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In this present work, we study heat transfer in a confined environment. We have to determine the thermal and dynamics fields of the cavity while observing the effect of the Rayleigh number which depends on the characteristics of the fluid and the temperatures imposed. The behavior of boundary layers in natural convection is analyzed along this square cavity. The central halves of its vertical walls are heated at different temperatures. The left active part is at a higher temperature than the one on the right wall. The remaining inactive parts and the horizontal walls (upper and lower) are adiabatic. The thermal and dynamic modeling of two-dimensional problem was done using a calculation code Fortran 90 and a visualization software ParaView based on the finite volume method. The equations governing this phenomenon of unsteady flow have thus been solved. This allows the modeling of both air flow and heat transfer with a numerical stabilization of the solution. So, we have presented our results of numerical simulations using a visualization tool. The results show the different velocity and temperature curves, velocity vectors and isotherms in laminar flow regime.

Convection is the transfer of thermal energy due to the motion of fluids. Natural convection is defined as the fluid flow and heat transport arising due to density and temperature differences in the fluid. This is different from forced convection where the transport is because of an external energy source like a fan or a pump. Natural convection is an important topic of scientific research with applications in nature and engineering. For example in nature, a better understanding of natural convection will help us to predict the flow of magma in the earth’s core and mantle [

There are mainly two configurations, the first being that of an enclosure containing a fluid and subjected to a vertical temperature gradient (for example Rayleigh-Benard convection), the second being that of a cavity with a horizontal temperature gradient. In this study, we focus on the case of a cavity with a horizontal temperature gradient. The case of vertical walls remains highly coveted in the industrial field because of its application to double skin facades, maintenance of equipment, electronic compounds, etc. Various geometric configurations have been studied theoretically, numerically or in the experimental design. The cavity studied has its thermally active vertical walls [

The chosen geometry is a square cavity differentially heated with a side of 0.05 m. It is formed by two vertical walls, each of which is heated at its center. The left active part has a higher temperature ( T r + Δ T / 2 ) than that of the right active part ( T r − Δ T / 2 ) as shown in

In

The dynamic boundary conditions are:

- to the walls, u = 0 m/s and v = 0 m/s.

The thermal boundary conditions give:

- for heated areas, we have:

for Ra = 1000 (Tg = 295.027 K and Td = 294.973 K),

for Ra = 10,000 (Tg = 295.27 K and Td = 294.73 K),

for Ra = 100,000 (Tg = 297.7 K and Td = 292.3 K),

for Ra = 1,000,000 (Tg = 322 K and Td = 268 K).

- for the adiabatic zone, φ = 0 W/m^{2}.

The flow is governed by the continuity equation, the equations of motion and the energy equation under the Boussinesq hypothesis, with: U r = α / A for velocity, A for length, A / U r = A 2 / α for time and θ = ( T − T r ) / Δ T corresponds to the dimensionless temperature.

∂ U ∂ X + ∂ V ∂ Y = 0 (1)

∂ U ∂ τ + U ∂ U ∂ X + V ∂ U ∂ Y = − ∂ P ∂ X + P r ( ∂ 2 U ∂ X 2 + ∂ 2 U ∂ Y 2 ) (2)

∂ V ∂ τ + U ∂ V ∂ X + V ∂ V ∂ Y = − ∂ P ∂ Y + P r ( ∂ 2 V ∂ X 2 + ∂ 2 V ∂ Y 2 ) + R a ⋅ P r ⋅ θ (3)

∂ θ ∂ τ + U ∂ θ ∂ X + V ∂ θ ∂ Y = ∂ 2 θ ∂ X 2 + ∂ 2 θ ∂ Y 2 (4)

We have the following boundary conditions (hydrodynamic and thermal) at the walls:

X = 0 , (Heated part of the left wall), U = V = 0 , θ = 1. (5)

X = 1 , (Heated part of the right wall), U = V = 0 , θ = 0. (6)

X = 0 and X = 1 , (Adiabatic parts of left and right walls), U = V = 0. (7)

∂ θ ∂ X = 0. (8)

Y = 0 and Y = 1 , (Lower and upper walls), U = V = 0. (9)

∂ θ ∂ Y = 0. (10)

The dimensionless numbers used are:

The Prandtl number is physically defined as ratio of momentum diffusivity to thermal diffusivity. It provides a measure of the efficiency of diffusion transport through the velocity boundary layer and the thermal boundary layer.

