Synopsis

After a two decades from its inception, the lattice Boltzmann method (LBM) has

now been established as a promising numerical tool of computational fluid dynam-

ics (CFD) for solving various problems of complex fluid flows of domestic as well

as engineering fields. The lattice Boltzmann method has derived from Boolean

variables based lattice gas automata (LGA). The lattice Boltzmann method is

considered as an alternative numerical tool to conventional CFD numerical tools,

which are based on the macroscopic continuum equations. The lattice Boltz-

mann method basically solves a kinetic and discrete velocity based Boltzmann

equation (in statistical physics). The application range of LBM involves complex

fluid flows such as porous structures, magneto hydrodynamic, reaction-diffusion,

diffusion-dispersion, suspension flows, compressible flows, multiphase flows, etc.

The advantages of the LBM are simplicity of coding and algorithm, ease in ap-

plication of boundary conditions (thus making LBM suitable for complex fluid

flow problems), ease of parallel computing, estimation of pressure field is very

adroitness as compared to conventional CFD tools, etc, (Chen and Doolen, 1998).

The investigation of thermally driven flows reveals that fluid flows and transport

processes generated or reformed by buoyancy are of theoretical and pragmatic

significance. Thus making this subject very popular among the researchers in

diverse fields of nuclear reactor systems, meteorology, geophysics, energy storage

and conservation, fire control, and chemical, food, and metallurgical industries, as

well as in the more conventional fields of the fluid and heat transfer sciences (Roy

and Basak, 2005).

v

Synopsis vi

Among others, the investigation of natural convection heat transfer in closed,

as well as open, ended cavities is considered as an important research field due

to the wide ranges of the industrially important applications including chemical

vapor deposition (Spall, 1996), cooling devices in electronic equipment (Bilgen and

Muftuoglu, 2008; Hsu and Wang, 2000; Du et al., 1998), polymer and material

processing (Hsiao, 2007; Habib et al., 2005), solar collectors (Hobbi and Siddiqui,

2009), electronic card arrays (Manca and Nardini, 2010), domestic refrigerators

and oven (Skok et al., 1990) etc. The heat transfer in cavity with one wall heated

other cooled/open to ambient represents the idealized case. In process industries,

the such structures are often exposed to non-linear heating. Thus study of partially

heated walls is necessary. In spite of wide occurrence of such structures, very

limited results are available for cavity with non-linear heating. Thus, in this

dissertation, an attempt have been made to fulfill the gap found in literature

for heat transfer in enclosure problems. The heat transfer in enclosure (closed

as well as open ended) is studied for laminar range of Rayleigh number (heat

intensity parameter for buoyancy driven flows), Prandtl numbers, heating size

and locations. The applicability of LBM for solving magneto hydrodynamic effect

on natural convection in a partially heated square cavity have been elucidated.

Another problem considered for this dissertation is flow past a built-in square/rect-

angular cylinder, which is important owing to its overwhelming theoretical and

practical relevance. The flow of fluids, especially Newtonian fluid, over bluff bodies

of different shapes (circular, square, elliptic and triangular cylinders) of asymmet-

ric shapes like spheres and spheroids, for instance, has been explored well over 100

years (Dhiman et al., 2006a,b, 2007; Sahu et al., 2009; Koteswara Rao et al., 2011;

Sharma et al., 2012). The study of flow past a square cylinder is very important

for knowledge of gross engineering parameters such as drag coefficient, Nusselt

number, wake size, etc., which are often used for the design of cooling towers, an-

tennas, chimneys, antennas, support structures, high rise building etc (Chatterjee

et al., 2009; Sharma et al., 2012). Though a reasonable amount of information

is available for flow past shapes other than circular cylinder, it neither extensive

nor comprehensible. In present work, the influence of higher blockage ratios on

momentum and heat transfer characteristics have been explored.

