TWO DIMENTIONAL ANALYSES OF

NATURAL CONVECTION IN AN

ENCLOSURE WITH DISCREET HEAT

GENERATION

2014

Ricardo Ribeiro Fernandes Mendes

Student number: 1330956

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Abstract:

The problem under consideration is the Rayleigh-Benard convection in a rectangular domain with

periodic boundary conditions at side walls and heating coming from discreet heat sources at

constant rate. The simulations were conducted for the case in which the sources length L was kept

constant and with equal value of the source spacing S. For the simulations conducted,

S=L=20mm, and two domain height were used, 15 and 30mm. Rayleigh number, Nusselt numbers

were calculated after the simulations, with Rayleigh number ranging from 100 and 40,000. Nusselt

numbers and maximum velocity vs Rayleigh number were plotted and analysed.

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Summary:

1 Introduciton............................................................................................................... 3

1.1 Overview of the Rayleigh-Benard convection 3

1.2 Pattern formation............................................................................................. 3

1.3 Literature Review............................................................................................. 4

2 Mathematical Model.................................................................................................... 6

2.1 Computational domain...................................................................................... 6

2.2 Approximation 7

2.3 Boussinesq approximation.. 7

2.4 Rayleigh number.............................................................................................. 7

2.5 Prandlt number and Nusselt number........... 8

2.6 Continuity equation. 8

2.7 Momentum equation.. 8

2.8 Thermal Diffusion Equation 9

2.9 Boundary conditions.. 9

3 Finite Elements Analyses. 11

3.1 Boundary layer regime.. 11

3.2 Discretisation scheme 11

4 Results 12

4.1 Maximum velocity and Nusselt number 12

5 Conclusion 20

Appendix A.. 21

References 23

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1- Introduciton:

1.1- Overview of the Rayleigh-Benard convection:

The Benard-Rayleigh convection, also known as natural convection, is a phenomenon in fluids

dynamics in which fluid motion is induced by thermal driven instabilities in the fluid. It has

applications in several areas of interest both in natural science and engineering. It is used to

understand continent drifting, star models, oceanic currents and weather patterns. It is relevant for

engineering systems such as microelectronic devices, solar energy systems, energy storage, material

processing, dynamics of chemical reactions, diffusion systems and cooling systems.

One of the reasons Rayleigh-Benard convection is relevant in a large range of application is that the

instabilities can occur even for small a temperature gradient. The fact that the convection largely

increases the heat transfer rate makes it interesting in cooling system applications, especially

microelectronic devices [5].

Benard-Rayleigh convection is also studied by its theoretical value. It is the most comprehensible

understood example of a non-linear pattern-forming system and shares a number of features with

other pattern-forming systems. It is an excellent model to try different approaches to non-linear

dynamics and self-organizing systems.

1.2- Pattern formation:

The Rayleigh-Benard convection is a model in which a fluid is heated from bellow, while kept at

constant temperature at the top. The temperature gradient induces variations in the density of the

fluid, making the fluid at the top heavier than the fluid on the bottom. If buoyance forces caused by

the density variation are large enough to overcome viscous forces, instability occurs within the fluid.

Convection rolls start to form as a consequence of this instability [3].

It is, therefore, of great importance to identify the conditions in which the Rayleigh-Benard convection

happens and the effects on the heat dissipation of the fluid. The convection can be described by

standard fluid dynamics equations and the analyses of the conditions in which it and its effects

happens can be described with the use of two dimensionless numbers: Rayleigh number and Nusselt

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number [5]. The description of the equations and the dimensionless numbers will be conducted in the

Mathematical Model section.

Figure 1 shows a schematization of the Rayleigh-Benard convection. The fluid is contained is

contained by two infinite horizontal plates and is heated from bellow and cooled from the top. The

height of domain is also an important parameter.

Figure 1: Schematization of Rayleigh-Benard convection, taken from http://www.isibrno.cz/

Buoyance driven instabilities can also be induced by internal or external heat generation. In the

present work, the instabilities were studied through the simulation of evenly distributed heat

sources in the bottom of the domain.