P r = ν α (11)

The Rayleigh number is physically the ratio of buoyancy and viscosity forces multiplied by the ratio of momentum and thermal diffusivities. Below the critical value of the Rayleigh number, heat transfer is primarily due to conduction. Above this critical value, heat transfer is due to convection.

R a = g β Δ T A 3 ν α (12)

The results produced by our calculation code were compared with some results available in the literature. Let’s consider the same model as the benchmark [

For the thermal and dynamic modeling of our problem, we first discretized the domain of study by a staggered cartesian mesh; the control volume of the component u is staggered in the direction x relative to the main control volume, that of the component v is staggered in the direction of y. Moreover, the staggered mesh technique consists in taking the vector and scalar variables in different nodes. This mesh is adapted to our choice of the distribution of the calculation points. Those of the temperature and the pressure are at the center of the meshes while those of the components u and v of the velocity are located on the faces of the control volume. We use the finite volume method which allows to

Authors | Dimensionless Values | |||
---|---|---|---|---|

U max ( 1 / 2 ; Y ) Y | V max ( X ; 1 / 2 ) X | N u max Y | N u min Y | |

Our Results | 35.519 0.840 | 68.306 0.060 | 7.804 0.085 | 0.809 0.984 |

G. De Vahl Davis [ | 34.73 0.855 | 68.59 0.066 | 7.717 0.081 | 0.729 1 |

P. M. Gresho [ | 34.620 0.856 | 68.896 0.0663 | 7.731 0.0746 | 0.7277 1.0 |

U max : the maximum horizontal velocity, V max : the maximum vertical velocity, N u max : the maximum value of the local Nusselt number and N u min : the minimum value of the local Nusselt number.

model both the flow of a fluid and the transfer of heat. This method relies on a discretization technique, which converts the differential equations to the partial derivatives into nonlinear algebraic equations, which can then be solved numerically. It was described in 1971 by Patankar and Spalding and published in 1980 by Patankar [

for Ra = 100,000, 1,000,000).

There is a change in the horizontal velocity curves from 41 × 41, 61 × 61 to 81 × 81. Hence considering the accuracy of the results required and computational time involved, a 41 × 41 grid size is chosen for computations with the two lower Rayleigh numbers. An 81 × 81 grid size is chosen also for the two upper Rayleigh numbers. The theoretical analysis of Rayleigh [

In the literature, some papers describe natural convection in rectangular enclosures [

The differential Equations (1)-(4) governing the physical situation are translated into algebraic equations using the finite volume scheme. The system of dimensionless algebraic equations with boundary conditions associated (5)-(10) is iteratively solved. For accurate numerical simulation, a mesh size of 41 × 41 was selected for computations with Rayleigh numbers Ra = (1000, 10,000) and a mesh size of 81 × 81 for Ra = (100,000, 1,000,000). We have developed a numerical code with FORTRAN 90. The governing equations are discretized with the Central Difference Scheme (CDS). This scheme (order 2) generates a linear interpolation to the problem boundary. An iterative process is employed to find the velocity and temperature fields. The process is repeated until the following convergence criterion for velocity and temperature is met. The calculation stopped when the follows inequalities were satisfied:

Max [ ∑ i , j | ϕ i , j n + 1 − ϕ i , j n | ∑ i , j | ϕ i , j n + 1 | ] ≤ 10 − 5 . (13)

The same convergence criterion is imposed in terms of relative error for velocity and temperature. In the above expression n is any time level and ϕ = U, V, θ. In this study, we used the finite volume method with quadrilateral control volumes and a staggered mesh. The latter is the subdivision of the field of study into longitudinal and transverse grids whose intersection represents a node, where the variables P and θ are located while the components U and V of the velocity vector are in the middle of the segments connecting two nodes adjacent. After discretization of the differential transport equations we obtain a system of nonlinear algebraic equations, these equations describe the discrete properties of the fluid at the nodes in the solution domain. For temporal discretization, we have used a numerical method of solving numerical differential equations, based on a multi-step method. The temporal integration was carried out on a mesh staggered according to an Adams-Bashforth scheme (order 2).

d Y d τ = f ( τ , Y ) , with τ = τ 0 , Y ( τ 0 ) = Y 0 . (14)

{ Y 0 ( given ) , Y 1 approached ( o n e - s t e p m e t h o d ) , Y n + 1 = Y n + Δ τ 2 ( 3 f ( τ n , Y n ) − f ( τ n − 1 , Y n − 1 ) ) . (15)

We performed our simulations by varying the Rayleigh number. We have analyzed below the velocity and temperature curves in the cavity for four different cases. Natural convection heat transfer is studied in a square enclosure for different thermally active locations with a Prandtl number Pr = 0.72 (air).

