Lid Driven Cavity CFD Simulation Report by S N Topannavar

by Mr.S.N.Topannavar, sub-center: KIT Kolhapur 1

A final assignment Report on

“Solution of Non-dimensional Navier-Stokes

equations for Flow and Heat transfer by using Scilab

CFD codes in LID DRIVEN CAVITY and study of

variation of non-dimensional, convergence criteria

parameters with different grid structure”

Submitted by

S.N.Topannavar [email protected]

Cell: +91 9480397798

Sub center: KIT, Kolhapur

Ten Day ISTE Main Workshop on Computational

Fluid Dynamics (CFD)

Conducted by

Indian Institute of Technology Bombay

mailto:[email protected]


Index Chapter Content Page No.

I Introduction 3-4

II Literature survey with conclusions 5-13

III Objectives of the work 14

IV Physical description of the problem and models for simulations 15-18

V Mathematical modeling with boundary conditions 19-20

VI Validation study 21

VII Results and Discussions 22-91

VIII Scilab CFD codes used in the problem and algorithm 92-125

IX Conclusion 126

X References 127


Chapter-I

Introduction Over the last three decades, the so-called lid-driven cavity flow problem has received considerable attention

mainly because of its geometric simplicity, physical abundance, and its close relevance to some fundamental

engineering flows. While some fundamental flow phenomena have become clear to us through two-

dimensional solutions. The recent progress in numerical analyses and computer hardware have made it

possible to numerically analyze unsteady flow problems by solving their corresponding Navier-Stokes

equations with a large number of grid points within a three-dimensional domain. In a parallel development, a

considerable number of experimental studies on this problem have been done since the early 1980s.

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.

Numerical simulation has full control on computing the parameters of problems of different complexities.

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

Flow in a 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) square domain of size L x L (1 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).

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 Four different cases like Isothermal fluid flow

(all walls of the are at same temperature in all time steps),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) and Natural

convection fluid flow ( where inertia is negligible therefore Reynolds not taken into consideration)

The study of the stability of two dimensional vortex flows of a viscous fluid is one of the fundamental

problems of hydrodynamics, which concerns the problems of control of separation flows. The topological

characteristics of a flow can be determined on the basis of solution of the Navier–Stokes equations.

However, the number of problems that can be exactly solved in this way is limited; this being so, numerical

methods are used in the majority of cases. The problem on the flow of an incompressible viscous fluid in a

rectangular cavity with a moving wall is a classical fluid-mechanics problem with closed boundaries. The


main structural peculiarities of this flow are characteristic of other separation flows having a more complex

geometry; therefore, solution of the problem on the indicated flow is used for testing and comparison of

different numerical methods of integrating the Navier–Stokes equations. Comprehensive data on the vortex

structure and characteristics of a flow in a rectangular cavity. On the basis of systematization and analysis of

the data, steady state convergence criteria 10-4

have been adopted for estimating the quality of the discrete

model used.

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. But we facilitated Microsoft Windows XP professional version 2002

operating system, Inter(R) Core(TM) Duo CPU T6670 @ 2.20 GHz processor with small memory of 1.96G,

2.19GHz speed RAM Dell Vastro lap top for large scale computations. Due to time restriction of submission

of the report we skipped some fine grid computation of simulations because, it is observed that some

simulations will take one and half day computation time.

.

mailto:[email protected]


Chapter-II

Literature survey with conclusions [1] A numerical study of the vertical flow structure in a confined lid-driven cavity which is defined by a

depth-to-width aspect ratio of 1:1 and a span-to-width aspect ratio of 3:1 SAR (Spanwise Aspect Ratio) ( L :

B . 1; 2; 3). A simple discretization technique ; third-order QUICK (Leonard 1979) upwind scheme,

formulated on the non-uniform basis, to the nonlinear advective fluxes was applied to study carefully

examined the computed data that the useful to gain an in-depth knowledge of the complex interactions

among secondary eddies, primary eddies, and spiraling span wise motions. Chief of conclusions drawn from

this study is to explain how the secondary eddies are intimately coupled with the primary re-circulating

flow. Also enlighten in this paper why spiraling vortices inside the upstream secondary eddy tend to

destabilize the incompressible flow system and aid development of laminar instabilities. Prior to describing

the appearance of TGL (Taylor-GoÈrtler) vortices are studied in detail how eddies of different sizes and

attributes are intimately coupled. And same is permitted a systematic approach to understanding the

complex interaction among spiraling eddies. The separation surface plotted in this paper furthermore helps

to show that fluid flows present in the narrow wavy trough of the separation surface have a higher

propensity to develop into TGL vortices.

Conclusions:

[1] The geometry of the cavity examined is extraordinarily simple; the flow physics in the cavity are

nevertheless rich. The physical complexity is attributable to the eddies which are characterized as possessing

different sizes and characteristics. Also, how interaction proceeds among the eddies is crucial to the

development into laminar instabilities. In the entire flow evolution, the transport mechanism is rooted

largely in the spiraling nature of the flow motion established inside the secondary eddies and, of course, in

the primary core. According to the finite volume solutions concluded with some important findings from the

numerical simulation. The three-dimensional lid-driven cavity flow is manifested by the presence of a span-

wise velocity component which arises due to the presence of two vertical end walls.

Accompanying the span-wise motion, the flow exhibiting the dominant recirculation flow pattern is prone to

spiral. It is interpreted that the presence of USE particles, which are engulfed from regions fairly near the

two end walls into the primary core and then spiral monotonically towards the symmetry plane, as being the

main cause leading to the flow instability because the two flow streams moving in opposite directions tend

to collide with each other at the symmetry plane. This instability causes the surface separating the primary

core and the upstream secondary eddy to detach from the upstream side wall. It is this distorted detachment

which disrupts the well-balanced force between the centrifugal and pressure-gradient forces established

inside the primary re-circulating cell. This paves the way for the onset of Taylor-GoÈrtler vortices.

As the end wall is approached, particles in the downstream secondary eddy begin to be engulfed into the

primary core and this is followed by suction of particles in the upstream secondary eddy, which is closer to

the end wall, into the primary core through the spiral-saddle point. There exists a higher possibility that

instabilities will result at spatial locations where the width of the upstream secondary eddy becomes

appreciably larger than the width of the downstream secondary eddy. Computational experience from this

study reveals that the size of the upstream secondary eddy and the contour lines of zero span-wise velocity at

the surface, and the separation surface are closely related.. In the vicinity of the distorted v = 0 contour

surface, the sign-switching span-wise velocity induces a free-shear vortex. The pressure field established to

support the existence of this vortex further affects the boundary layer of the outward-running spiraling flow

in the sense that a wall-shear vortex is formed near the floor of the cavity. This pair of well-established

vortices, referred to as Taylor-GoÈrtler vortices, bursts from the spatial location which has the local

maximum kinetic energy.


[3] The work on stabilized finite element formulation proposed by Tezduyar applied to solve steady

viscoplastic incompressible flows on unstructured grids. The formulation, originally proposed for

Newtonian fluids, allows that equal-order-interpolation velocity-pressure elements are employed,

circumventing the Babuska-Brezzi stability condition by introducing two stabilization terms. The first term

used is the streamline upwind/Petrov-Galerkin (SUPG) introduced by Brooks and Hughes and the other one

is the ressurestabilizing/ Petrov-Galerkin (PSPG) stabilization proposed initially by Hughes for Stokes flows

and generalized by Tezduyar to the Navier–Stokes equations. The inexact-Newton methods associated with

iterative Krylov solvers have been used to reduce computational efforts related to non-linearities in many

problems of computational fluid dynamics, offering a trade-off between accuracy and the amount of

computational effort spent per iteration.

A parallel edge-based solution of three dimensional viscoplastic flows governed by the steady Navier–

Stokes equations is presented. The governing partial differential equations are discritized using the SUPG

(streamline upwind/Petrov-Galerkin)/PSPG (pressure stabilizing/Petrov-Galerkin) stabilized finite element

method on unstructured grids. The highly nonlinear algebraic system arising from the convective and

material effects is solved by an inexact Newton-Krylov method. The locally linear Newton equations are

solved by GMRES with nodal block diagonal pre-conditioner. Matrix-vector products within GMRES are

computed edge-by-edge (EDE), diminishing flop counts and memory requirements. A comparison between

EDE and element-by-element data structures is presented. The parallel computations were based in a

message passing interface standard. Performance tests were carried out in representative three dimensional

problems, the sudden expansion for power-law fluids and the flow of Bingham fluids in a lid-driven cavity.

Results have shown that edge based schemes requires less CPU time and memory than element based

solutions. The SUPG/PSPG finite element formulation with the inexact nonlinear method.

Conclusions:

[3] The nonlinear character due to the non-Newtonian viscous and convective terms of the Navier–Stokes

equations was treated by an inexact-nonlinear method allowing a good tradeoff between convergence and

computational effort. At the beginning of the solution procedure the large linear tolerances produced fast

nonlinear steps, and as the solution progresses, the inexact nonlinear method adapts the tolerances to reach

the desired accuracy. The linear systems of equations within the nonlinear solution procedure were solved

with the nodal block diagonal preconditioned GMRES. An edge-based data structure was introduced and

successfully employed to improve the performance of the matrix-vector products within the iterative solver.

The results showed that the computing time when using EDE data structure was on the average 2.5 times

faster than for those problems using standard EBE. The computations were performed in a message passing

parallelism environment presenting good speedup and scalability.

[4] A general, efficient, accurate and reliable algorithm developed with emphasis on high Reynolds number

flows that still maintains a simple algorithmic structure and which is not hampered by the diffusive time step

limit. A new semi-implicit finite element algorithm for time-dependent viscous incompressible flows. The

algorithm is of a general type and can handle both low and high Reynolds number flows, although the

emphasis is on convection dominated flows. An explicit three-step method is used for the convection term

and an implicit trapezoid method for the diffusion term. The consistent mass matrix is only used in the

momentum phase of the fractional step algorithm while the lumped mass matrix is used in the pressure

phase and in the pressure Poisson equation. An accuracy and stability analysis of the algorithm is provided

for the pure convection equation. Two different types of boundary conditions for the end-of-step velocity of

the fractional step algorithm have been investigated. Numerical tests for the lid-driven cavity at Re = 1 and

Re= 7500 and flow past a circular cylinder at Re =100 are presented to demonstrate the usefulness of the

method.