1.3- Literature review:

Sir Benjamin Thomson (1797) is the regarded as the first one to recognize a relation between heat

motion in a fluid and fluid motion. Later, J. Thomson (1881-82) pointed out evidence of convective

motion [4]. The first experiments to present a complete description of convection in a flow were

conducted by Henri Benard on the beginning of the twentieth century. He observed pattern formation

of a thin layer of fluid subjected to a small temperature gradient from bellow [1]. Benard observed

that the fluid motion only happened after the temperate gradient reached a critical value. Before that

value, viscous forces prevented fluid motion. When the critical value was reached, the fluid motion he

observed the formation of convection rolls, called Benard cells.

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Lord Rayleigh carried out the analytical treatment of the phenomenon observed by Benard and

proposed a theory of feedback coupling driven by buoyance [2]. As these works provides the

foundation for the study of the thermal instability problem, it has been called Rayleigh-Benard

convection.

However, the instability in the experiment conducted by Benard was actually caused by surface

tension. In 1958, J. Pearson pointed out the role the thermal Marangoni effect in Benard experiments

[2]. The term Benard-Marangoni convection is related to the phenomenon in which instability is

caused by surface tensions.

Jeffery stated that the convection happens for Rayleigh number (Ra) greater than 1708, after

investigating the Rayleigh-Benard convection [6]. Jeffery (1926, 1928) and Low (1929) investigated

the effects of Prandlt number on the instability [6-8]. Chandrasekhar used infinitesimal perturbations

to formulate a comprehensible theory on Rayleigh-Benard convection [9]. Utilizing visual methods,

Chandra conducted experiments with smoke [10] that deviated from Jeffery by 20 % [6]. These

experiments showed that the depth of the domain influenced the nature of the instability.

Perekattu and Balaji conducted a numerical and analytical study on the influence of internal heat

generation on the occurrence of convection for a shallow fluid layer, differentially heated from bellow

[11]. Puigjaner at al investigated the dynamics of Rayleigh-Benard convection in a cubic cavity for

moderate Rayleigh numbers [12]. Vedantam at al carried out three-dimensional numerical

simulations in a rectangular enclosure for a fluid with low Prandlt number [13]. Baletta and Nield

studied the effects of pressure and viscous dissipations in the Benard-Rayleigh convection [14,15].

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2- Mathematical Model:

2.1- Computational domain:

Figure 2 illustrates the problem under discussion. The fluid is contained in a rectangular domain of

length L1 and height H. Periodic conditions were applied to the model, to avoid interference from the

side walls. The fluid is heated from bellow by sources of equal length L, distributed though the

bottom wall. The distance between the sources is S.

The heat transfer qL at the sources is constant. The spaces between the sources were

considered adiabatic, with qS=0. The top wall was kept at constant temperature T0.

2.2- Approximations:

The fluid is assumed to homogeneous and incompressible.

Boussinesq approximations are used for the fluid density.

Kinematic viscosity, dynamic viscosity and thermal diffusivity of the fluid are assumed to be

constant.

Radiation effects are negligible due to the low emissivity of the material used in the

investigation.

The mean temperature of the flow

.

L S

H

qL qS

g

x y

Periodic

L1

T=T0

Figure 2: Computational domain of the problem studied

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2.3- Boussinesq Approximation:

Boussinesq approximation consists in the assumption that density variations are small enough to be

neglected everywhere except in buoyancy. Therefore density variations are only considered in y-

direction motion equation.

Let be the temperature at the top layer and be the density there. Let be a small variation in the density, caused by a variation in the temperature.

= + (1)

= + (2)

= 1 (3)

Where is the coefficient of thermal expansion of the fluid.

2.4- Rayleigh Number:

Rayleigh Number (Ra) is the dimensionless number that relates the ratio between buoyance forces

and viscous forces. The Rayleigh Number is given by the following equation.

=

(4)

Where is the acceleration of gravity, is the coefficient of thermal expansion of the fluid, is the temperature on the surface of the heat source, is the temperature on the upper wall, is the characteristic length of the domain, is the thermal diffusivity of the fluid and is the kinematic viscosity of the fluid.

For the present work, was determined by the simulations and the domain height was determined as the characteristic length.

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2.5- Prandlt number and Nusselt number:

The Prandlt number (Pr) is the ratio between viscous diffusion rate and thermal diffusion rate .

