The aim of this paper is to analyse the natural convective heat transfer generated by a source with a height of

The natural convection in a square cavity plays a very interesting role in a lot of engineering applications, such as the solar energy system, the cooling of the electronic circuits, the conditioning of the air, and many others; therefore, it is very important for the applied research. The technical literature presents a lot of studies on natural convection in a square cavity, and many of them analyse the convective phenomenon thanks to numerical simulation.

The work by Ramos and Milanez [

Cesini et al. [

In this paper, the cavity testing is a square enclosure, and we analysed the influence of the position of a heated strip on the natural convection heat transfer; the heated strip was placed on the bottom of the cavity.

Two different positions of the hot strip are analysed:

The symmetrical configuration was studied both
through a holographic interferometer to evaluate its thermal behaviour and
through a 2D-PIV system to determine dynamic structures such as the stream functions,
the velocity vectors, and the velocity maps. These experimental results were
used to validate a numerical model made by a finite volume code (Fluent
6.3.26). The achievement of a good agreement between the experimental data and
the numerical ones allows us studying the asymmetrical position through the
numerical code. In particular, the finite volume software was used to obtain
the local and average Nusselt numbers on the hot surfaces. While the dynamical
behaviour for

The experimental research analysed different
Rayleigh numbers in particular from

The main components of the experimental
apparatus, represented in Figure

Holographic interferometry system.

The cavity structure.

It has a square transversal section of 0.05 m
side, and it is heated from below. The heat hot source is maintained at a
temperature

The dimension of the test cell along the longitudinal direction is 0.42 m, so it is possible to neglect the end effects and to consider the problem as a two-dimensional one.

The temperature on the heat source and on the vertical walls is measured through copper-constantan thermocouples that are located 1 mm under the surface of the test volume. Three of them are put on the hot source, and the other ten thermocouples are set on the vertical walls. The last one is positioned in the middle of the air region, and it gives us a reference temperature to analyse the interferograms. The difference between the three values of the temperature on the hot plate is about 0.2 K, consequently it is possible to consider an isotherm situation; on the lateral wall the maximum difference between the five thermocouples is 0.4 K.

The light source is an argon-ion laser with a nominal power rating of 4 W, with etalon for the 514.5 nm wavelength. Both the object and the reference beams have a maximum diameter of 0.15 m.

The optical setup allows us the use of the double-exposure or of the real-time holographic interferometry techniques, for the steady-state and the temporal evolution measurements of heat transfer processes. The real-time technique has also been used to check the presence of plume oscillations.

The test cell, the acquisition system, and the thermostatic elements are the same used for the holographic setup.

The particle image velocimetry (PIV) is
finalized to make instantaneous velocity measurements in the cross-section of a
flow. The PIV system provides the velocity fields with the distance made by the
particles-seeded in the flow. The seeding particles nature must be chosen in
order to provide a real description of the airflow. For a more detailed explanation of the PIV technique, it is possible to
see Raffel et al. [

In these experiments, oil particles were used and they were nebulised by
an air compressor system; the diameter of the particle was about 1

The data analysis was performed using the software package “Dynamic
Studio,” provided by Dantec Dynamics, Bristol, UK (2006) [

The images of the area of the flow field, lighted by a light sheet, are
captured by a CCD camera (Hamamatsu
camera C8484-05C with

The light sheet gets in through the upper part of the enclosure made up of Plexiglas, and it lights up a cross-section of the cavity in the middle of the cell. The camera, that is located perpendicularly to the light sheet, captures the images, which are sent to a computer to be processed.

A small pane of glass to permit an optical access to the camera covers
the front sidewall of the cavity; instead the rear side is rigged with a 0.05 m
square piece of wood which has an opening in the middle. Through it, we
sprinkle the oil particles for 20 seconds before the beginning of each test.
The top and the bottom surfaces of the enclosure are made up of Plexiglas with
a thickness of 0.03 m to neglect the leaks of heat through these surfaces. The
PIV equipment is shown in Figure

2D-PIV system.

The processing technique used in this study is Dantec adaptive-cross correlation,
described by Dantec Dynamics (2006) [

The overlap of the interrogation area in the horizontal and vertical
directions is 50%. Furthermore, a moving average validation is applied; this
method validated vectors based on comparison between neighbouring vectors: an
averaging area of

This method substituted outliers due to false correlations resulting from reflections at the cell walls and from the other invalid vectors.

