numerical analysis of non-isothermal forced convection fluid flow and mixed convection fluid flow in...

8/20/2019 Numerical Analysis of Non-isothermal forced Convection fluid flow and Mixed Convection fluid flow in a Lid Driven…

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IPASJ International Journal of Mechanical Engineering (IIJME)Web Site: http://www.ipasj.org/IIJME/IIJME.htm

A Publisher for Research Motivation........ Email: [email protected]

Volume 3, Issue 9, September 2015 ISSN 2321-6441

Volume 3, Issue 9, September 2015 Page 16

ABSTRACT

Mixed convection from a uniform and non uniform sinusoidal heat source on the bottom of a rectangular cavity is studied

numerically. Two-dimensional forms of non-dimensional Navier-Stokes equations are solved by using control volume based

finite volume technique with staggered grid . Three typical values of the Reynolds numbers are chosen as Re = 10, 100, and

2000 and steady, laminar results are obtained in the values of Richardson number as Ri = 0, 1 and 10, 100 and the values of

Prandtl numbers is taken as Pr = 0.71, 7 and 10. The parametric studies for a wide range of governing parameters show

consistent performance of the present numerical approach to obtain as stream functions and temperature profiles. Heat

transfer rates at the heated walls are presented based on the value of Re and Pr. The computational results indicate that the

heat transfer is strongly affected by Reynolds number and Richardson number. In the present investigation, bottom wall is(a)

uniformly heated & (b) non –uniformly heated while the two vertical walls are maintained at constant cold temperature and

the top wall is well insulated. A complete study on the effect of Ri shows that the strength of circulation increases with the

increase in the value of Ri irrespective of Re and Pr. As the value of Ri increases, there occurs a transition from conduction to

convection dominated flow at Ri =1. A detailed analysis of flow pattern shows that the natural or forced convection is based on

both the parameters Ri and Pr.

Keywords— Mixed convection; Rectangular cavity; Lid driven cavity; Uniform heating ;Non uniform heating;

Reynolds number; Richardson number and Prandtl number. Staggered Grid

1.INTRODUCTION

Convection is the heat transfer mechanism affected by the flow of fluids. The amount of energy and matter are

conveyed by the fluid can be predicted through the convective heat transfer. The convective heat transfer splits into two

branches; the natural convection and the forced convection. Forced convection regards the heat transport by induced

fluid motion which is forced to happen. This induced flow needs consistent mechanical power. But natural convection

differs from the forced convection through the fluid flow driving force which happens naturally. The flows are driven

by the buoyancy effect due to the presence of density gradient and gravitational field. As the temperature distribution in

the natural convection depends on the intensity of the fluid currents which is dependent on the temperature potential

itself, the qualitative and quantitative analysis of natural convection heat transfer is very difficult. Numerical

investigation instead of theoretical analysis is more needed in this field. Two types of natural convection heat transfer

phenomena can be observed in the nature. One is that external free convection that is caused by the heat transfer

interaction between a single wall and a very large fluid reservoir adjacent to the wall. Another is that internal free

convection which befalls within an enclosure. Mathematically, the tendency of a particular system towards natural

convection relies on the Grashoff number, (Gr=gβ(T H -T C )L3/ν2), which is a ratio of buoyancy force and viscous force.

The parameter β is the rate of change of density with respect to the change in temperature (T H -T C ), and ν is viscosity.

Thus, the Grashoff number can be thought of as the ratio of the upwards buoyancy of the heated fluid to the internal

friction slowing it down. In very sticky, viscous fluids, the fluid movement is restricted, along with natural convection.