Finite element method for predicting time-dependent viscous incompressible flows over a wide range of

inertial conditions has been presented. The method is mainly aimed at solving convection dominated flows

and employs an explicit three-step algorithm for the convection terms, which gives not only high accuracy


but also high efficiency since it allows large Courant numbers. To further improve accuracy for this kind of

flows, the consistent mass matrix has also been included. Two variants of the method have been used; one

fully explicit scheme with lumped mass matrix and one semi-implicit scheme with the consistent mass

matrix in the momentum phase of the fractional step algorithm but with the lumped mass matrix in the

pressure phase and in the pressure Poisson equation. The latter of these variants requires an extra system of

linear equations to be solved at every time step. This was done in a simple and efficient way by using just a

few Jacobi iterations and it was shown that this worked well even for very low Reynolds number flows. Two

different kinds of velocity boundary conditions for the end-of-step velocity of the fractional step algorithm

have been investigated, one which excludes checker boarding (type 1 B.C.) and one simpler version which

does not exclude the checkerboard mode (type 2 b.c.). The type 1 B.C. was found to be slightly more

accurate and it was also found to initiate the vortex shedding behind the circular cylinder much earlier than

the type 2 b.c.

Conclusions:

[4] The simple algorithmic structure and that no extra terms or new higher-order derivatives are needed. In

spite of the simplicity, the method is of a general nature and can easily handle complex geometries.

Numerical tests show good agreement with other numerical solutions and experimental data and suggest that

the proposed method is competitive in terms of both accuracy and efficiency.

[5] Fixed point iteration idea employed to linearize the coarse and fine scale sub-problems that arise in the

variational multi scale frame work and it lead to a stabilized method for the incompressible Navier–Stokes

equations. In the current work we present a consistent linearization of the nonlinear coarse and fine scale

sub-problems, and substitution of the fine scales extracted from the fine-scale problem into the coarse-scale

variational form leads to the new stabilized method. The solution of the fine-scale or the sub-grid scale

problem which is an integral component of the proposed procedure for developing stabilized methods

automatically yields an explicit definition of the stabilization operator τ. Another significant contribution of

the paper is a numerical technique for evaluating the advection part of the stabilization operator τ that brings

in the notion of up-winding in the resulting method. Presented a variational multi-scale-based stabilized

formulation for the incompressible Navier–Stokes equations. A novel feature of our method is that fine

scales are solved in a direct nonlinear fashion. Consistent linearization of the nonlinear equations in the

context of the variational multi scale framework leads to the design of the stabilization terms in the new

method

A variational multi-scale residual-based stabilized finite element method for the incompressible Navier–

Stokes equations. Structure of the stabilization terms is derived based on the two level scale separation

furnished by the variational multi-scale framework. A significant feature of the new method is that the fine

scales are solved in a direct nonlinear fashion, and a definition of the stabilization tensor τ is derived via the

solution of the fine-scale problem. A computationally economic procedure is proposed to evaluate the

advection part of the stabilization tensor. The new method circumvents the Babuska–Brezzi condition and

yields a stable formulation for high Reynolds number flows. A family of equal-order pressure-velocity

elements comprising 4- and 10-node tetrahedral elements and 8- and 27-node hexahedral elements is

developed. Convergence rates are reported and accuracy properties of the method are presented via the lid-

driven cavity flow problem. Presented the strong form and the classical weak form of the incompressible

Navier–Stokes equations. Consistent linearization of the nonlinear equations performed in the vartiational

multi-scale setting leads to the new multi-scale /stabilized formulation that is developed. The structure of the

stabilization tensor and a numerical scheme to evaluate its advection part are presented; a convergence study

for a family of 3D tetrahedral and hexahedral elements. An extensive set of numerical simulations of lid-

driven cavity flows for various Reynolds number are also presented.


Conclusions:

[5] The VMS based stabilized form possesses additional stabilization terms than are present in the classical

stabilization methods alone. An important feature of the new method is that a definition of the stabilization

operator τ appears naturally via the solution of the fine-scale problem. This stabilization operator is a second

order tensor and leads to a full matrix that brings in cross coupling effects in the stabilization terms. A

computationally economic scheme is proposed that incorporates up-winding effects in the calculation of the

advection part of the stabilization operator τ. Good stability and accuracy properties of the new method are

shown for a family of linear and quadratic tetrahedral and hexahedral elements.

[6]A scalable numerical model to solve the unsteady incompressible Navier–Stokes equations is developed

using the Galerkin finite element method. The coupled equations are decoupled by the fractional-step

method and the systems of equations are inverted by the Krylov subspace iterations. The data structure

makes use of a domain decomposition of which each processor stores the parameters in its sub-domain,

while the linear equations solvers and matrices constructions are parallelized by a data parallel approach.

The accuracy of the model is tested by modeling laminar flow inside a two-dimensional square lid-driven

cavity for Reynolds numbers at 1,000 as well as three-dimensional turbulent plane and wavy Couette flow

and heat transfer at high Reynolds numbers. The parallel performance of the code is assessed by measuring

the CPU time taken on an IBM SP2 supercomputer. The speed up factor and parallel efficiency show a

satisfactory computational performance.

The innovative parallel solution techniques are adopted for exploring the power of parallel computing.

Domain decomposition or the Schwarz method that is commonly adopted by CFD analysts. The discretized

information is distributed to each processor which is responsible for the calculation in the corresponding

sub-domain. The boundary conditions are obtained from the neighboring sub-domains during computations.

To facilitate parallel computation the domain is usually divided into several sub-domains according to the

structure of the mesh. A semi-implicit second-order accurate fractional-step method is used to decouple

unsteady incompressible Navier–Stokes equation. The quasi-minimal residual (QMR) and the conjugate

gradient (CG) methods are used to solve the non-symmetric and symmetric systems of equations,

respectively. These are non-stationary iterations that involve some constants to be calculated at each

iteration. Typically these constants are calculated by either taking products of matrices and vectors, or inner

products of the vectors. Hence, the iterations are parallelized once the above two products are able to do so.

A data parallel approach is adopted to perform these two parallelizations in the study.

Conclusions:

[6]A computation model based on equal-order FEM interpolating polynomials is developed for solving both

velocity and pressure of the Navier–Stokes equations. The governing equations are decoupled by a four-step

fractional method. The spatial domain is solved by the Galerkin FEM while the temporal domain is

integrated by the Crank–Nicolson scheme, both of second-order accuracy. The main advantage of the

current model is its simplicity in prescribing the boundary conditions for the velocity and pressure

formulation. The proposed solution procedure is parallelized for porting on distributed-memory machines.

Those expensive computational loads such as data storage, matrices/vectors constructions and linear

equations solvers are parallelized by employing either domain decomposition or data parallel approaches.

The developed parallel model is implemented on a distributed memory IBM SP2 supercomputer which is a

SPMD type model. Improvement on solution accuracy of an equal-order FEM is shown by comparing

laminar flow solution inside a two-dimensional square cavity at a Reynolds number of 1000. In addition, the

current model is validated by a three-dimensional DNS of fluid turbulence in plane Couette flow at a

Reynolds number of 5000. The capability of the current numerical scheme in large-scale scientific

computation is further demonstrated through DNS of turbulent Couette flow over wavy surface. The parallel

performance of the proposed parallel strategy is tested by analyzing the CPU time taken on an IBM SP2

supercomputer. Two scales, namely the small and large, of computations consisting of millions of elements

are performed on different numbers of processors and improved computational performance is obtained


upon parallelism. By measuring the speed up factor and the parallel efficiency, the large scale calculation

shows better parallel performance and scalability compared with its small scale counterpart.

[7]The solution accuracy is compared with the existing interpolation functions such as the discretized Dirac

delta function and the reproducing kernel interpolation function. The finite element shape function is easy to

implement and it naturally satisfies the reproducing condition. They are interpolated through only one

element layer instead of smearing to several elements. A pressure jump is clearly captured at the fluid–solid

interface. Two example problems are studied and results are compared with other numerical methods. A

convergence test is thoroughly conducted for the independent fluid and solid meshes in a fluid–structure

interaction system. The required mesh size ratio between the fluid and solid domains is obtained. The

discretized Dirac delta function used in the immersed boundary method and the reproducing particle method

used in the immersed finite element method satisfy this condition. Propose and implemented a finite element

interpolation function for non-uniform background fluid grid to capture a sharper fluid–structure interface

than the reproducing kernel interpolation function used in the immersed finite element method and the Dirac

delta function used in the immersed boundary method. The solutions are examined thoroughly and

compared with other published results. The convergence test will be performed and a range of allowable

mesh size ratios between the fluid and solid domains will be identified.

A comprehensive convergence test is performed using this example. We pay special attention to the

allowable fluid– solid mesh size ratios that can be used to yield convergent solutions. For a coupled fluid–

structure problem, the convergence rate is computed independently with Lagrangian mesh element size and

Eulerian grid spacing. Since there is no analytical solution for this problem, the errors of fluid velocity and

solid displacement are calculated based on the solution obtained from a finely discretized system. The

convergence of the solid displacement is calculated by refining the Lagrangian mesh while keeping the

Eulerian mesh fixed at a refined state. Similarly, the convergence of the N-S solver is studied by refining the

fluid mesh while keeping the solid mesh at a very fine resolution. Both components are performed with

uniform mesh spacings for consistencies. Errors in the fluid velocity and solid displacement are calculated in

L2 norms for steady state solutions.