During our tests, we evaluated also the repeatability of the experiment. We repeated the same test with the same initial boundary condition at different Rayleigh numbers for each geometrical configuration. Then after 45 minutes from the beginning, we stored the modulus of the velocity vector at different heights: in particular at 16,1 mm, 26,1 mm, and 36,1 mm from the floor of the enclosure. During these checks, the same velocity vector values were obtained. There is only a small difference between the three tests near the left side of the cavity. This fact is due to the reflections of the laser light near the contours of the enclosure that create some noise in the PIV image, and so there is a lack of precision in the velocity vectors.

Similar results are obtained in each test that is made to determinate the repeatability of our experiments.

The Fluent solution methods are known in the scientific background. Particularly, the numerical simulation is developed through the finite volume code Fluent 6.3.26 using the Boussinesq approximation for the air and a two-dimensional model.

The numerical results are carried out with segregated solvers [

As for the spatial discretization, in our tests a second-order up-wind implicit scheme is employed for the conservation equations; the pressure interpolation is provided with the body-force-weighted scheme, and the pressure-velocity coupling is achieved using the simple algorithm.

The diffusion terms are central differenced with a second-order accuracy.

In the performed analysis, the cavity was reproduced with the real dimensions, and the temperature of the lateral walls and of the heated strip is assigned. In the Fluent model, we introduced also the Plexiglas elements used to make the upper and the lower surfaces of the experimental enclosure. It was necessary to simulate the conductive heat transfer between these elements and the cold wall.

A mesh structure with quadrilateral cell elements is performed. A study of the mesh was carried out preliminarily in order to obtain the lowest number of cells necessary to perform an analysis with results that are independent from the choice of the cell number.

At this step, we noticed that a good mesh was necessary to solve
important dynamic structures. Particularly, thanks to the PIV system; we saw
very small vortexes on the upper surface of the hot strip with

During the analysis, we used the steady procedure and the admitted value
for all residuals to obtain a convergence that was

The local and average Nusselt numbers on the hot surfaces, the velocity fields, the stream functions, and the velocity vectors distribution inside the cavity obtained numerically were compared to the experimental results.

The real-time interferograms are obtained with a finite-fringe field, while an infinite-fringe field is used in the double exposure interferograms; so doing the fringe pattern shows directly the distribution of the isothermal lines.

The evaluation of the interferograms is achieved thanks to the help of a
travelling microscope to obtain the intensity distribution. Density and
temperature distribution are obtained by the usual methods of inversion [

The study deals with the effects of the position of the heat source; its
length is 1/5 of the side of the enclosure, and the height is

The symmetrical position was studied thanks to the PIV system and the
holographic interferometry while the asymmetrical one was analysed thanks to
the 2D PIV and through the numerical code to determinate the heat transfer on
the hot surfaces. The numerical code was validated with the comparison between
the numerical and experimental Nusselt numbers for

The dimensionless parameters used are

In both two cases, the hot strip is located on the bottom of the
enclosure with a distance from the sidewalls of

The local Nusselt number, Nu(

The average Nusselt number,

While for the lateral surfaces, we used the following expression:

Analysing the interferograms obtained through the double exposure
technique, the local and the average Nusselt numbers on the hot surface for

In the second column of Figure

Comparison between numerical and experimental isothermal lines at

Average Nusselt numbers as function of Rayleigh numbers on the hot surfaces
for

By studying in depth the image of Figure

Moreover, at this step of the study, the
agreement between the numerical Nusselt numbers and the experimental ones
obtained for the symmetrical position (

Comparison between average experimental Nusselt numbers and numerical ones
on the lateral sides for

Ra | Nu rg exp | Nu rg num | Δ% | Nu lf exp | Nu rg num | Δ% |
---|---|---|---|---|---|---|

1.24 | 9.04 | 8.98 | 0.61 | 9.07 | 8.98 | 1.00 |

1.46 | 9.38 | 9.40 | 9.32 | 9.39 | ||

1.76 | 9.98 | 9.89 | 0.90 | 9.9 | 9.89 | 0.14 |

2.05 | 10.4 | 10.31 | 0.87 | 10.41 | 10.31 | 0.87 |

2.25 | 10.86 | 10.57 | 2.68 | 10.8 | 10.57 | 2.16 |

Comparison between average experimental Nusselt numbers and numerical ones
on the superior side for

Ra | Nu up exp | Nu up num | Δ% |
---|---|---|---|

1.24 | 3.36 | 3.67 | |

1.46 | 3.42 | 3.74 | |

1.76 | 3.525 | 3.82 | |

2.05 | 3.72 | 3.89 | |

2.25 | 3.93 | 3.93 |

After this important validation, the numerical
code was used to simulate the natural convective heat transfer for the
asymmetrical position (