In the extreme case of infinite viscosity, the fluid could not move and all heat transfer would be through conductive

heat transfer. Convection and Conduction Heat Transfer Forced convection is often encountered by engineers designingor analyzing heat exchangers, pipe flow, and flow over flat plate at a different temperature than the stream (the case of

a shuttle wing during re-entry, for example). Note that the characteristic velocity considered here for natural convection

is equal to α/L; thus, the diffusion-coefficient is Pr for momentum and 1 for energy equation. However, in any forced

Numerical Analysis of Non-isothermal forced

Convection fluid flow and Mixed Convectionfluid flow in a Lid Driven Rectangular Cavity

with uniform and non uniform heating of bottom

wall by Finite Volume method in a Staggered

Grid

Shantanu Dutta

NSHM Knowledge Campus Durgapur, West Bengal District: Burdwan -713212


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convection situation, some amount of natural convection is always present. When the natural convection is not

negligible, such flows are typically referred to as mixed convection. When analyzing potentially mixed convection, a

parameter called the Richardson number (Ri= Gr/ Re2) parametizes the relative strength of free and forced convection.

The Richardson number is the ratio of Grashoff number and the square of the Reynolds number, which represents the

ratio of buoyancy force and inertia force, and which stands in for the contribution of natural convection. When Ri>>1,natural convection dominates and when Ri


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flow and heat transfer in solar ponds [5], dynamics of lakes [6], thermal-hydraulics of nuclear reactors [7], industrial

processes such as food processing, and float glass production [8]. The specialty of the uniform and non-uniform bottom

heating involves the realistic heating processes for practical applications. The non-uniform boundary conditions

typically represented by sine functions, are found applications involving glass melting [8a]. The interaction of the

shear driven flow due to the lid motion and natural convective flow due to the buoyancy effect is quite complex andwarrants comprehensive analysis to understand the physics of the resulting flow and heat transfer process.

1.4 Motivation behind the selection of problem

We know that a scalable numerical model to solve the unsteady incompressible Navier–Stokes equations has been

developed using the Galerkin finite element method. Tanmay Basak, Ram Satish Kaluri, and A. R. Balakrishnan[9]

studied the effects of Thermal Boundary conditions on Entropy generation during Natural Convection . By using

penalty finite element analysis ,Mixed convection flows in a lid-driven square cavity filled with porous medium are

studied numerically by Tanmay Basak, S. Roy, A.J. Chamkha [10]. by a Peclet number based analysis for lid-driven

porous square cavities with various heating of bottom wall .

Due to the relatively inexpensive high speed computers, numerical simulation approach, such as computational fluid

dynamics (CFD), is widely adopted for investigating realistic and research problems and validation of several papers

solved by FEM. Numerical simulation has full control on computing the parameters of problems of differentcomplexities. Therefore, it is able to provide a compromising solution among cost, efficiency and complexity to

engineering problems. Although high speed computers and robust numerical techniques have been developed rapidly,

the computation of turbulence at high Reynolds number using direct numerical simulation (DNS) is too expensive for

practical problems. The large-eddy simulation (LES) is an alternative that demands relatively less computational load.

However, it still requires huge amount of computation resources for simulations conducted on sequential computers.

The recent advance of supercomputers provides a possibility for conducting these large scale computations. Sequential

computer codes could be parallelized directly by compilers but it is unable to fully utilize supercomputers. Therefore,

innovative parallel solution techniques are necessary for exploring the power of parallel computing. To facilitate

parallel computation the domain is usually divided into several sub-domains according to the structure of the mesh.

1.5 Main objectives of the work

The investigation is carried out in a two dimensional lid driven rectangular enclosure filled with fluid of different

Prandlt number. The side walls are kept adiabatic and the bottom wall of the cavity is kept at uniform/non-uniform heatflux. The cooled top wall having constant temperature will move with a constant velocity. The specific objectives of the

present research work are as follows: To study the influence of grid sizes, computational time steps on the convergence

of the governing equation codes and to catch the oscillations in the contours of different governing parameters; we

categorized problem into three models; firstly, a coarse grids i.e. 12 x 12 model secondly, a medium grids i.e. 32 x 32

model and finally, a fine grids i.e. 52 x 52 model. To study the x and y velocity component contours and steam

functions in the conservation of mass and momentum; we employed Non-dimensional Navier-Stokes solver with

Reynolds number, Prandl number, Grashoff number and Richardson number for the simulation of the problem. And

whole study is mainly concentrated on two different cases like Non-isothermal forced convection fluid flow ( where

Grashoff number is almost negligible because buoyancy induced flow exists), Mixed Convection fluid flow ( where

buoyancy as well as inertia induced; the Grashoff number is non zero value).