Conclusions:

[7] The interpolation functions used in the immersed boundary method and the immersed finite element

method, i.e. the discretized Dirac delta function and the reproducing kernel function. Proposed a

straightforward finite element interpolation function that is capable of producing sharper interface that

preserves the accuracy in interface solutions and to be used on unstructured background fluid meshes. The

finite element interpolation function naturally satisfies the reproducing condition and it is easy to implement.

Comparing to the previously mentioned techniques, the thickness of the interface can be narrowed by

approximately 65% when using uniform grids, and can be improved even further when non uniform or

unstructured grids are used. Through the example problems, we performed a thorough convergence test and

examined the mesh size compatibility requirement for the fluid and solid domains. We found a mesh size

ratio of 0.5 is required for the fluid and solid discretization to avoid numerical issues. If the fluid mesh size

is less than half of the solid mesh size, then a leaking phenomenon would occur and lead the solutions to

diverge. This value is consistent for several mesh resolutions. We also observed a relatively large volume

change when the solid comes near a moving fluid boundary that generates large velocity gradient. A volume

correction algorithm is imposed to enforce this incompressibility constraint. This correction algorithm can

dramatically improve the durability of the incompressibility assumption and enhance the performance of the

simulation. In summary, this paper introduces a finite element interpolation function to be used in the

immersed finite element method and closely examines and resolves several detailed numerical issues that are

present in the current non-conforming techniques. It provides a more accurate and a more reliable approach

to be used in the simulations of fluid–structure interactions.

[8] The fluid-structure interaction in fully nonlinear setting, where different space discretization can be used.

The model problem considers finite elements for structure and finite volume for fluid. The computations for


such interaction problem are performed by implicit schemes, and the partitioned algorithm separating fluid

from structural iterations. The formal proof is given to find the condition for convergence of this iterative

procedure in the fully nonlinear setting. Several validation examples are shown to confirm the proposed

convergence criteria of partitioned algorithm. The model problem for testing the novel paradigm of solution

procedure based upon the direct coupling of different codes developed for a particular sub problem (i.e.

either solid or fluid mechanics) into a single code. In particular, we seek to provide the guarantees for the

robustness of such a computation approach in fully nonlinear setting, where implicit schemes are used for

each sub problem, and we derive (by a formal proof) the convergence criterion for partitioned scheme

iterations.

For solving fluid–structure interaction problems are mostly oriented towards the monolithic schemes, where

both sub-problems are discretized in space and time in exactly the same manner resulting with a large set of

(monolithic) algebraic equations to be solved simultaneously with no need to distinguish between the ―fluid‖

and the ―structure‖ part. Provided the unified discretization basis for monolithic approach, the most frequent

choice is to use the stabilized finite elements for fluids (first proposed by Hughes and co-authors followed

by Tezduyar and many other works The main advantage of code-coupling approach for fluid– structure

interaction concerns the fact that the coupling is limited only to the fluid–structure interface. Therefore, the

main difficulty is reduced to enforcing the interface matching with respect to two different discretization

schemes, finite element versus finite volume, as well as two different time integration schemes and different

time steps. We thus split the presentation of our work in two parts, pertaining, respectively, to time and to

space discretization for fluid and for structure and their matching at the interface. We will deal with the

interface matching for different space discretization, along with other related issues pertaining to the

computational efficiency enhancements by nested parallelization. In present paper (Part I), we discuss how

to accommodate any particular (implicit) scheme that ensures the unconditional stability for either fluid or

structure motion computation, and how to ensure that the unconditional stability extends to partitioned

solution of the fluid–structure interaction problem. By considering equal time step size for fluid and

structure, this direct force-motion transfer algorithm is named conventional serial staggered (DFMT-

CSS).Also consider the so-called Sub-cycled conventional staggered scheme (DFMT-SCSS) where time

steps selected for integration of fluid flow and structure motion are not the same size.

Conclusions:

[8] Examined partitioned solution approach for nonlinear fluid–structure interaction problems. The

partitioned approach is preferred for its modularity and the possibility of re-using existing software

developed for each sub-problem (see Part II). The partitioned approach used here is based on the DFMT.

Both explicit and implicit coupling algorithms for multi-physics problems are detailed. An explicit strategy

leads to the so-called ―added mass effect‖, and for that justifies the use of more costly implicit solvers for

the case of incompressible fluid flows. In this work, the problem of enforcing the fluid– structure interface

matching is handled by the fixed-point strategy (DFMT-BGS) with an adaptive relaxation parameter. This

strategy shows a sufficiently robust performance, especially for the example where the flow is not highly

constrained by incompressibility. In fact, we showed by direct proof the stability of the implicit DFMT-BGS

algorithm which is valid for the fully nonlinear fluid–structure interaction problem.

[9] New adaptive Lattice Boltzmann method (LBM) implementation within the Peano framework, with

special focus on nano-scale particle transport problems. With the continuum hypothesis not holding anymore

on these small scales, new physical effects—such as Brownian fluctuations—need to be incorporated. We

explain the overall layout of the application, including memory layout and access, and shortly review the

adaptive algorithm. The scheme is validated by different benchmark computations in two and three

dimensions. An extension to dynamically changing grids and a spatially adaptive approach to fluctuating

hydrodynamics, allowing for the thermalisation of the fluid in particular regions of interest, is proposed.

Both dynamic adaptivity and adaptive fluctuating hydrodynamics are validated separately in simulations of

particle transport problems. The application of this scheme to an oscillating particle in a nano-pore illustrates

the importance of Brownian fluctuations in such setups. Presented an approach to nano-flow simulations in

complex and/or moving geometries.


Conclusions:

[9] Implementated a block-structured adaptive Lattice Boltzmann solver, including its memory access,

adaptivity concept and intermolecular collision models. In order to profit from both the simple-to-use

adaptivity concept of the Peano framework and the simple and computationally cheap Lattice Boltzmann

update rule, we proposed the usage of an application-specific grid management system handling the

memory-intensive storage of the particle distribution functions. This scheme avoids costly copy operations

between the Peano-internal stacks on the one hand, but leaves the handling of the adaptive grids as well as

the parallelization to the Peano kernel on the other hand. We verified and validated our adaptive

implementation by different benchmark computations using adaptive and non-adaptive grids in two and

three dimensions. Furthermore, the extension of the adaptive scheme to dynamically changing grids has

been presented, allowing for the simulation of moving structures within the flow. The new scheme was

validated for particle transport problems which are of major concern in our work. An additional focus of

research was on nano-flow simulations where Brownian motion effects play a crucial role. The modeling of

the respective Brownian fluctuations, however, comes at high computational costs as huge numbers of

Gaussian random numbers are required in this case. We proposed a multiscale approach, allowing for

fluctuating effects within the fluid on fine grid levels only. On coarser grid levels, the fluctuations are cut off

and a simple BGK collision kernel is applied. We used this cut-off approach to simulate the diffusion of an

isolated spherical particle. The short-time diffusion of the particle is slightly underestimated by the method,

the long-term behavior is captured correctly. Finally, combined our dynamic refinement approach and the

cut-off mechanism for thermal fluctuations to simulate a particle which is exposed to oscillating pressure

fields within a nano-pore. Similar to previous results, diffusive effects due to thermal fluctuations dictate the

magnitude and the direction of the particle drift. Both our new cut-off approach and a completely

thermalised fluid model show a behavior of the particle drift which is different to non-fluctuating

simulations. This illustrates the importance of Brownian motion on the nano-scale for our flow scenarios. As

part of future work, further studies in two and three dimensions will be carried out to completely understand

the short-time behavior of the particle motion within the nano-pores. Within this contribution, we restricted

our numerical experiments to the simulation of several periods in the particle oscillations. Simulations

overmuch longer time intervals might be required to completely evaluate the motion of the particle inside

the pore structures. Therefore, new methods need to establish. We currently work on a hybrid approach to

include both thermal fluctuations on the short and long time scale in our simulations.

[10]A newly developed LES flow solver to compute a true three-dimensional flow applied. The research

also investigates the behavior of turbulence statistics by comparing transient simulation results to available

data based on experiments and simulations. An extensive discussion on the results such as energy spectrum,

velocity profiles and time trace of velocities is carried out in the research as well. Based on the results

obtained, the application of the flow solver for a turbulent three-dimensional driven cavity flow by using

three grids with varying densities is proven. In addition, the research successfully verifies that in many

instances computational results agreed reasonably well with the reference data, and the changes in the

statistical properties of turbulence with respect to time are closely related to the changes in the flow structure

and strength of vortices. The focus of this study is on the prediction of a sub-grid scale Reynolds shear stress

profiles and the results show that the standard model is able to reproduce general trends measured from

experiments. Furthermore, in certain areas inside the cavity the computed shear stress values are in close

agreement with experimental data. The dynamics of the statistical properties of turbulence as these vortices

and secondary flow develop. A further novel aspect of this work is to obtain some insight into accuracy of

shear stress computations using the baseline Smagorinsky model, for flows with spatially and temporally

varying turbulence structures . Published data from Prasad and Kosef and Migeon et al. are used to validate

the code

Conclusions:

[10] Investigated in detail the dynamics of the statistical properties of turbulence as Taylor-Gortler vortices

and secondary flows develop. In particular, the w v stress profiles, which are particularly difficult to predict,


are studied. Furthermore, some insight into the accuracy of shear stress computations using the baseline

Smagorinsky model, for flows with spatially and temporally varying turbulence structures has been

obtained. This paper has demonstrated the application of the flow solver for a turbulent three-dimensional

driven cavity flow by using three uniform grids with varying densities. In many instances, computational

results agree reasonably well with the reference data. A number of important conclusions can be drawn from

this case: In general, the profiles from computation follow the trend exhibited by the reference data. The

numerical setup was, to some extent, quantitatively successful in predicting the w v stress profiles. In

addition, to predict the w v stress profiles more accurately, the value of the Smagorinsky constant, Cs, must

be varied with location inside the domain. The turbulence kinetic energy spectrum plots show the presence

of inertial sub-range eddies though the level of energy may vary with respect to the location of the

monitoring points. The turbulence kinetic energy plots show that this parameter is produced in the region

where the Taylor-Görtler-like vortices reside.