Average numerical Nusselt numbers for

Ra | Nu rg | Nu lf | Nu up |
---|---|---|---|

1.24 | 8.98 | 9.02 | 4.05 |

1.46 | 9.41 | 9.45 | 4.10 |

1.76 | 9.91 | 9.97 | 4.16 |

2.05 | 10.34 | 10.42 | 4.20 |

2.25 | 10.61 | 10.69 | 4.23 |

Numerical isothermal lines for

Moreover, it is possible to find another
difference between the two positions on the upper surface in the development of
the natural convective heat transfer. For

Finally, the numerical and experimental
relationships between the Rayleigh numbers and the average Nusselt numbers were
evaluated. The correlation is

Coefficients for the correlationship number 5.

Nu up exp ( | 0.1714 | 0.2523 | 0.91 | |

Nu up num ( | 0.9496 | 0.1153 | 0.99 | |

Nu up num ( | 1.7525 | 0.0715 | 3.60% | 0.97 |

Nu rg exp ( | 0.2488 | 0.3057 | 0.89% | 0.99 |

Nu rg num ( | 0.3624 | 0.2737 | 0.45% | 0.99 |

Nu rg num ( | 0.3338 | 0.2807 | 1.29% | 0.99 |

Nu lf exp ( | 0.3668 | 0.2727 | 2.17% | 0.93 |

Nu lf num ( | 0.3592 | 0.2744 | 0.56% | 0.99 |

Nu lf num ( | 0.3082 | 0.2878 | 0.31% | 0.99 |

In the following step of the study, the 2D-PIV
system was used to analyse the dynamic structures connected to the natural
convective heat transfer in these two geometrical configurations. The first
results are shown in Figures

Velocity vectors and streamfunctions for the symmetrical configuration obtained through PIV.

Velocity vectors and streamfunctions for the asymmetrical configuration obtained through PIV.

Numerical stream functions.

Through these images, it is also possible to
understand why the Nusselt numbers on the left side of the source for

During each test, it was possible to see that the dynamic structures (distribution of the velocity vectors and streamfunctions) did not change with the Rayleigh number. Instead, the modulus of the velocity vectors grows up according to the growth of the Rayleigh number.

The velocity maps are shown in Figures

Velocity maps in m/s for

Velocity maps for

Finally, the vortical structures, shown in Figure

A particular of the vortical structures on the upper surface of the strip
for

For

In this paper, the influence of the position of a heat source on the
natural convective heat transfer in a square cavity of side

The analysis was conducted experimentally both through the holographic
interferometry and through the particle image velocimetry for Rayleigh number
from

Two different positions of the source were analysed: the symmetrical
position and an asymmetrical one. In the first case, the distance between the
right side of the strip and the right side of the cavity is

Through the holographic interferometry, the
heat transfer was analysed. In particular, the local and average Nusselt
numbers on the hot surfaces were evaluated. The natural convection is more
efficient on the lateral sides of the strip. On the upper surface, the average
Nusselt numbers at the same Rayleigh numbers are lower than on the lateral
ones. This is confirmed by the numerical code and is connected to two particular
vortical structures analysed through the PIV system. Then through the
experimental data, the numerical code was validated and used to analyse the
asymmetrical configuration. Two important differences are evident making a
comparison with the symmetrical position. There is a small difference between
the average Nusselt number on the right side and the one on the left side at
the same Rayleigh number. In particular, the left surface has a higher Nusselt
number than the other one. It is possible to understand this event by analysing
the velocity maps obtained through the PIV system. In the left zone of the
enclosure for

The second difference is about the average Nusselt numbers on the upper
surface. In the asymmetrical configuration, their values grow up in comparison to
the symmetrical position. This fact deals with the behaviour of the two small
cylinders on this surface. For

Modulus of the gravity vector (

Time
between two laser pulses (

Square cavity side (m)

Thermal conductivity (

Heat source length (m)

Heat source height (m)

Distance of the hot strip from the sidewall (m)

Prandtl number

Local Nusselt number

Average Nusselt number on the heat source

Rayleigh number

Temperature (K)

Cartesian coordinate

Dimensionless Cartesian coordinate

Thermal
expansion coefficient (

Dimensionless position of the heat source

Dimensionless length of the heat source

Dimensionless height of the heat source

Dimensionless temperature

Cinematic viscosity (

Density (

Cold wall

Hot wall

Experimental

Numerical

Right side of the hot strip

Left side of the hot strip.