2. LITERATURE REVIEW There have been many investigations in the past on mixed convective flow in lid-driven cavities. Many different

configurations and combinations of thermal boundary conditions have been considered and analyzed by various

investigators. Torrance et al. [11] investigated mixed convection in driven cavities as early as in 1972. Papaniclaou and

Jaluria [12] carried out a series of numerical studies to investigate the combined forced and natural convective cooling

of heat dissipating electronic components, located in rectangular enclosures, and cooled by an external through flow of

air. The results indicate that flow patterns generally consists of high of low velocity re-circulating cells because of

buoyancy forces induced by the heat source. Koseff and Street [13] studied experimentally as well as numerically the

recirculation flow patterns for a wide range of Reynolds (Re) and Grashof (Gr) numbers. Their results showed that the

three dimensional features, such as corner eddies near the end walls, and Taylor- Gortler like longitudinal vortices,

have significant effects on the flow patterns for low Reynolds numbers. Khanafer and Chamakha [14] examined

numerically mixed convection flow in a lid-driven enclosure filled with a fluid saturated porous medium and reported

on the effects of the Darcy and Richardson numbers on the flow and heat transfer characteristics. G. A. Holtzman et. al[15] have studied laminar natural convection in isosceles triangular enclosures heated from below and symmetrically

cooled from above..Moallemi and Jang [16] numerically studied mixed convective flow in a bottom heated square

driven cavity and investigated the effect of Prandtl number on the flow and heat transfer process. They found that the


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effects of buoyancy are more pronounced for higher values of Prandtl number. They also derived a correlation for the

average Nusselt number in terms of the Prandtl number, Reynolds number, and Richardson number. Mohammad and

Viskanta [17] performed numerical investigation and flow visualization study on two and three-dimensional laminar

mixed convection flow in a bottom heated shallow driven cavity filled with water having a Prandtl number of 5.84.

They concluded that the lid motion destroys all types of convective cells due to heating from below for finite sizecavities. They also implicated that the two-dimensional heat transfer results compare favorably with those based on a

three-dimensional model for Gr/Re< 1. Later, Mohammad and Viskanta [18] experimentally and numerically studied

mixed convection in shallow rectangular bottom heated cavities filled with liquid Gallium having a low Prandtl number

of 0.022. They found that the heat transfer rate is rather insensitive to the lid velocity and an extremely thin shear layer

exists along the major portion of the moving lid. The flow structure consists of an elongated secondary circulation that

occupies a third of the cavity. Mansour and Viskanta [21] studied mixed convective flow in a tall vertical cavity where

one of the vertical sidewalls, maintained at a colder temperature than the other, was moving up or downward thus

assisting or opposing the buoyancy. They observed that when shear assisted the buoyancy a shear cell developed

adjacent to the moving wall while the buoyancy cell filled the rest of the cavity. When shear opposed buoyancy, the heat

transfer rate reduced below that for purely natural convection. Iwatsu et al. [22] and Iwatsu and Hyun [23] conducted

two-dimensional and three-dimensional numerical simulation of mixed convection in square cavities heated from the

top moving wall. Mohammad and Viskanta [24] conducted three-dimensional numerical simulation of mixedconvection in a shallow driven cavity filled with a stably stratified fluid heated from the top moving wall and cooled

from below for a range of Rayleigh number and Richardson number. Prasad and Koseff [25] reported experimental

results for mixed convection in deep lid driven cavities heated from below. In a series of experiments which were

performed on a cavity filled with water, the heat flux was measured at different locations over the hot cavity floor for a

range of Re and Gr. Their results indicated that the overall (i.e. area-averaged) heat transfer rate was a very weak

function of Gr for the range of Re examined (2200 < Re < 12000). The data were correlated by Nusselt number vs