[11] An experimental study is presented for a flow field in a two dimensional wavy channels by PlY. This

flow has two major applications such as a blood flow simulation and the enhancement of heat transfer in a

heat exchanger. While the numerical flow visualization results have been limited to the fully developed

cases, existing experimental results of this flow were simple qualitative ones by smoke or dye streak test

Therefore, the main purpose of this study is to produce quantitative flow data for fully developed and

developing flow regimes by the Correlation Based Correction PlY (CBC PlY) and to conjecture the analogy

between flow characteristics and heat transfer enhancement with low pumping power. Another purpose of

this paper is to examine the onset position of the transition and the global mixing, which results in transfer

enhancement. PlY results on the fully developed and developing flow in a wavy channel at Re=500, 1000

and 2000 are obtained. For the case Reynolds Number equals 500, the PlY results are compared with the

finite difference numerical solution.

The practical use of the particle image velocimetry (PIV), a whole flow field measurement technique,

requires the use of fast, reliable, computer-based methods for tracking velocity vectors. The full search block

matching, the most widely studied and applied technique both in the area of PIV and Image Coding &

Compression is computationally costly. Many alternatives have been proposed and applied successfully in

the area of image compression and coding, i. e. MPEG, H. 261 etc. Among others, the Three Step Search

(TSS) (Jain, 1981), the New Three Step Search (NTSS) (Li et al., 1994), the Hierarchical Projection Method

(HPM) (Sauer and Schuartz, 1996), the FFT-Direct Hybrid Method (HYB) and the Two Resolution Method

(TRM) (Anandan, 1989) are introduced. A Correlation Based Correction technique (CBC) (Hart, 2000) is

also appreciated and found to be most accurate and adequate for this flow. For the cases Reynolds number

fRe) of 500, 1000 and 2000. Developing and fully developed flow data are obtained by CBC PIV with one

window shifting. The global mixing phenomenon; which results in the increase in heat and mass transfer and

drag, can be identified through the investigation of developing flow in beginning modules. At Re above 500,

promotion to turbulence is prominent. While it happens at Re above 2300 in a straight channel. The three-

point Gaussian fit is used for a sub pixel estimator, and the Local Median Filter (LMF) is chosen to validate

a vector field. (Kim, 1999)

Conclusions:

[11]Unlike a simple dye or smoke streak visualization, the PlY analysis can resolve the exact flow structure,

even in turbulent flow situation. It can also deal with the unsteady behavior of global mixing. In this paper,

fully developed and developing flow data in a wavy channel of Re 500, 1000, 2000 are obtained through the

CBC PlY measurements. The analogy between flow characteristics and the enhancement of heat and mess

transfer in a wavy channel can be visible through the RMS distribution near wall. The onset point of the

global mixing is clearly identified through instantaneous velocity and RMS intensity distributions of a

couple of beginning modules. It happens at 4th wavy module for Re= 500, and 2nd module in case Re

equals 1000, which are in a good agreement of Rush et aI.'s prediction from dye streak visualization. The

phase averaging of PIV results will give a more precise insight of flow structure, like instability and

shedding vortices etc.


[12] Analyze the fluid flow with moving boundary using a finite element method. The algorithm uses a

fractional step approach that can be used to solve low-speed flow with large density changes due to intense

temperature gradients. The explicit Lax-WendrofT scheme is applied to nonlinear convective terms in the

momentum equations to prevent checkerboard pressure oscillations. The ALE (Arbitrary Lagrangian

Eulerian) method is adopted for moving grids. The numerical algorithm in the present study is validated for

two-dimensional unsteady flow in a driven cavity and a natural convection problem. To extend the present

numerical method to engine simulations, a piston-driven intake flow with moving boundary is also

simulated. The density, temperature and axial velocity profiles are calculated for the three-dimensional

unsteady piston-driven intake flow with density changes due to high inlet fluid temperatures using the resent

algorithm. The calculated results are in good agreement with other numerical and experimental ones.

Conclusions:

[12] Used a fractional step method with equal-order interpolation functions for the velocity components and

pressure. The explicit Lax-Wendroff scheme was applied to the nonlinear convective terms in the

momentum equations and the ALE (Arbitrary Lagrangian-Eulerian) method was adopted for treating the

moving boundary. To validate the present algorithm, several problems have been calculated and compared

with other results. As a result, the calculation results have shown good agreement with other results. In order

to extend the present numerical method to engine simulations, we also investigated the basic behavior of the

unsteady flow generated by an impulsively started piston movement in a piston-cylinder assembly, yielding

flow separation and spatially moving vortices. The numerical results indicate that the present calculation

procedure can be used to predict the behavior of periodic intake/exhaust flows and is applicable to a wide

range of problems. Although the discussion has been restricted to laminar flows governed by the Navier-

Stokes equations, the methodology proposed can readily be extended to accommodate the Reynolds-

averaged equations and turbulence models. [13] The bifurcation of the lines of a viscous-fluid flow in a rectangular cavity with a moving cover has been

investigated for different ratios between the sides of the cavity and different Reynolds numbers on the basis

of the qualitative theory of dynamic systems. The critical parameters of the problem, at which the type of

singular points changes and other topological characteristics of a vortex flow in the indicated cavity have

been determined and the corresponding bifurcation diagrams have been constructed. The topological

characteristics of a flow can be determined on the basis of solution of the Navier–Stokes equations. a flow in

a rectangular cavity was investigated for different ratios between the cavity sides and different velocities of

travel of the upper and lower walls. The finite-element method with bunching of nodes of a grid in the

neighborhood of local stagnation points was used for discretization of Navier–Stokes equations. The

investigations were carried out for fairly small Reynolds numbers (Re < 100). Reasonably exact results were

not obtained by the finite-element method because it, when used for solving fluid-mechanics problems,

provides a lower accuracy than the finite-difference method.

Conclusions:

[13] The dependence of the change in the type of singular points of a vortex flow (bifurcation of streamlines)

in a rectangular cavity on the ratio between the cavity sides and on the Reynolds number has been

investigated on the basis of numerical simulation of this flow. The data obtained can be used for determining

the topological characteristics and features of separation flows in cavities of more complex geometries.


Chapter-III

Objectives of the work

1) To study the fluid dynamics in the lid-driven cavity.

2) To study the effect of governing parameters in terms of non dimensional numbers in Navier-Stokes

equation for four different cases that are Isothermal, Forced convection, mixed convection and

Natural convection.

3) To study the effect of grid sizes (coarse, medium and fine) to catch actual characteristics of the fluid

flow in above said four different cases.

4) To study the effect of non dimensional numbers on the temperature in above said cases except first

case.

5) To study the stream function contours for different non dimensional number in all above said cases.

6) To study the time steps for different parameters in said cases.

7) To study the convergence criteria in all above said cases.

8) To study the vertices movement in the cavity for different non dimensional numbers for said cases.

9) To study the computation time for said cases with different parameters.

10) To study the code validation.


Chapter-IV

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) square domain of size L x L (1

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.

Fig.1 Lid driven cavity

To perform a non dimensional CFD simulation for various values of non-dimensional governing parameters

such as Reynolds number, Prandl number, Grashoff number etc.; the following models are made for four

different situations (cases) as mentioned in the following tables.

Bottom stationary solid wall X

Y

Cavity

Right side stationary solid

wallwall

Left side stationary solid wall

Top long horizontal moving belt (lid) with velocity Uo units

1 u

nit

1 unit

Temperature Boundary

Conditions:

Top wall at TH temperature and

Left, Right & Bottom walls are at

TC temperature

Boundaries of the domain for iterations steps

(i,j)= (1,1) at left bottom corner; (i,j)= (imax,1) at right bottom corner

(i,j)= (1,jmax) at left top corner; (i,j)= (imax, jmax) at right top corner


Case-I: Isothermal fluid flow

Note: All walls of the cavity are at constant temperature i.e

TH=TC=Constant

Model

No.

No. of Grids in x direction

X

No. of Grids in y direction

Reynolds Number

I1 32X32 100

I2 52X52 100

I3 32X32 400

I4 52X52 400

I5 32X32 1000

I6 52X52 1000

Case-II: Forced convection fluid flow

Note: Non-isothermal i.e top wall at TH and all other walls at TC

temperatures. This corresponds to non-dimensional temperature

,-

- and is

a buoyancy induced flow therefore in all models Grashoff number is

taken as zero i.e Gr=0

Model

No.

Prandl

Number

Reynolds

Number

No. of Grids in x direction X


F1 1 100 12X12

F2 1 100 32X32

F3 1 400 12X12

F4 1 400 32X32

F5 0.5 100 12X12

F6 0.5 100 32X32

F7 0.5 400 12X12

F8 0.5 400 32X32

F9 1.2 100 12X12

F10 1.2 100 32X32

F11 1.2 400 12X12

F12 1.2 400 32X32

Case-III: Mixed convection fluid flow

Note: All the conditions are same as in Case-II except the Grashoff

number is finite values i.e +ve or –ve and ML=Mixed flow Low

Grashoff number model Gr=1X105

Model

No.

Prandl

Number

Reynolds

Number



ML1 1 100 12X12

ML 2 1 100 32X32

ML 3 1 400 12X12

ML 4 1 400 32X32

ML 5 0.5 100 12X12

ML 6 0.5 100 32X32

ML 7 0.5 400 12X12

ML 8 0.5 400 32X32

ML 9 1.2 100 12X12

ML 10 1.2 100 32X32

ML 11 1.2 400 12X12

ML 12 1.2 400 32X32



Note: MM=Mixed flow Medium Grashoff number model Gr=1x106

Model

No.