= (5)

The Prandlt number investigate for this work was 0.71.

The Nusselt number (Nu) is the dimensionless number that expresses the heat transfer from the

sources:

= (6)

Where is the heat transfer coefficient and is the thermal conductivity.

2.6- Continuity equation:

The general continuity equation is:

+ = 0 (7)

= + (8)

Where is the velocity vector and is the fluid density

As the fluid is incompressible, = in the entire domain. The continuity equation becomes:

= 0 (9)

2.7- Momentum Equation:

Navier-Stokes equation for a incompressible flow:

+ = + + (10)

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= (11)

Appling the Boussinesq approximation, we can write:

+ =

+ + (1 )

(12)

2.8- Thermal Diffusion Equation:

The energy equation, if we consider only convection and conduction, is:

+ = (13)

Where is the thermal conductivity of the fluid, and is the specific heat of the fluid.

2.9- Boundary Conditions:

Boundary conditions at the upper wall are given by:

, = 0 (14) , = 0 (15) , = (16)

Boundary conditions at the bottom wall are given by:

, 0 = 0 (17) , 0 = 0 (18)

Heat flux on the heat sources is ! =

(19)

Heat flux on the space between the sources is 0

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! = = 0

(20)

Boundary conditions at the side walls are periodic

0,! = "1,! (21) 0,! = "1,! (22)

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3- Finite Elements Analyses:

3.1- Boundary layer flow regime:

Due to the low Grashof number of the simulations to be realized (

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4- Results:

4.1- Maximum velocity and Nusselt number:

The purpose of the present work is to study convection of atmospheric air heated from bellow by

discrete heat sources for Prandlt number = 0.71. In previous works, the critical Rayleigh number for a fluid heated by a uniform heat sources has been identified as 1708 {16}. Papnicolaou and

Gopalakrishna stated that the critical Rayleigh number for layer of fluid heated by discrete heat

sources depends on domain height, length of heat sources and side wall effects (5). For this work,

side wall effects were removed with the use of periodic boundary conditions and the length of the

heat sources was kept constant. Therefore, only the effects of the domain height were studied.

Figure 3: Maximum velocity vs. Rayleigh number

Figure 3 shows a clear a shift in the graphic. While for h=30cm, the onset of the convection occurred

at a Rayleigh number even higher than 1708, for h=15cm, it happened for 900. Furthermore, for similar Rayleigh numbers, the maximum velocity is higher for the smaller domain high.

Tables 1 and 2 shows the values of maximum velocity and Rayleigh number obtained through the

simulations for domain height of 30 and 15 millimeters, respectively.

0

0.01

0.02

0.03

0.04

0.05

0.06

100.00 1,000.00 10,000.00 100,000.00

Ma

xim

um

ve

lov

ity

(m

/s)

Rayleigh number

H=30mm

H=15mm

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Table 1: Maximum velocity and Rayleigh number for H=30mm

Top wall temperature

(K)

Maximum temperature

(K)

Heat flux

(W/m2)

Maximum velocity

(m/s)

Rayleigh

number

300 300.35 0.5 0.00026 877.20

300 300.71 1 0.00061 1,757.36

300 301.17 2 0.00586 2,855.61

300 301.54 3 0.00866 3,754.15

300 302.19 5 0.01301 5,351.46

300 305.14 15 0.02584 12,311.26

300 307.79 25 0.03404 18,394.92

300 310.28 35 0.04037 23,912.60

300 312.99 45 0.04230 29,780.98

300 315.17 55 0.04993 34,316.49

300 318.16 70 0.05578 40,322.19

Table 2: Maximum velocity and Rayleigh number for H=15mm

Top wall temperature

(K)

Maximum temperature

(K)

Heat flux

(W/m2)

Maximum velocity

(m/s)

Rayleigh

number

300 300.4205 1 0.00028 129.92

300 301.2709 3 0.00092 386.78

300 302.5593 5 0.00411 778.89

300 303.3553 8 0.00410 1,009.07

300 304.7645 12 0.00655 1,432.88

300 306.9767 20 0.01090 2,063.22

300 309.3243 30 0.01574 2,681.82

300 311.9362 42 0.02035 3,379.54

300 315.6652 60 0.02592 4,316.43

300 319.6473 80 0.03104 5,269.82

Figure 4 shows the Nusselt number as a function of the Rayleigh number for both H=30mm and

H=15mm.