Reynolds number, as well as Stanton number vs Reynolds number relations. They observed that the heat transfer is

rather insensitive to the Richardson number. Hsu and Wang [26] investigated the mixed convective heat transfer where

the heat source was embedded on a board mounted vertically on the bottom wall at the middle in an enclosure. The

cooling air flow enters and exits the enclosure through the openings near the top of the vertical sidewalls. The results

show that both the thermal field and the average Nusselt number depend strongly on the governing parameters, position

of the heat source, as well as the property of the heat-source-embedded board. Aydin and Yang [27] numerically studied

mixed convection heat transfer in a two dimensional square cavity having an aspect ratio of 1. In their configuration

the isothermal sidewalls of the cavity were moving downwards with uniform velocity while the top wall was adiabatic.

A symmetrical isothermal heat source was placed at the otherwise adiabatic bottom wall. They investigated the effects

of Richardson number and the length of the heat source on the fluid flow and heat transfer. Shankar et al. [28]

presented analytical solution for mixed convection in cavities with very slow lid motion. The convection process has

been shown to be governed by an inhomogeneous bi harmonic equation for the stream function. Oztop and Dagtekin

[29] performed numerical analysis of mixed convection in a square cavity with moving and differentially heated

sidewalls. Sharif [30] investigates heat transfer in two-dimensional shallow rectangular driven cavity of aspect ratio 10

and Prandtl number 6.0 with hot moving lid on top and cooled from bottom. They investigated the effect of Richardson

number and inclination angle. G. Guo and M. A. R. Sharif [31] studied mixed convection in rectangular cavities at

various aspect ratios with moving isothermal sidewalls and constant heat source on the bottom wall. They plotted the

streamlines and isotherms for different values of Richardson number and also studied the variation of the average Nu

and maximum surface temperature at the heat source with Richardson number with different heat source length. Theysimulated streamlines and isotherms for asymmetric placements of the heat source and also the effects of asymmetry of

the heating elements on the average Nu and the maximum source length temperature.

3.PHYSICAL DESCRIPTION OF THE PROBLEM AND MODELS FOR SIMULATIONS

To study the effect of non dimensional parameters in fluid flow characterization; the lid driven cavity is one of the most

widely used benchmark problems to test steady state incompressible fluid dynamics codes. Our interest will be to

present this problem as a benchmark for the steady and unsteady state solution. In order to demonstrate the grid

independence, code validation and other details like time step, grid size and steadiness criteria; we taken 2D Cartesian

(x, y as horizontal and vertical components) rectangular domain of size 2L x L (2 unit x 1 unit for simplicity of the

problem) with bottom, left and right boundaries as solid walls stationary; whereas top wall is like a long conveyor-belt,

moving horizontally with a constant velocity (Uo=1unit for the simplicity of the problem) shown in Fig.1. in this

problem the non-isothermal forced/mixed convective flow is generated due to difference in the temperature of the bottom as compared to other walls of the cavity; bottom wall at TH and other walls at TC. This corresponds to a non-

dimensional temperature, θ=(T-TC)/(TH-TC), equal to 1 at bottom and 0 at other walls of the lid driven cavity.

The buoyancy induced flow, taken as negligible (with Gr=0) for forced convection, is considered finite for the mixed


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convection problem Two dimensional steady, mixed convection heat transfers in a two-dimensional rectangular cavity

with constant heat flux from heated bottom wall while the isothermal moving top wall has been studied numerically

Fig 1

4.MATHEMATICAL FORMULATION

A two-dimensional rectangular cavity is considered for the present study with the physical dimension as shown in

Fig.1. The bottom wall of the cavity is maintained at a uniform temperature/non uniform temperature varying in a

sinusoidal manner and the upper wall is well insulated. The two vertical walls are maintained at lesser(cold)

temperature . It may be noted that the bottom wall is maintained at a higher temperature to induce buoyancy effect. The

top wall is assumed to slide from left to right with a constant speed U0. The flow is assumed to be laminar and the fluid

properties are assumed to be constant except for the density variation which is modelled according to Boussinesq

approximation while viscous dissipation effects are considered to be negligible. The viscous incompressible flow and

the temperature distribution inside the cavity are governed by the Navier–Stokes and the energy equations, respectively.

