

The present work is an exploration of the routes towards the chaos of a natural thermal convection in a porous medium.

The phenomena of natural convection are described by the general equations of fluid dynamics complemented by the energy conservation equation deduced from the first principle of thermodynamics.

We consider Newtonian and incompressible fluid and two-dimensional flow. All the physical properties of fluid are constant except the density (approximation of Boussinesq). The medium is considered isotropic, no chemical reaction in the porous medium, thermo-physical properties are considered constant and the local thermal equilibrium is assumed. The Darcy-Brinkman model while neglecting the radiation, the viscous dissipation and the work of the pressure forces in the heat equation governs the motion of the fluid in the porous medium.

The governing equations of natural convection are written in non-dimensional form as follows:

The governing equations are solved by the finite volume technique. The semi-implicit ADI method was applied. The algebraic equations systems are solved, for each time step, by Tri Diagonal Matrix Algorithm.

In order to validate our results a comparison was made with those obtained by LAURIAT and PRASAD. The parameters chosen are ε = 0.4, Ra = 10^{7} and 10^{5}, Da = 10^{−1} and 10^{−6}, Pr = 1 and 10^{−2}. The results are summarized in in

Mesh | 61 | 81 | 101 | Ref |
---|---|---|---|---|

Hot Nusselt | 1.0655706 | 1.0684997 | 1.0704805 | 1.07 |

Deviation% | 0.413 | 0.14 | 0.044 |

Mesh | 61 | 81 | 101 | Ref |
---|---|---|---|---|

Hot Nusselt | 2.2321446 | 2.1800326 | 1.9468758 | 1.94 |

Deviation% | 15.05 | 12.37 | 0.35 |

We carried out several mesh and time step sensitivity tests to obtain an optimal mesh and time step for the calculation of the stationary and periodic solutions.

The calculation of the relative error for different dynamic parameters considering the influence of the mesh and the time step led us to determine the stationary solution (^{6}, a time step dτ = 9 10^{−5} and a mesh of (111 × 111).

The phase portrait is a spiral, which tends towards a fixed limit point. It is an attractor for which all the values of a parameter tend to a constant value when the time tends to infinity. The effects of non-linearity diminishes and the trajectories of the points of the system tend towards a basin of attraction in the form of a spiral, which ends by point. This explains the stationary state of the system.

The evolution as a function of time of a dynamic parameter such as the Nusselt number, the function of the current and the speeds shows that the dynamic system undergoes oscillations having large amplitudes then they attenuate after some dimensionless time until stability is achieved (

The temporal evolution of maximal stream-function begins with a transient regime and then tends towards the steady state. This stationary state is characterized by the damping of the oscillations having zero amplitude.

To confirm the stationarity of the dynamic system, it is necessary to have recourse to the spectral analysis (

We increase the Rayleigh number while keeping the mesh and the time step that we have previously designated and we plot the phase portraits (

We keep increasing the Rayleigh number while keeping the mesh and the time step and we plot the phase portraits. We notice the appearance of strange geometry of attractor, sign of the progressive change in the state of our system as a function of the Rayleigh number, up to get a persistent chaos for a Rayleigh number equal to 10^{11} (

The appearance of strange shape of attractors in the phase space (

In

The cell who occupies the diagonal is very intense which means that the transfer is high in this part of the cavity.

The cellular, which occupy the right side of the upper part of the cavity as well as the left side of the lower part of the cavity, rotate in the same direction. This phenomenon is manifested in the trace of the

The boundary layer formed on the hot and cold sides of the cavity is very important this shows that the heat transfer is ensured by convection.

For the periodic solution (

We presented in this work a numerical study on the unstationary natural convection in an inclined square, closed and totally porous enclosure. We used the Darcy-Brinkman model and the Boussinesq approximation to establish our equations. The SIMPLE algorithm solves the governing equations governing the phenomenon.

To determine the states of the system, we have used the representation of the attractors in a phase space and the representation of the temporal evolution of dynamics parameters.

The study was carried out for a porous medium having porosity equal to 0.45 and Darcy number equal to 10^{−3}.

The numerical study allowed us to obtain the following results:

− The fixed-point attractor corresponding to the stationary solution is obtained for a Rayleigh number equal to 8.106, a time step equal to 9 × 10^{−6}, and a mesh equal to 111 × 111. The tilt promotes thermal transfer into the cavity due to the buoyant forces.

− The chaos occurs when the number of Rayleigh reaches the value of 10^{11}. The structure of current lines is a disorder causing disorder in the cavity characterized by an excess of energy which is localized throughout the cavity.

This work was done at the LTPMP Laboratory, USTHB, Algeria.