In the first case, we have the following solutions for different thermally active locations.

parts than on those which are inactive (adiabatic). In

In a second case, we have the following results for different thermally active locations.

active walls. It is observed that the fluid particle moves with greater velocity for the middle active locations and the velocity is less for the top/bottom active locations. With a larger temperature variation and therefore a higher Rayleigh number, we observe in

In the third case, we have the following findings for different thermally active locations.

In the fourth case, we have the following results for different thermally active locations.

active walls. It is observed that the fluid particle moves with greater velocity for the middle active locations and the velocity is less for the top/bottom active locations. The flow regime always remains laminar with higher velocity and temperature variations (

Figures 11-14 show the impact of variation in the Rayleigh number. With an increase in the Rayleigh number from 1000 to 1,000,000, we have some noticeable changes. There are fluctuations in the velocity profiles and in particular for

the maximum values u_{max} and v_{max}. The velocity components (_{max} moves to the vertical walls with the increase of the Rayleigh number. The centro-symmetry’s property is retained throughout the range of the Rayleigh number studied. We observe a symmetry of the curves of velocity with respect to the

center of the cavity. For our model, the variation of the Rayleigh number represents a variation proportional to the temperature difference between the right and left active parts. In the zone 0.0125 m ≤ y ≤ 0.0375 m, the temperature increases progressively (

The results are displayed graphically in terms of velocity vectors and isotherms. In

in an intense way at the corners of the cavity. In

In

As it approaches the hot wall, the particle gains heat (its kinetic energy), heats up (becomes lighter) and rises along the hot wall with a trajectory that will be further modified by the presence of the upper horizontal wall. Along the latter, the fluid particle cools as it approaches the cold wall: we have the formation of a circulation cell. The warm air rises and the cold air descends. In

We have studied the behavior of boundary layers in natural convection along a cavity. The central parts of the vertical walls are at imposed temperatures (horizontal gradient). We have seen that from Ra = 1000 to Ra = 1,000,000, the flow regime is laminar and evolves remarkably. Analysis of the results obtained with different ordinates showed that for different heights the values of velocity and temperature fluctuated due to the convection which creates upward and downward movements of the air. Moreover, we have shown that there are recirculations and a property of centro-symmetry in the flow. The vertical velocity profiles show a correspondence with the results obtained in the form of velocity vectors.

We know that in physics, particularly in solar energy, for collectors due to shading it is only the unshaded part of the wall that is thermally active. Buoyancy-driven flow in a square cavity with vertical sides which are differentially and partially heated is a very important way to understand the thermal and dynamic effects in many practical problems.

We thank all the contributing members of SOLMATS group for their support, their assistance and their encouragement.

We author(s) declare that we have no competing interests.

Kane, M.K., Mbow, C., Sow, M.L. and Sarr, J. (2017) A Study on Natural Convection of Air in a Square Cavity with Partially Thermally Active Side Walls. Open Journal of Fluid Dynamics, 7, 623-641. https://doi.org/10.4236/ojfd.2017.74041

A: Side of the cavity m

g: Gravity intensity m/s^{2}

l: Length of adiabatic parts m

P: Dimensionless pressure

Pr: Prandtl Number

Ra: Rayleigh Number

s: Length of active parts m

T: Dimensional temperature K

Td: Dimensional temperature of right active part K

Tg: Dimensional temperature of left active part K

Tr: Reference temperature K

u: Dimensional horizontal velocity m/s

U: Dimensionless horizontal velocity

U_{r}: Reference velocity m/s

v: Dimensional vertical velocity m/s

V: Dimensionless vertical velocity

x: Dimensional abscissa m

X: Dimensionless abscissa

y: Dimensional ordinate m

Y: Dimensionless ordinate

α : Thermal diffusivity m^{2}/s

β : Coefficient of thermal expansion 1/K

Δ : Variation

θ : Dimensionless temperature

ν : Kinematic viscosity m^{2}/s

τ : Dimensionless time

ϕ : Dimensionless variable

φ : Heat flux density W/m^{2}

Max or max Maximum value

min Minimum value