The aim of the current work is to investigate the steady state solutions and hence, we have considered the time

independent differential governing equations. Similar procedure was also followed in the recent work on mixed

convection. A number of earlier works was based on steady state solutions which were obtained via steady

mathematical model. The governing equations are non-dimensionalized to yield


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The dimensionless variables and parameters are defined as follows

Here x and y are the distances measured along the horizontal and vertical directions, respectively; u and v are the

velocity components in x and y directions, respectively; T denotes the temperature; p is the pressure and ρ is the

density;

Th and Tc are the temperature at the hot and cold walls, respectively; L is the length of the side of the square cavity; X

and Y are dimensionless coordinates varying along horizontal and vertical directions, respectively; U0 is the velocity

of the upper wall; U and V are dimensionless velocity components in the X and Y directions, respectively; θ is the

dimensionless temperature; P is the dimensionless pressure; Gr, Re and Pr are Grashoff, Reynolds and Prandtl

number, respectively.

4.1 Solution Methodology

First of all we solve (a) The FVM Continuity equation. (b)Then the Transport Equation by assigning Grids .(c) Then

we solve the X – Y Momentum lost by the fluid in the C.V.(as we choose). (d) Next we formulate a method of

calculating the internal energy lost by the fluid in the C.V.(e) Then we take up the calculation of Viscous force acting

in X & Y direction .(f)Then we Calculate the Conduction Heat Transfer. (g) Then we calculate the Pressure forces in X

& Y direction .(h) Then we calculate the X & Y Momentum Equation. (I) Then we calculate the Energy Equations .All

these calculations are pretty simple if we can assign the Staggered grid properly.

Here is also a method of calculating a code for temperature.

Further the method of pressure correction is of prime Importance . we can adopt either semi –Implicit or semi –

Explicit method of solving this problem.

4.2 Philosophy of Pressure Correction Method

•

We predict a velocity using the previous time step pressure

•

But it does not satisfies the mass balance.

•

Thus, we need to correct the velocity which is only possible by varying/correcting the pressure inside the CV based

on the mass imbalance

•

If more mass of fluid is getting out as compared to that coming in, then the pressure inside the C.V should

decrease, i.e, pressure correction should be negative and vice-versa.

•

By tuning the pressure in each cell center based on the magnitude and direction of mass imbalance, the predicted

velocity will change and new mass imbalance will be generated.


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The pressure and velocity correction is continued iteratively till mass balance and a divergence free velocity field is

obtained.

Also note that : As there is no explicit equation for the calculation of pressure, there are two class of methods for the

calculation of pressure from the continuity equation

Pressure Correction Method

Projection Method.

The pressure and velocity correction is continued iteratively till a divergence free velocity field is obtained. Then

only the R.H.S of above equations becomes the velocity and pressure for the next time step

4.3 Semi-Explicit Method: Solution Algorithm

5. RESULT AND DISCUSSION The computational domain consists of 12×12 main grid points. Numerical solutions are obtained for various values of

Ri = 0 - 10, Pr = 0.71–10 and Re = 10–2x103 with uniform and non –uniform heating of the bottom wall where the

two vertical walls are cooled and the top wall is well insulated with a horizontal velocity, U=1. To ensure the

convergence of the numerical solution to the exact solution, the grid sizes can be optimized and then the results can

be presented in which are independent of grid sizes(also called Grid independent study) .