The thermal field by using LBM can be simulated by using three approaches, viz.,

Synopsis vii

multispeed, passive scalar and double distribution approach. In present work, ther-

mal lattice Boltzmann method (TLBM) based on simplified double distribution

function model of (He et al., 1998; Peng et al., 2003b) is used for solving thermal

field. The present in-house TLBM code has been developed in C++ programming

language. The basic validation of the present LBM code is ascertained by solving

basic problems of 2D lid driven cavity and flow between parallel walls. Further, the

extensive bench-marking has been carried out by comparing the present numeri-

cal results values with previous studies. The grid and domain size have significant

effect on the accuracy of the solution, thus, appropriate values of these have been

chosen after a thorough study, thus elucidating the influence of these parameters

on the accuracy of the results.

The new extensive results are obtained for the range of flow governing parameters

for the gross engineering parameters such as drag coefficient (in case of flow past a

square/rectangular cylinder) and average Nusselt number. The physical insights of

the system are gained by detailed analysis of flow and thermal field. In particular,

the stream-function, isotherms and vorticity patterns and local variation of Nusselt

number are examined. In this thesis, the extensive results are presented for the

following ranges of conditions.

Problem Physical parametersDifferentially heated cavity∗ 0.71 ≤ Pr ≤ 100; 104 ≤ Ra ≤ 106

Partially-differentially heated cavity∗ Pr = 0.71; 104 ≤ Ra ≤ 106

Heater size=H/2, Cooler size=H/2Partially heated cavity∗ Pr = 1; 103 ≤ Ra ≤ 105; 0 ≤ Ha ≤ 120

Angle of magnetic field (0o, 45o, 90o)Partially heated open ended cavity∗ 0.71 ≤ Pr ≤ 7; 103 ≤ Ra ≤ 106

Heating locations (Middle, top, bottom)Heater size (H/4,H/2,3H/2)

Flow past square cylinder# Pr = 1; 5 ≤ Re ≤ 40; β = 1/8, 1/12, 1/16, 1/20Flow past rectangular cylinder# Pr = 1; 5 ≤ Re ≤ 40; β = 1/8, 1/12, 1/16

aspect ratio (a=1,2,4,6)( * Natural convection), (# Forced convection)

The five problems have been considered in this thesis are described in brief as

below.

Synopsis viii

1. Natural convection in differentially heated square cavity: Effect of

Prandtl number

The influence of wide range of Prandtl numbers (0.71 ≤ Pr ≤ 100) on natural

convective heat transfer in heated closed cavity have been elucidated by using

thermal lattice Boltzmann method (TLBM) for laminar range of Rayleigh number

(104 ≤ Ra ≤ 106). Natural convection effect increases with the increase in Prandtl

number (Pr) for all values of the Rayleigh number (Ra) due to the increased

viscous force effect in comparison to inertial force. As thermal diffusion is inversely

proportional to Prandtl number, velocity is more diffused than thermal energy.

For Ra = 104, dominant heat transfer mechanism is conductive for Pr ≥ 10. For

higher Prandtl numbers, temperature distribution weakens and isotherms are more

stratified towards the hot wall indicating predominant momentum boundary layer.

The average Nusselt number (dimensionless heat transfer coefficient) of isothermal

wall (x = 0) is seen to increase with the Prandtl and Rayleigh numbers.

2. Natural convection in partially-differentially heated square cavity

The problem basically studied to explore the influence of one wall of cavity exposed

to contrast thermal conditions. The one wall of cavity is equally exposed to hot and

cold conditions and other wall exposed to ambient. The flow governing parameters

used for numerical experimentation are Rayleigh number (104 ≤ Ra ≤ 106) with

heater size is half of characteristics length (H/2) with air (Pr = 0.71) as a working

fluid. The results indicated the formation of convection cell near lower part of

mixed heated wall (X = 0) of cavity is observed after Ra ≥ 104, as low temperature

fluid retained in that region. The size of convection cell increases with the increase

in Rayleigh number (Ra). The average Nusselt number (Nu) and overall Nusselt

number (N̂u) value show linear increase with Rayleigh number. At Ra = 105, the

rate of heat transfer of both vertical walls is almost same, which is indicated by

nearly same values of average Nusselt number (Nu).