Prandl

Number

Reynolds

Number



MM 1 1 100 12X12

MM 2 1 100 32X32

MM 3 1 400 12X12

MM 4 1 400 32X32

MM 5 0.5 100 12X12

MM 6 0.5 100 32X32

MM 7 0.5 400 12X12

MM 8 0.5 400 32X32

MM 9 1.2 100 12X12

MM 10 1.2 100 32X32

MM 11 1.2 400 12X12

MM 12 1.2 400 32X32


Note: MH=Mixed flow High Grashoff number model Gr=2X106

Model

No.

Prandl

Number

Reynolds

Number



MH 1 1 100 12X12

MH 2 1 100 32X32

MH 3 1 400 12X12

MH 4 1 400 32X32

MH 5 0.5 100 12X12

MH 6 0.5 100 32X32

MH 7 0.5 400 12X12

MH 8 0.5 400 32X32

MH 9 1.2 100 12X12

MH 10 1.2 100 32X32

MH 11 1.2 400 12X12

MH 12 1.2 400 32X32



number is finite values i.e +ve or –ve and ML=mixed flow low Grashoff

number model Gr=-2X106; MNL: Mixed convection Less Negative

Grashoff number.

* Due to shortage of time to submit the report the simulation has not

been done, because that simulations are taking days together.

Model

No.

Prandl

Number

Reynolds

Number



MNL1 1 100 12X12

MNL 2* 1 100 32X32

MNL 3 1 400 12X12

MNL 4 1 400 32X32

MNL 5 0.5 100 12X12

MNL 6* 0.5 100 32X32

MNL 7 0.5 400 12X12


MNL 8 0.5 400 32X32

MNL 9* 1.2 100 12X12

MNL 10 1.2 100 32X32

MNL 11 1.2 400 12X12

MNL 12 1.2 400 32X32



number is finite values i.e +ve or –ve and ML=mixed flow low Grashoff

number model Gr=-1X105; MNH: Mixed convection High Negative

Grashoff number

Model

No.

Prandl

Number

Reynolds

Number



MNH1 1 100 12X12

MNH 2 1 100 32X32

MNH 3 1 400 12X12

MNH 4 1 400 32X32

MNH 5 0.5 100 12X12

MNH 6 0.5 100 32X32

MNH 7 0.5 400 12X12

MNH 8 0.5 400 32X32

MNH 9 1.2 100 12X12

MNH 10 1.2 100 32X32

MNH 11 1.2 400 12X12

MNH 12 1.2 400 32X32

Case-IV: Natural convection fluid flow

Note: For natural convection, the flow is only due to buoyancy with no

forced flow. Thus, the lid is also taken as stationary here. Thus, the

physical situation corresponds to a buoyancy induced flow in a

differentially heated closed square cavity. The left-wall is maintained at

TH, right-wall at TC and the remaining walls are insulated; all walls are

stationary.

Model

No.

Prandl

Number

Rayleigh

Number



N 1 0.71 103 12X12

N 2 0.71 103 32X32

N 3 0.71 104 12X12

N 4 0.71 104 32X32


Chapter-V

Mathematical modeling with boundary conditions We consider the incompressible viscous fluid flow with constant density, viscosity and thermal conductivity

in the absence of an applied body force. A set of non-dimensional governing equations following elliptic-

parabolic characterization are as follows:

General mass conservation equation:

In the eqn. (1) u and v are velocity components along x and y directions respectively. are free

steam velocities along x and y directions and total length of the domain respectively. are non-

dimensional velocities along the x and y directions respectively; and similarly are the non

dimensional coordinates.

General momentum equations:

In the eqn. (2) are non-dimensional temperature, Reynolds number and Prandl number

respectively. is the bulk mean temperature in case of internal flow and free stream temperature in case of

external flow situations. is the known temperature of the wall or surface. T is the temperature to be

find during iteration and time steps. are viscosity, specific heat and thermal conductivity of the

fluid respectively and these are taken as constant for our problem. This equation includes temperature term

therefore it is used to discritize non-isothermal cases that are case-II to IV in our CFD simulations.

In the eqn. (3); is non-dimensional pressure and is ; where is free stream known pressure and

P is pressure to be find for each step of the CFD simulation. This equation does not include any temperature

terms; therefore this can be used to solve isothermal type problems i.e case-I in our CFD simulation.


Boundary conditions:

The boundary condition equations from (4) to (7) are incorporated to solve isothermal type of problems;

where the temperature remains constant i.e is in case-I in our CFD simulations. And the boundary condition

equations from (8) to (11) are incorporated to solve non-isothermal type of problems; where the temperature

varying with respect to time and coordinates; that are case-II to IV in our CFD simulations. In equations (4)

to (6) and (8) to (10); the velocity components on left, right and bottom surfaces are taken as zero unit along

both x and y directions because of viscous effect of the solid stationary walls on the fluid particles. Because

the top surface is continuously moving in the horizontal direction only i.e x direction with a velocity ‗u‘; the

fluid particles close to the bottom surface of the top plate are affected by the motion of the plate; therefore it

is considered in the equations (7) and (11) as unit for simplification of the problem.


Chapter-VI

Validation study As a first step towards numerical investigation of the physical problem, we first justify our computer code.

A validation study is often proceeding by an analytical assessment of the problem. To achieve this goal and,

furthermore, estimate the spatial rate of convergence of the scheme employed, we consider the following

transport eqn. (12) for a scalar Φ in a simple domain of two dimensions; .

In eqn.(12) left side first unsteady term will calculate scalar quantity for all time steps and second and third

terms in the same side will calculate advection quantities on all geometric coordinates. The first term on the

right side calculates diffusion coefficients.

Where are the pressures at east and north nodes respectively and is the pressure at the node

where the pressure has to find. The eqn. (13) to (16) are used as pressure correctors to remove the difficulty

in the pressure term for linear interpolation is solved by taking staggered grid solution. Prediction error due

to oscillatory velocities and the so called false diffusion error grossly pollute the flow physics over the entire

domain. Remedy for such discretization error is to apply pressure correction equations in semi explicit

QUICK wind scheme SOU at the boundaries. The above shown equations are used to calculate the velocities

along x and y direction by considering adjacent nodes pressure for the next time step.

As is usual, we assessed the employed QUICK-type upwind discretization scheme by examining the

prediction nodal errors. Tests on various grids were conducted to assure that the solution converged. With

grid spacing being continuously refined, we could compute the rate of convergence from the computed. The

test case considered and the results obtained thus far confirm the applicability of the QUICK scheme to

multidimensional analyses. We now turn to examining whether or not linearization procedures and the zero-

divergence constraint condition will cause the rate of convergence to deteriorate. To answer this question,

we solved a Navier-Stokes problem in the same domain as that considered in the previous benchmark test by

GHIA et al. (1982) JOURNAL OF COMPUTATIONAL PHYSICS VOL. 48, pp.387-411 @ Re = 100 we

are assured that the proposed scheme is also applicable to analysis of incompressible Navier-Stokes

equations. The good agreement from two analytical tests, as demonstrated provides us with strong

confidence to proceed with investigation of the time-history of the flow evolution, which is driven by a

constant upper lid, in the rectangular cavity.


Chapter-VII

Results and Discussions

Case-I: Isothermal fluid flow

Stream function contours 32X32 Grid structure 52X52 Grid structure

Fig.I 1.1

Re:100 Computation time:5 hours Time steps:49

No.of iterations in the mass conservation loop:1938

Fig.I 2.1



Fig.I 3.1



Fig.I 4.1



Fig.I 5.1

Re:1000 Computation time:3.5 hours Time steps:42


Fig.I 6.1

Re:1000 Computation time:15.5 hours Time steps:39 No.of iterations in the mass conservation loop:2254

It is observed from the above shown stream function contours that as the magnitude Reynolds number

increases then the inertia forces will increase in the cavity therefore more turbulences will formed in the

fluid flow; hence for more values Reynolds number, the solution will take less time steps, less computational

time and less number of iterations in the mass conservation loop for the convergence even though change in


the grids in x and y directions for all figures of Case-I. Higher magnitude of Reynolds number means; top

surface of the cavity moving with a higher velocity.

As we observed from the stream contour figures‘ I 1.1, I 3.1 and I 5.1; the lower value stream contours

decreases with increasing the magnitude of the Reynolds number means top lid moving with higher velocity

and the veracity will shift towards top right corner. Same effect can be observed in the higher grid points

also.

It is observed from the figures‘ I1.1 & I1.2, I1.3 & I1.4 and I1.5 & I1.6 of steam function contours that as

the grid size increases from 32X32 to 52X52 structure in the x and y directions; the simulation catches lower

values of stream functions towards bottom surface of the top lid and also we can observed the sharp changes

in the physical shape of the verticity near at top right corner of the cavity for 52X52 grid size.

V velocity along the horizontal centerline

32X32 Grid structure 52X52 Grid structure

Fig.I 1.2



Fig.I 2.2

Re:100 Computation time:21 hours Time steps:50 No.of iterations in the mass conservation loop:2624

Fig.I 3.2



Fig.I 4.2



Fig.I 5.2



Fig.I 6.2


As we observed from the velocity along the horizontal centerline figures‘ I 1.2, I 3.2 and I 5.2; the deviation

in the velocity result increases with increasing magnitude of the Reynolds number with the published results

for 32X32 as well as in 52X52 grid structure; therefore even more denser grid structure is required to catch


the small in the velocities in the center horizontal line. It is also observed from both grid structures; the more

closure velocity profile with the published results for 52X52 structures; and still finer grid structure is

required to match with published results.

And also observed from same figures; as the magnitude of the Reynolds number increases from 100 to 1000,

the inside surfaces of the left and right side walls effect decreases and velocity of the fluid particles

increases; it enhances little more turbulence in the fluid particles. Hence these turbulences cannot catch in

the higher magnitude Reynolds number; therefore the centerline velocity profile is almost horizontal in case

of 1000 Reynolds number and small change in case of Reynolds number 400; but sharp changes in Reynolds

number 100.