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SET 1:(Here fluid is Air)

Fig2(a) &(b) Re No.: 10,Pr No : 0.71, Gr No; 0, Ri=0 Temperature & Steam Contour Plots



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Discussion: Analysing the three Fig namely 2( a) & (b),3(a) & (b), 4(a) & (b) for Temperature Contour plot and Stream

function plot (a) shows that the effect of lid-driven flow predominates the forced convection for Gr = 0, Pr = 0.71 and

Re = 10, i.e fig 2.for the value of Ri = 0, the effect of buoyancy force is gradually weaker as compared to the lid drivenforce. All amount of fluid is pulled up towards the left corner due to drag force created by the motion of the upper lid in

fig 3 & 4 as observed from the temperature contour plots. The clockwise circulation is found to be predominant as

compared to anticlockwise circulation. The primary circulation occupies the major portion of the cavity which is also

observed from the plots. As a result, the stream function contours near to the upper lid are not perfectly in oval shape is

observed and in some cases circular.

The value of Reynolds number is very low for Fig 2 and heat transfer is conduction dominated within the cavity. Due to

dominant conduction mode, all the isotherms are smooth symmetric curve that span the entire cavity. In Fig.2 (b) it

shows the stream contour plots for Ri = 20. That means strength of buoyancy approaches the lid driven force . The

max value is 0.14 The stream lines at the centre of the cavity are mostly elliptic in shape and far away from the centre

are changes it shape. Also interesting to note is that for higher values of Reynolds number keeping the Prandlt number

fixed and Grashoff number fixed ,the maximum value of stream function for the circulation cell is 0.13 and the size of

the circulation cells is larger than that with Re = 100. Due to enhanced circulation and thermal mixing at the right half,

the isotherms are pushed towards the left-side wall. The dominant natural convection is attributed to the asymmetric

isotherms.

SET2 ( Here Fluid is Water)


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Fig5(a) &(b) Re No.: 10,Pr No : 7, Gr No; 0, Ri=0Temperature & Steam Contour Plots



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Discussion: Analysing the three Fig namely 5( a) & (b),6(a) & (b), 7(a) & (b) for Temperature Contour plot and Stream

function plot (a) shows that the effect of lid-driven flow predominates the forced convection for Gr = 0, Pr = 7 and Re =

10, i.e fig 5.for the value of Ri = 0, the effect of buoyancy force is gradually weaker as compared to the lid driven force. Note that here also all amount of fluid is pulled up towards the left corner due to drag force created by the motion of the

upper lid in fig 6 & 7 as observed from the temperature contour plots. However the contour lines are more uniform here

in SET 2 compared to SET 1.The clockwise circulation is found to be predominant as compared to anticlockwise

circulation. The primary circulation occupies the major portion of the cavity which is also observed from the plots. As a

result, the stream function contours near to the upper lid are not perfectly in oval shape is observed and in some cases

circular.

The value of Reynolds number is very low for Fig 5 and heat transfer is conduction dominated within the cavity. Due to

dominant conduction mode, all the isotherms are smooth symmetric curve that span the entire cavity. In Fig5 (b) it

shows the stream contour plots for Ri = 20. That means strength of buoyancy approaches the lid driven force . The

max value is 0.13(not much change as compared to SET 1). The stream lines at the centre of the cavity are mostly

elliptic in shape and far away from the centre are changes it shape. Also interesting to note in SET 2 is that the

temperature contour lines are only predominant only near the bottom wall because the fluid here is water for fig 6(a) .Due to enhanced circulation and thermal mixing at the right half, the finer isotherms(compared to SET 1) are pushed

towards the left-side wall. The dominant natural convection is attributed to the asymmetric isotherms.