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Figure 4: Nusselt number vs Raylegh number

Figure 4 shows that the onset for the instability happens for a smaller Rayleigh number for a lower

domain height. It also shows that the Nusselt number grows almost linearly in relation to the

logarithmic of the Rayleigh number once the regime becomes convection dominated.

Figures 5 and 6 shows, respectively, velocity profile and temperature contour for H=30mm and

various Rayleigh number. Figures 7 and 8 shows velocity profile and temperature contours for

H=15mm and various Rayleigh number.

The Benard cells are clearly discernible from Rayleigh number 2,855.61 for H=30mm and for every

Rayleigh number for H=15, even for low velocity speeds. For lower Rayleigh number there is only

signficant variation at temperature countors for the lower half of the fluid. As the Rayleigh

numbers increases, there are higher temperatures near the top wall and the high temperature

region near the bottom wall shrinks to the centre of the heating cells. It happens due the effects

of the Benard cells. As figures 9 and 10 shows, at higher Rayleigh numbers, the larger velocities

happens above the heating cells centre for, were the temperature gradient is greater.

Alternativelly, for lower Rayleigh number, the higher velocities happens above the edges of the

heating cells, as figures 11 and 12 shows. The shift in the velocity profile is particulary easy to see

at figure 7.

2

3

4

5

6

7

8

9

100.00 1,000.00 10,000.00 100,000.00

Nu

sse

lt n

um

be

r

Rayleigh number

H=30mm

H=15mm

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*

()

&

+

Figure 5: Velocity profile for H=30mm, (a) Ra= 877.20 (b) Ra= 2,855.61 (c) Ra=

12,311.26 (d) Ra= 23,912.60 (e) Ra= 40,322.19

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*

()

&

+

Figure 6:Temperature contours for H=30mm, (a) Ra= 877.20 (b) Ra= 2,855.61 (c) Ra=

12,311.26 (d) Ra= 23,912.60 (e) Ra= 40,322.19

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*

()

&

+

Figure 7: Velocity profile for H=15mm, (a) Ra= 129.92 (b) Ra= 778.89 (c) Ra= 1,432.88 (d) Ra=

2,681.82 (e) Ra= 4,316.43

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Figure 8: Temperature contours for H=15mm, (a) Ra= 129.92 (b) Ra= 778.89 (c) Ra= 1,432.88

(d) Ra= 2,681.82 (e) Ra= 4,316.43

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Figure 9: Velocity profile and temperature

countour for H=30mm and Ra=40,322.19

Figure 10: Velocity profile and temperature

countour for H=15mm and Ra=4,316.43

Figure 11: Velocity profile and temperature

countour for H=30mm and Ra=877.20

Figure 12: Velocity profile and temperature

countour for H=15mm and Ra=129.92

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5- Conclusion:

The simulations conducted on fluent gave results coherent with previous work, regarding the range of

domain highs and Rayleigh numbers studied, however, was not large enough for a proper

observation of the phenomenon under investigation. In future investigation, the range should be

extended for a better observation of the effects of domain highs on the onset of the convection.

The accuracy of the simulation could also be improved. There are two ways in which it could be

achieved. The investigation of a possible turbulence model could improve the accuracy of velocity and

temperature values. Another possible improvement could come from a double phase interpolation.

The grid should be remeshed after an initial interpolation, with refining in function of the temperature

gradient.

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Appendix A: Tables

Table 3: Rayleigh number calculations

Top wall

temperature

(K)

Maximum

temperature

(K)

Kinematic

viscosity

(m2/s)

Thermal

diffusivity

(m2/s)

Characteristic

Length (m)

Thermal

expansion

coefficient

(1/K)

Gravity

acceleration

(m/s2)