U velocity along the vertical centerline


Fig.I 1.3


Fig.I 2.3



Fig.I 3.3



Fig.I 4.3



Fig.I 5.3 Re:1000 Computation time:3.5 hours Time steps:42 No.of iterations in the mass conservation loop:1072

Fig.I 6.3 Re:1000 Computation time:15.5 hours Time steps:39 No.of iterations in the mass conservation loop:2254


It is observed from the figures I1.3, I3.3 and I5.3 of horizontal velocity component (U) along the vertical

centerline that the fluid particles will changes their flow direction from right side to the left side at the 0.15

unit distance from bottom surface along the y direction of the cavity for 1000 Reynolds number in the

published results and that same distance will increases for lower Reynolds numbers i.e 0.28 for Re=400 and

0.55 for Re=100. And is also observed that that sharp change in the velocity profile for higher Reynolds

numbers i.e for 400 and 100 but in lower values of the same number smooth changes will happen i.e for 100.

As we observed from the figures I1.3 to I6.3 that our simulation velocity profile is close with published

profile for lower Reynolds number i.e 100 but this closeness is decreases with higher value of the Reynolds

number; this is because our grid structure is not sufficient to catch the higher velocities of the particles of

higher Reynolds number. Therefore finer grid structure may be necessary to catch the higher Reynolds

number turbulences.

V velocity contours 32X32 Grid structure 52X52 Grid structure

Fig.I 1.4


Fig.I 2.4


Fig.I 3.4



Fig.I 4.4



Fig.I 5.4


Fig.I 6.4




It is observed from the figures from I1.4to I6.4 that primary higher velocity verticity at top left corner and

secondary lower velocity verticity at the top right corner are changing their characterization when the

magnitude of the Reynolds numbers varies. The magnitude of the primary and secondary verticities will

decreases with increasing Reynolds number. This is because; when the velocity of the top lid increases then

the turbulence in the fluid flow will also increases; so more distribution of the velocity contours. Commonly

in all figures it is also observed that the moment of the lid is highly at top left corner fluid particles of the

cavity on vertical velocity contours whereas it reverses at right top corner fluid particles of the cavity.

By comparing the figures for 32X32 grid structure I1.4, I3.4. I5.4 and 52X52 grid structure I2.4, I4.4, I6.4;

more uniform v-velocity contours in higher grid structures. This is because more fluid particles will cover in

dense grid structure. As the Reynolds number increases from 100 to 400 for both grid structure; the higher

value velocity contours in Re=400 than Re=100 in the domain. It is also observed from the figures; as the

Reynolds number increases from 100 to 1000 through 400, the magnitude of the primary and secondary

verticities is decreasing and it may disappear for even higher values of the Reynolds number.

U velocity contours


Fig.I 1.5



Fig.I 2.5



Fig.I 3.5



Fig.I 4.5



Fig.I 5.5



Fig.I 6.5



It is observed from the figures I1.5 to I6.5 that the higher value U velocity contours at the top surface of the

domain because top lid is moving horizontally. As the Reynolds number is low higher value U-velocity

contours is more at bottom of the lid whereas these are decreases as the Reynolds number increases in both

32X32 and 52X52 grid structure. The lower velocity verticity is observed in all figures at the right side of

the domain and it is towards top right corner as the Reynolds number increases. And also the magnitude of

the verticity also decreases with increasing Reynolds number.

It is also clearly observed from two different grid structures; the change of the characteristic of the verticity

different in 52X52 than the 32X32. The effect of the top lid velocity caught in dense grid structure than the

coarse grid structure. Because less viscosity effect at the bottom side top lid; the particles immediately

bottom of the plate are in the same velocity of the plate i.e the highest velocity in the problem. For denser

grid structure; the verticity at the top right corner of the domain may disappear. Case-II: Forced convection fluid flow (where Gr=0)

U velocity contours 12X12 Grid structure 32X32 Grid structure

Fig.F 1.1

Re:100 Pr:1 Computation time:15 mins Time steps:92


Fig.F 2.1

Re:100 Pr:1 Computation time:11 hours Time steps:126


Fig.F 3.1

Re:400 Pr:1 Computation time:23 mins Time steps:125 No.of iterations in the mass conservation loop:353

Fig.F 4.1

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Fig.F 5.1

Re:100 Pr:0.5 Computation time:20 mins Time steps:64 No.of iterations in the mass conservation loop:456

Fig.F 6.1

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87



Fig.F 7.1

Re:400 Pr:0.5 Computation time:15 mins Time steps:104


Fig.F 8.1

Re:400 Pr:0.5 Computation time: 12hours Time steps:115


Fig.F 9.1

Re:100 Pr:1.2 Computation time: 25min Time steps:109


Fig.F 10.1



Fig.F 11.1

Re:400 Pr:1.2 Computation time: 20minTime steps:126


Fig.F 12.1

Re:400 Pr:1.2 Computation time: 16hours Time steps:165 No.of iterations in the mass conservation loop:1938

It is observed from the figures from F 1.1 to F12.1; higher velocity verticity is small and it is shifting

towards top right corner of the cavity of the domain for dense grid structure than the coarse grid structure.

The same verticity is shifting towards bottom of the cavity for higher Reynolds number i.e 400 as compared

to the 100. The time steps required for to converge the solution are more for higher Reynolds number (400)

than the lower (100) for the same Prandl number irrespective of the grid structure. As the Prandl number

increases from 0.5 to 1.2 through 1; the solidity of the fluid is more, therefore the velocity of the top lid is

more in case of lower Prandl number (0.5) than the higher number (1.2). The physical characteristic of the

verticity is slightly different in case of higher Reynolds number towards the bottom surface of the cavity. As

we know top lid is moving; therefore the higher velocity contours are appears at the top.



Fig.F 1.2


Fig.F 2.2



Fig.F 3.2


Fig.F 4.2

Re:400 Pr:1 Computation time: 16 hours Time steps:158 No.of iterations in the mass conservation loop:1938

Fig.F 5.2 Re:100 Pr:0.5 Computation time:20 mins Time steps:64


Fig.F 6.2 Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87


Fig.F 7.2 Re:400 Pr:0.5 Computation time:15 mins Time steps:104


Fig.F 8.2 Re:400 Pr:0.5 Computation time: 12hours Time steps:115 No.of iterations in the mass conservation loop:1938


Fig.F 9.2



Fig.F 10.2



Fig.F 11.2



Fig.F 12.2



It is observed from the figures of higher Reynolds number (400); the higher V-velocity verticity at the top

left corner is stretched towards right surface of the cavity as compared to the lower number (100) and this

character is clearly observed in the dense grid structure(32X32) than the coarse grid structure (12X12). The

magnitude of the verticity is slightly higher in case of lower grid structure (12X12) than the higher (32X32)

irrespective of Prandl and Reynolds number. In all figures from F 1.2 to F 12.2 one higher primary verticity

at the top left corner and another lower secondary verticity at the right top corner of the cavity. It is observed

from higher Reynolds number contours (400) having lower velocity contours than the lower Reynolds

number (100) irrespective of Prandl number and grid structure. It observed from the figures; as the Reynolds

number increases from 100 to 400 the magnitude of the primary higher verticity increase with decreasing

secondary lower verticity. The magnitude of the secondary verticity is decreasing with Prandl number for

same Reynolds number irrespective of the grid structure. The number of time steps required are more for

higher Reynolds number than the lower.


Fig.F 1.3



Fig.F 2.3




Fig.F 3.3


Fig.F 4.3

Re:400 Pr:1 Computation time: 16 hours Time steps:158


Fig.F 5.3


Fig.F 6.3



Fig.F 7.3



Fig.F 8.3



Fig.F 9.3

Re:100 Pr:1.2 Computation time:25min Time steps:109


Fig.F 10.3

Re:100 Pr:1.2 Computation time:22 hours Time steps:137



Fig.F 11.3



Fig.F 12.3



It is observed from the stream line contours; the verticity is shifting towards top right corner of the cavity in

case of dense grid structure than the coarse grid structure and its magnitude also decreasing with dense grid

structure. It is observed from the figures‘ F1.1 & F1.2, F1.3 & F1.4 and F1.5 & F1.6 of steam function

contours that as the grid size increases from 32X32 to 52X52 structure in the x and y directions; the

simulation catches lower values of stream functions towards bottom surface of the top lid and also we can

observed the sharp changes in the physical shape of the verticity near at top right corner of the cavity for

52X52 grid size.

Temperature contours 12X12 Grid structure 32X32 Grid structure

Fig.F 1.4



Fig.F 2.4



Fig.F 3.4



Fig.F 4.4




Fig.F 5.4



Fig.F 6.4



Fig.F 7.4


Fig.F 8.4


Fig.F 9.4

Re:100 Pr:1.2 Computation time: 25min Time steps:109 No.of iterations in the mass conservation loop:456

Fig.F 10.4



Fig.F 11.4

Re:400 Pr:1.2 Computation time:20min Time steps:126


Fig.F 12.4



It observed from the figures; the characteristic of the temperature contours at the top right corner of the

cavity is changes as the Reynolds number changes from 100 to 400. As the grid structure increases from

12X12 to 32X32; the sensitive variation in the temperature are also catches and it is clearly seen in the

figures. It is also observed from the figures; as the Prandl number increases from 0.5 to 1.2, the higher

temperature contours are available near bottom of the top lid.