SET3( Here Fluid is SAE oil)

Fig8(a) &(b) Re No.: 10,Pr No : 10, Gr No;100, Ri=1 Temperature & Steam Contour Plots


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Fig9(a) &(b) Re No.: 10,Pr No : 10, Gr No;1000, Ri=10 Temperature & Steam Contour Plots

Fig10(a) &(b) Re No.: 2000,Pr No : 10, Gr No:10000, Ri=0.0 Temperature & Steam Contour Plots


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Fig11(a) &(b) Re No.: 2000,Pr No : 10, Gr No: 4E6, Ri=1.0 Temperature & Steam Contour Plots

Fig12(a) &(b) Re No.: 2000,Pr No : 10, Gr No;4E8, Ri=100 Temperature & Steam Contour Plots


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Discussions: In SET3 more number of cases have been taken up. As Ri increases, the effect of buoyancy increases

leading to an increase in the strength of circulation. In Fig 8(a) & 9(a) due to increase in circulation strength, the

isotherms are stretched along the side walls and heat is transferred mostly by convection for higher value of Pr. The

increased effect of Re for fig(10)- fig12 reveals from the temperature contour lines which have also been studied in the

present investigation for fixed value of Pr and Ri that the heat transfer rate is very high at the edges of the bottom walland it decreases to zero at the Center of the cavity.The maximum value of stream contour lines is lesser as compared

to Prandlt No 7 ( or SET 2). The shape is more elliptical and uniform than the previous cases(Set 1). It is also

observed that the effect of natural convection decreases and forced convection increases with the increase of Re. It has

also been observed that for higher value of Pr, the effect of heating is more pronounced near the bottom and left walls

as the formation of thermal boundary layers is restricted near the bottom and left wall for uniform heating case.

Now we take up some results for non- uniform heating of bottom wall.

Fig13(a) &(b) Re No.: 10,Pr No : 0.71, Gr No; 100, Ri=1.0 Temperature & Steam Contour Plots

Fig14(a) &(b) Re No.: 100,Pr No : 0.71, Gr No; 100, Ri=0.0 Temperature & Steam Contour Plots


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Fig15(a) &(b) Re No.: 2000,Pr No : 0.71, Gr No; 4E8 , Ri=100 Temperature & Steam Contour Plots

16(a) &(b) Re No.: 2000,Pr No : 7 Gr No; 4E7, Ri=10.0 Temperature & Steam Contour Plots


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Fig17(a) &(b) Re No.: 2000,Pr No 10, Gr No; 4E7 , Ri=10.0 Temperature & Steam Contour Plots

Discussions: Fig. 13a–b to 17 a-b display flow and temperature distributions for non-uniform heating of bottom wall.

For Re No.: 10,Pr No : 0.71, Gr No; 100, Ri=1.0 what we observe is interesting that temperature profiles are smooth

and symmetric based on conduction dominant heat transport satisfying the sinusoidal equation. The stream lines also

describe an uniform elliptical pattern covering most of the cavity.For the case of Re No.: 100,Pr No : 0.71, Gr No; 100, Ri=0.0 i.e. with increase of Reynolds number and the

temperature contour lines The lid velocity plays a dominant role and the temperature profile is found to be asymmetric.

The stronger clockwise circulation cell near the right wall leads to greater thermal mixing near the right half. In

addition, the circulation cell near the left wall is weaker. Therefore, the thermal mixing is weak near the left wall and

stronger thermal boundary layer is found to be developed near the lef t wall. It is found that |ψ|max is around 0.11 for Pr

=0.71 for Fig 13 whereas that is 0.12 for Fig 14.

For the cases of Re No.: 2000,Pr No : [0.71, 7,10,] Gr No; 4E8 , Ri=100. with increase of Reynolds number in range of

2000 , and high value of Grashoff’s number The lid velocity no longer plays a dominant role and the temperature

profile is found to be symmetric and the lines move towards the top lid because Forced convection effects are usually

insignificant and natural convection plays a dominant role. Three types of Prandlt number has been studied here and

what we observe is that with increase of Prandlt number, the temperature contour lines are concentrated near the top

lid. |ψ|max is around 0.029 for Pr =0.71 for Fig 15 whereas that is 0.03 for Fig 16. The convective heat transport is

dominant at a high Richardson number (Ri =102). The fluid is cooler near the top wall and thus cold fluid isrecirculated in the zone near the top wall around the center of vortex which corresponds to |ψ|max ≈0.03. Similar to

earlier cases, dominant effect on con-vective heat transport reduces the local buoyancy effect and the flow strength with

Pr =10 (Pe =103) is found to be smaller than that with Pr =0.7 (Pe =70).