3. Magneto-hydrodynamic natural convection in partially-differentially

heated square cavity

In this problem, the influence of cooler size (H, H/2,H/4,H/6), Hartmann number

(0 ≤ Ha ≤ 120) and angle of magnetic field (θ = 0o, 45o, 90o) on natural convection

heat transfer in differentially heated cavity is elucidated. The cavity considered

is partially heated at middle location (H/4 ≤ T ≤ 3H/4) at one wall while other

Synopsis ix

wall is partially cooled. In particular, the effect of four cooling conditions have

been explored on rate of heat transfer. It is observed that the rate of heat transfer

increases with Hartmann number, while, the angle of magnetic field has marginal

influence on heat transfer rate.

4. Natural convection in partially heated open ended square cavity

The natural convection heat transfer analysis in a partially heated open ended

square cavity have been studied for elucidating the influence of heater size and

heating location. Two studies have been carried out for partially heated open

ended cavity, (a) Effect of three heating locations (middle, top, bottom) and heater

size (H/4, H/2, 3H/4) for Pr=0.71 and (b) Effect of Prandtl number (0.71 ≤Pr ≤ 7) on partially heated open ended cavity (heated at the middle location of

vertical wall), on heat transfer characteristics. Linear dependence of the average

Nusselt number (Nu) on the Rayleigh number is observed, irrespective of the

heating locations and heater size. However, average Nusselt number (Nu) shows

a proportional dependence for the bottom and middle locations and inversely

proportional dependence for the top heating location on the heater size, i.e., an

increasing value of Lh enhanced Nu for the bottom and middle locations and

deteriorated Nu for the top heating location. Over the range of Rayleigh number,

middle partial heating location shows higher heat transfer rate followed by bottom

and top heating locations. The significant convection losses are indicated in the

top heating location as Nu decreased with increase in the heater size.

While the results of another case indicated the strong influence of Prandtl numbers

on rate of heat transfer, the increase in Prandtl and Rayleigh number increases

the average Nusselt number values. In the end, the a closure relationship between

average Nusselt number with Prandtl and Rayleigh numbers have developed in

standard form.

5. Forced convection heat transfer from built-in square cylinder: Effect

of wall confinement

The effect of wall confinement on the momentum and heat transfer characteris-

tics have been studied herein. In particular, the influence of four blockage ra-

tios (β = 1/8, 1/12, 1/16, 1/20) and Reynolds numbers (5, 10, 20, 40) on fluid flow

behavior have been explored. The physical insights near the surface of square

Synopsis x

cylinder have been achieved by delineating the stream-lines, vorticity, isotherms,

pressure as well as viscous coefficient and local Nusselt number for range of con-

ditions. The results of this study indicated that the increase in blockage ratio

causes marginal increase in re-circulation length for considered range of Reynolds

number. The drag values are found to be in inverse proportion with blockage ratio

and Reynolds number. Furthermore, at a constant Reynolds number, increase in

blockage ratio causes crowding isotherms along cylinder. Higher surface pressure

coefficient (CP ) values are obtained for front face of cylinder for low blockage ra-

tio. Thus, increasing blockage ratio reduces CP values along the cylinder surface.

Linear increase in average Nusselt number (Nu) is observed with Reynolds num-

ber and lower blockage ratio. Thus, increase in blockage ratio impedes rate of

heat transfer. The Colburn heat transfer factor jH is dominated more by block-

age ratio than Reynolds number. Finally, an empirical correlations relating total

drag coefficient (CD) and average Nusselt number (Nu) with blockage ratio (β)

and Reynolds number (Re) have been developed for its possible use in engineering

design purpose.

The above mentioned study has been extended to rectangular cylinder by changing

the longitudinal length. In particular, the influence of aspect ratio of rectangu-

lar cylinder (a=2,4,6), blockage ratio (β = 1/8, 1/12, 1/16) at constant Prandtl

number of Pr=1 on forced convection heat transfer have been delineated. It is

observed that drag as well as average Nusselt number have linear dependence with

aspect ratio.

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