U velocity along vertical centerline


Fig.F 1.5


Fig.F 2. 5



Fig.F 3. 5



Fig.F 4. 5



Fig.F 5. 5



Fig.F 6. 5

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 7. 5



Fig.F 8. 5

Re:400 Pr:0.5 Computation time:12 hours Time steps:115 No.of iterations in the mass conservation loop:1938


Fig.F 9. 5

Re:100 Pr:1.2 Computation time: 25min Time steps:109


Fig.F 10. 5



Fig.F 11. 5


Fig.F 12. 5



It is observed from the figures; the horizontal velocity component (U) along the vertical centerline that the

fluid particles will changes their flow direction from right side to the left side at the 0.15 unit distance from

bottom surface along the y direction of the cavity for 1000 Reynolds number in the published results and

that same distance will increases for lower Reynolds numbers i.e 0.28 for Re=400 and 0.55 for Re=100. And

is also observed that that sharp change in the velocity profile for higher Reynolds numbers i.e for 400 and

100 but in lower values of the same number smooth changes will happen i.e for 100.

As we observed from the figures F1.3 to F6.3 that our simulation velocity profile is close with published

profile for lower Reynolds number i.e 100 but this closeness is decreases with higher value of the Reynolds

number; this is because our grid structure is not sufficient to catch the higher velocities of the particles of

higher Reynolds number. Therefore finer grid structure may be necessary to catch the higher Reynolds

number turbulences.

V velocity along horizontal centerline


Fig.F 1.6



Fig.F 2. 6




Fig.F 3. 6


Fig.F 4. 6



Fig.F 5. 6


Fig.F 6. 6

Re:100 Pr:0.5 Computation time: 8.5 hours Time steps:87 No.of iterations in the mass conservation loop:1801

Fig.F 7. 6



Fig.F 8. 6



Fig.F 9. 6 Re:100 Pr:1.2 Computation time: 25min Time steps:109


Fig.F 10. 6 Re:100 Pr:1.2 Computation time: 22hours Time steps:137


Fig.F 11. 6 Re:400 Pr:1.2 Computation time:20min Time steps:126


Fig.F 12. 6




I is observed from same figures; as the magnitude of the Reynolds number increases from 100 to 1000, the

inside surfaces of the left and right side walls effect decreases and velocity of the fluid particles increases; it

enhances little more turbulence in the fluid particles. Hence these turbulences cannot catch in the higher

magnitude Reynolds number; therefore the centerline velocity profile is almost horizontal in case of 1000

Reynolds number and small change in case of Reynolds number 400; but sharp changes in Reynolds number

100.

Temperature along vertical centerline 12X12 Grid structure 32X32 Grid structure

Fig.F 1.7



Fig.F 2. 7



Fig.F 3. 7



Fig.F 4. 7



Fig.F 5. 7



Fig.F 6. 7



Fig.F 7. 7



Fig.F 8. 7




Fig.F 9. 7 Re:100 Pr:1.2 Computation time: 25min Time steps:109


Fig.F 10. 7 Re:100 Pr:1.2 Computation time: 22hours Time steps:137


Fig.F 11. 7


Fig.F 12. 7



It is observed from the figures; the validation of the simulated data will be more for coarse grid structure

(12X12) than the dense grid structure(32X32) compared to the published data and same effect can be

observed for the lower value Reynolds number(100) than the higher value Reynolds number (400). It is also

observed that the sharp changes in the temperature from higher Prandl number (1.2) than the lower(0.5).

Case-III: Mixed convection fluid flow (where Gr=105)


Fig.ML 1.1

Re:100 Pr:1 Time steps: 49

Fig.ML 2.1

Re:100 Pr:1 Time steps:78


Fig.ML 3.1


Fig.ML 4.1


Fig.ML 5.1

Re:100 Pr:0.5 Time steps:44

Fig.ML 6.1


Fig.ML 7.1


Fig.ML 8.1


Fig.ML 9.1 Re:100 Pr:1.2 Time steps:52



Fig.ML 11.1 Re:400 Pr:1.2 steps:103


The U velocity contour verticity is moving towards bottom surface of the cavity as the Prandl number

decreases from 1.2 to 0.5 with higher Reynolds number irrespective of the grid structure. This is because the

top lid velocity increases with liquidity of the fluid inside the cavity.


Fig.ML 1.1 Re:100 Pr:1 Time steps:49

Fig.ML 2.1



Fig.ML 4.1


Fig.ML 6.1




Fig.ML 7.1


Fig.ML 8.1






It is observed from the above shown figures; for lower Reynolds number i.e 100 with lower grid structure

showing three verticities; two at left and right faces and one in between them. But these three verticities are

disappears in case of lower Prandl number (0.5) with higher grid structures (32X32). This is because as the

viscosity will decreases then the Prandl number also decreases with increasing liquidity of the fluid inside

the cavity; this makes uniform in a short duration and it is clearly catches in dense grid structure.


U velocity along vertical centerline 12X12 Grid structure 32X32 Grid structure

Fig.ML 1.1


Fig.ML 2.1


Fig.ML 3.1


Fig.ML 4.1 Re:400 Pr:1 Time steps: 103

Fig.ML 5.1


Fig.ML 6.1


Fig.ML 7.1


Fig.ML 8.1


Fig.ML 9.1


Fig.ML 10.1



Fig.ML 11.1

Re:400 Pr:1.2 steps:103

Fig.ML 12.1


It is observed from the above sown graphs that that the Prandl number increases ( from 0.5 to 1.2) with

lower Reynolds number; the fluid particles will takes a sharp at very close to the left side wall of the cavity.

This is because the top lid is moving with lower velocity and liquidity of the fluid is less for less Reynolds

number (100).

V velocity along horizontal centerline 12X12 Grid structure 32X32 Grid structure

Fig.ML 1.1


Fig.ML 2.1


Fig.ML 3.1


Fig.ML 4.1


Fig.ML 5.1


Fig.ML 6.1



Fig.ML 7.1


Fig.ML 8.1


Fig.ML 9.1


Fig.ML 10.1


Fig.ML 11.1

Re:400 Pr:1.2 steps:103

Fig.ML 12.1


It is observed from the figure; the fluid with lower Reynolds number (100) with lower Prandl number (0.5)

will take two sharp turns at the bottom side of the cavity wall and these sharp turns will shift towards top

surface for higher Reynolds number (400) with higher Prandl number (1.2). But more variations in the

graphs when the grid structure changes from coarse to dense.


Fig.ML 1.1


Fig.ML 2.1



Fig.ML 3.1


Fig.ML 4.1










Fig.ML 12.1 Re:400 Pr:1.2 Time steps125:

It is observed from the figure; two verticities are formed for lower Reynolds number (100) fluid in less

dense grid structure (12X12) and almost only one verticity for verticity for higher Reynolds number (400)

for the same grid structure but in dense grid structure (32X32) only one verticity in all the Reynolds number.


Fig.ML 1.1

Re:100 Pr:1 Time steps: 49 Fig.ML 2.1



Fig.ML 4.1









Fig.ML 11.1 Re:400 Pr:1.2 steps:103 Fig.ML 12.1


It is observed from the figures; the temperature contours with higher Reynolds number are having different

characteristic and magnitude at right top corner of the cavity. This is because the top lid is moving with

higher velocity and it will affect the fluid particles at bottom side of the top surface.



Fig.ML 1.1


Fig.ML 2.1


Fig.ML 3.1


Fig.ML 4.1


Fig.ML 5.1


Fig.ML 6.1


Fig.ML 7.1


Fig.ML 8.1


Fig.ML 9.1


Fig.ML 10.1



Fig.ML 11.1

Re:400 Pr:1.2 steps:103

Fig.ML 12.1


It is observed from the above shown figures; the simulation results are closer with published data for higher

value of the Reynolds number (400) than the lower (100).

Case-III: Mixed convection fluid flow (where Gr=10

6)


Fig.MM 1.1 Re:100 Pr:1 Time steps: 46

Fig.MM 2.1 Re:100 Pr:1 Time steps:81


Fig.MM 4.1


Fig.MM 5.1 Re:100 Pr:0.5 Time steps:58







Fig.MM 11.1 Re:400 Pr:1.2 steps:84


It is observed from the figure; more vertices are found lower Reynolds number. This is because as the

Grashoff number increases, the temperature difference between the cavity walls also increases; this increases

the kinetic energy of the fluid particles.






Fig.MM 4.1











It is observed that the Groshoff number increases with the increasing kinetic energy of the fluid particles due

to its higher temperature; it creates more number of verticities in the cavity. The verticities at left and right

side walls of the cavity will shift towards bottom wall for lower Prandl number. This is because the liquidity

of the fluid particles decreases with increasing Prandl number.



Fig.MM 1.1


Fig.MM 2.1


Fig.MM 3.1


Fig.MM 4.1


Fig.MM 5.1


Fig.MM 6.1



Fig.MM 7.1

Re:400 Pr:0.5 Time steps:64 Fig.MM 8.1


Fig.MM 9.1


Fig.MM 10.1


Fig.MM 11.1

Re:400 Pr:1.2 steps:84

Fig.MM 12.1




Fig.MM 1.1


Fig.MM 2.1


Fig.MM 3.1


Fig.MM 4.1



Fig.MM 5.1


Fig.MM 6.1


Fig.MM 7.1


Fig.MM 8.1


Fig.MM 9.1


Fig.MM 10.1


Fig.MM 11.1


Fig.MM 12.1


Stream function contours


Fig.MM 1.1

Fig.MM 2.1


Re:100 Pr:1 Time steps: 46 Re:100 Pr:1 Time steps:81


Fig.MM 4.1







Fig.MM10.1 Re:100 Pr:1.2 Time steps:85




Temperature contours





Fig.MM 4.1






Fig.MM 8.1


Fig.MM 9.1


Fig.MM 10.1



Fig.MM 12.1 Re:400 Pr:1.2 Time steps: 99

Temperature along vertical centerline


Fig.MM 1.1


Fig.MM 2.1





Fig.MM 5.1


Fig.MM 6.1


Fig.MM 7.1


Fig.MM 8.1


Fig.MM 9.1


Fig.MM 10.1


Fig.MM 11.1


Fig.MM 12.1


Case-III: Mixed convection fluid flow (where Gr=2x106)