Rayleigh

number

300 300.3549 1.60E-05 2.24E-05 0.03 3.34E-03 9.81 877.20

300 300.711 1.60E-05 2.24E-05 0.03 3.34E-03 9.81 1,757.36

300 301.1692 1.61E-05 2.26E-05 0.03 3.34E-03 9.81 2,855.61

300 301.5371 1.61E-05 2.26E-05 0.03 3.34E-03 9.81 3,754.15

300 302.1911 1.61E-05 2.26E-05 0.03 3.34E-03 9.81 5,351.46

300 305.1355 1.62E-05 2.28E-05 0.03 3.34E-03 9.81 12,311.26

300 307.7949 1.63E-05 2.30E-05 0.03 3.34E-03 9.81 18,394.92

300 310.2827 1.65E-05 2.31E-05 0.03 3.34E-03 9.81 23,912.60

300 312.985 1.66E-05 2.33E-05 0.03 3.34E-03 9.81 29,780.98

300 315.1743 1.67E-05 2.35E-05 0.03 3.34E-03 9.81 34,316.49

300 318.1568 1.68E-05 2.37E-05 0.03 3.34E-03 9.81 40,322.19

300 300.4205 1.60E-05 2.24E-05 0.015 3.34E-03 9.81 129.92

300 301.2709 1.61E-05 2.26E-05 0.015 3.33E-03 9.81 386.78

300 302.5593 1.61E-05 2.26E-05 0.015 3.33E-03 9.81 778.89

300 303.3553 1.62E-05 2.27E-05 0.015 3.33E-03 9.81 1,009.07

300 304.7645 1.62E-05 2.27E-05 0.015 3.33E-03 9.81 1,432.88

300 306.9767 1.63E-05 2.28E-05 0.015 3.32E-03 9.81 2,063.22

300 309.3243 1.64E-05 2.31E-05 0.015 3.30E-03 9.81 2,681.82

300 311.9362 1.65E-05 2.33E-05 0.015 3.29E-03 9.81 3,379.54

300 315.6652 1.67E-05 2.35E-05 0.015 3.28E-03 9.81 4,316.43

300 319.6473 1.69E-05 2.38E-05 0.015 3.27E-03 9.81 5,269.82

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Table 4: Nusselt number calculations

Top wall

temperature

(K)

Maximum

temperature

(K)

Heat

flux

(W/m2)

Characteristic

Length (m)

Medium

Temperature

(K)

Thermal convection

coefficient (W/m2*k)

Thermal

diffusion

coefficient

(W/m*k)

Nusselt number

300 300.3549 0.5 0.03 300.17745 2.817695125 2.62E-02 3.23E+00

300 300.711 1 0.03 300.3555 2.812939522 2.62E-02 3.22E+00

300 301.1692 2 0.03 300.5846 3.421142662 2.63E-02 3.91E+00

300 301.5371 3 0.03 300.76855 3.903454557 2.63E-02 4.46E+00

300 302.1911 5 0.03 301.09555 4.563917667 2.63E-02 5.21E+00

300 305.1355 15 0.03 302.56775 5.841690196 2.64E-02 6.63E+00

300 307.7949 25 0.03 303.89745 6.414450474 2.65E-02 7.27E+00

300 310.2827 35 0.03 305.14135 6.807550546 2.66E-02 7.69E+00

300 312.985 45 0.03 306.4925 6.931074317 2.66E-02 7.81E+00

300 315.1743 55 0.03 307.58715 7.249098805 2.68E-02 8.12E+00

300 318.1568 70 0.03 309.0784 7.710609799 2.68E-02 8.62E+00

300 300.4205 1 0.015 300.21025 4.756242568 2.62E-02 2.72E+00

300 301.2709 3 0.015 300.63545 4.721063813 2.63E-02 2.70E+00

300 302.5593 5 0.015 301.27965 3.907318407 2.63E-02 2.23E+00

300 303.3553 8 0.015 301.67765 4.76857509 2.63E-02 2.72E+00

300 304.7645 12 0.015 302.38225 5.037254696 2.63E-02 2.87E+00

300 306.9767 20 0.015 303.48835 5.733369645 2.64E-02 3.26E+00

300 309.3243 30 0.015 304.66215 6.434799395 2.66E-02 3.63E+00

300 311.9362 42 0.015 305.9681 7.037415593 2.66E-02 3.96E+00

300 315.6652 60 0.015 307.8326 7.660291602 2.68E-02 4.29E+00

300 319.6473 80 0.015 309.82365 8.143612608 2.69E-02 4.54E+00

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