6.VALIDATIONS

The computer code that is has been run has been validated with the solutions are available in the literatures. There are

some possibilities of validating the numerical code. One possibility is to compare the numerical results obtained by our

code with benchmarks available in the technical literature according to different works by different Authors. Another

option is to simulate a similar problem investigated by other authors with well accepted available results. Here the code

has been validated by using lid-driven cavity benchmark problem by O. Botella and R. Peyret (1998). Further, the

computational model is validated for mixed convection heat transfer by comparing the results of correlation on mixed

convection in lid-driven square cavity with uniform heat flux in bottom wall performed by Tanmay Basak and I. Pop

(2009). The right, top and left walls are insulated. The change of cavity dimensions does not change the results .


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7.FINAL CONCLUSIONS

A numerical investigation on mixed convection in a rectangular cavity with various boundary conditions was carried

out using a finite volume method in a staggered grid. The prime objective of the investigation is to study the effect of

uniform and non uniform heating of the bottom, on the flow and heat transfer characteristics due to mixed convectionin cavity. It is evident from SET 1-3, i.e. Figs. 2 – 12(Uniform heating of bottom wall) which are for fixed Pr, for

every SET ( Air, water, SAE Oil) the strength of circulation increases with the increase in Ri. As Ri increases, the

effect of buoyancy increases leading to an increase in the strength of circulation. Due to increase in circulation strength,

the isotherms are stretched along the side walls and heat is transferred mostly by convection for higher value of Pr. The

effect of Re has also been studied in the present investigation for fixed value of Pr and Ri in all the three SET’s. It is

observed that the effect of natural convection decreases and forced convection increases with the increase of Re. It has

also been observed that for higher value of Pr, the effect of heating is more pronounced near the bottom and left walls

as the formation of thermal boundary layers is restricted near the bottom and left wall for uniform heating cases. The

heat transfer rate is very high at the edges of the bottom wall and it decreases at the center of the cavity.

Further laminar convection in a two-dimensional, horizontally driven rectangular enclosure with a prescribed

sinusoidal temperature heat source mounted on the bottom wall is simulated numerically in this work. Mixed

convection arises as the buoyancy-induced cold flow from the source interacts with an externally induced cold air flow

from the surroundings. The heat transfer results explain the importance of the non-dimensional parameters like

Reynolds number and Richardson number in the natural and mixed convection regime for both uniform and non-

uniform heat source. The effects of these parameters on the flow fields are also investigated in detail for all the cases

and results and discussions enumerated . The governing parameter affecting heat transfer is the Richardson number

and Reynolds number. As we know that for Ri > 1, the heat transfer is dominated by natural convection. When Ri < 1,

the flow and heat transfer are dominated by forced convection. The mixed regime is obtained when Ri = 1. The results

show the same.

8.ACKNOWLEDGEMENT

The author wishes to acknowledge Department of Mechanical Engineering, IIT Bombay, India, for technical help

throughout this work.

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AUTHOR[SHANTANU DUTTA] He Received his B E (Mechanical) degree from Jalpaiguri Govt.

Engineering college, North Bengal University in 1996. He received his M Tech degree in

Mechanical Engineering( Design and Production Specalisation) from National Institute of

Technology Durgapur in 2014. He also obtained his Post graduate degree in Materials and

Logistics Management from , Indian Institute of Materials Management Mumbai in 2002. Currently

he is Pursuing his Ph.d program in Micro Fluidics and Computational Heat Transfer from NIT Durgapur. He has got

14 years of Industry experience and 5 ½ years of Teaching Experience . Further he was associated with research

projects in IIT Kharagpur in the past on Fluid Mechanics area.

numerical analysis of non-isothermal forced convection fluid flow and mixed convection fluid flow in...

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