Fig.MH 1.1


Fig.MH 2.1



Fig.MH 3.1 Re:400 Pr:1 Time steps:82

Fig.MH 4.1


Fig.MH 5.1


Fig.MH 6.1


Fig.MH 7.1


Fig.MH 8.1


Fig.MH 9.1 Re:100 Pr:1.2 Time steps:53


Fig.MH 11.1


Fig.MH12.1 Re:400 Pr:1.2 Time steps:120



Fig.MH 1.1 Re:100 Pr:1 Time steps: 53


Fig.MH 3.1


Fig.MH 4.1



Fig.MH 6.1


Fig.MH 7.1

Re:400 Pr:0.5 Time steps:62 Fig.MH 8.1




Fig.MH 10.1


Fig.MH 11.1


Fig.MH 12.1




Fig.MH 1.1



Fig.MH 3.1


Fig.MH 4.1


Fig.MH 5.1


Fig.MH 6.1



Fig.MH 7.1


Fig.MH 8.1


Fig.MH 9.1


Fig.MH 10.1


Fig.MH 11.1


Fig.MH 12.1




Fig.MH 1.1


Fig.MH 2.1


Fig.MH 3.1


Fig.MH 4.1



Fig.MH 5.1


Fig.MH 6.1


Fig.MH 7.1


Fig.MH 8.1


Fig.MH 9.1


Fig.MH 10.1


Fig.MH 11.1


Fig.MH 12.1







Fig.MH 3.1



Fig.MH 5.1


Fig.MH 6.1


Fig.MH 7.1



Fig.MH 9.1

Re:100 Pr:1.2 Time steps:53 Fig.MH10.1


Fig.MH 11.1 Re:400 Pr:1.2 steps:87

Fig.MH 12.1








Fig.MH 4.1




Fig.MH7.1 Re:400 Pr:0.5 Time steps:62



Fig.MH 9.1



Fig.MH 11.1


Fig.MH 12.1




Fig.MH 1.1


Fig.MH 2.1


Fig.MH 3.1


Fig.MH 4.1


Fig.MH 5.1


Fig.MH 6.1



Fig.MH 7.1


Fig.MH 8.1


Fig.MH 9.1


Fig.MH 10.1


Fig.MH 11.1


Fig.MH 12.1


Case-III: Mixed convection fluid flow (where Gr=-10

5)


Fig.MNH 1.1


Fig.MNH 2.1


Fig.MNH 4.1


Fig.MNH 3.1 Re:400 Pr:0.5 Time steps:78


Fig.MNH 5.1 Re:100 Pr:1 Time steps: 54

Fig.MNH 6.1 Re:100 Pr:1 Time steps:68



Fig.MH 9.1





The U velocity contour verticity is moving towards bottom surface of the cavity as the Prandl number

decreases from 1.2 to 0.5 with higher Reynolds number irrespective of the grid structure. This is because the

top lid velocity increases with liquidity of the fluid inside the cavity.



Fig.MNH 1.2


Fig.MNH 2.2


Fig.MNH 3.2

Re:400 Pr:0.5 Time steps:78 Fig.MNH 4.2



Fig.MNH 6.2







Fig.MNH 11.2 Re:400 Pr:1.2 steps:48


It is observed from the above shown figures; for lower Reynolds number i.e 100 with lower grid structure

showing three verticities; two at left and right faces and one in between them. But these three verticities are

disappears in case of lower Prandl number (0.5) with higher grid structures (32X32). This is because as the

viscosity will decreases then the Prandl number also decreases with increasing liquidity of the fluid inside

the cavity; this makes uniform in a short duration and it is clearly catches in dense grid structure.


Fig.MNH 1.3


Fig.MNH 2.3


Fig.MNH 3.3


Fig.MNH 4.3



Fig.MNH 5.3


Fig.MNH 6.3


Fig.MNH 7.3


Fig.MNH 8.3


Fig.MNH 9.3


Fig.MNH 10.3


Fig.MNH 11.3


Fig.MNH12.3 Re:400 Pr:1.2 Time steps:52

It is observed from the above sown graphs that that the Prandl number increases ( from 0.5 to 1.2) with

lower Reynolds number; the fluid particles will takes a sharp at very close to the left side wall of the cavity.

This is because the top lid is moving with lower velocity and liquidity of the fluid is less for less Reynolds

number (100).



Fig.MNH 1.4


Fig.MNH 2.4


Fig.MNH 3.4


Fig.MNH 4.4


Fig.MNH 5.4


Fig.MNH 6.4


Fig.MNH 7.4


Fig.MNH 8.4


Fig.MNH 9.4


Fig.MNH 10.4



Fig.MNH 11.4

Re:400 Pr:1.2 steps:48 Fig.MNH 12.4 Re:400 Pr:1.2 Time steps:52

It is observed from the figure; the fluid with lower Reynolds number (100) with lower Prandl number (0.5)

will take two sharp turns at the bottom side of the cavity wall and these sharp turns will shift towards top

surface for higher Reynolds number (400) with higher Prandl number (1.2). But more variations in the

graphs when the grid structure changes from coarse to dense.


Fig.MNH 1.5

Re:100 Pr:0.5 Time steps:243 Fig.MNH 2.5


Fig.MNH 3.5


Fig.MNH 4.5


Fig.MNH 5.5


Fig.MNH 6.5



Fig.MNH 7.5

Re:400 Pr:1 Time steps:57 Fig.MNH 8.5


Fig.MNH 9.5



Fig.MNH 11.5

Re:400 Pr:1.2 steps:48 Fig.MNH 12.5


It is observed from the figure; two verticities are formed for lower Reynolds number (100) fluid in less

dense grid structure (12X12) and almost only one verticity for verticity for higher Reynolds number (400)

for the same grid structure but in dense grid structure (32X32) only one verticity in all the Reynolds number.


Fig.MNH 1.6


Fig.MNH 2.6



Fig.MNH 3.6


Fig.MNH 4.6





Fig.MNH 8.6




Fig.MNH 11.6 Re:400 Pr:1.2 steps:48



It is observed from the figures; the temperature contours with higher Reynolds number are having different

characteristic and magnitude at right top corner of the cavity. This is because the top lid is moving with

higher velocity and it will affect the fluid particles at bottom side of the top surface.


Fig.MNH 1.7


Fig.MNH 2.7


Fig.MNH 3.7


Fig.MNH 4.7


Fig.MNH 5.7


Fig.MNH 6.7


Fig.MNH 7.7


Fig.MNH 8.7


Fig.MNH 9.7


Fig.MNH 10.7



Fig.MNH 11.7


Fig.MNH 12.7


It is observed from the above shown figures; the simulation results are closer with published data for higher

value of the Reynolds number (400) than the lower (100).

Case-III: Mixed convection fluid flow (where Gr=-2X106)

Note: Due to time restriction to submit the report online; it not possible do some simulation in mixed fluid flow

with 100 Reynolds number with 32X32 grid structure in the VII chapter ‗Results and discussion‘ ; because

these simulations will take days together. Therefore the some contours and graphs are predicted based on the

trends of the fluid flow and characteristics of the fluid flow already simulated in the previous cases.


Fig.MNL 1.1 Re:100 Pr:0.5 Time steps:217

Fig.MNL 2.1 Re:100 Pr:0.5 Time steps:


Fig.MNL 4.1



Fig.MNL 5.1


Fig.MNL 6.1 Re:100 Pr:1 Time steps:

Fig.MNL 7.1 Re:400 Pr:1 Time steps:340 Fig.MNL 8.1


Fig.ML 9.1











Fig.MNL 5.2 Re:100 Pr:1 Time steps:320



Fig.MNL 8.2 Re:400 Pr:1 Time steps: 54


Fig.ML 9.2

Re:100 Pr:1.2 Time steps: Fig.MNL 10.2 Re:100 Pr:1.2 Time steps:52

Fig.MNL 11.2 Re:400 Pr:1.2 steps:344



Fig.MNL 1.3



Fig.MNL 3.3


Fig.MNL 4.3



Fig.MNL 5.3



Fig.MNL 7.3


Fig.MNL 8.3


Fig.MNL 9.3



Fig.MNL 11.3

Re:400 Pr:1.2 steps:344




Fig.MNL 1.4



Fig.MNL 3.4


Fig.MNL 4.4


Fig.MNL 5.4



Fig.MNL 7.4


Fig.MNL 8.4


Fig.MNL 9.4




Fig.MNL 11.4

Re:400 Pr:1.2 steps:344

Fig.MNL 12.4



Fig.MNL 1.5



Fig.MNL 3.5



Fig.MNL 5.5

Re:100 Pr:1 Time steps:



Fig.MNL 7.5

Re:400 Pr:1 Time steps:340 Fig.MNL 8.5


Fig.MNL 9.5



Fig.MNL 11.5

Re:400 Pr:1.2 steps:344











Fig.MNL 8.6








Fig.MNL 1.7



Fig.MNL 3.7


Fig.MNL 4.7


Fig.MNL 5.7

Re:100 Pr:1 Time steps:


Fig.MNL 7.7


Fig.MNL 8.7



Fig.MNL 9.7




Fig.MNL 12.7


Case-IV: Natural convection fluid flow


Fig. N 1.1

Gr:103 Pr:0.71 Time steps: 204

Fig. N 2.1


Computation time: 21hours

Fig. N 3.1

Gr:104 Pr:0.71 Time steps:230

Fig. N 4.1 Gr:104 Pr:0.71 Time steps: 1198




Fig. N 1.2


Fig. N 2.2



Fig. N 3.2


Fig. N 4.2





Fig. N 1.3


Fig. N 2.3



Fig. N 3.3


Fig. N 4.3





Fig. N 1.4


Fig. N 2.4



Fig. N 3.4


Fig. N 4.4





Fig. N 1.5


Fig. N 2.5



Fig. N 3.5


Fig. N 4.5






Fig. N 1.6


Fig. N 2.6



Fig. N 3.6 Gr:104 Pr:0.71 Time steps:230

Fig. N 4.6 Gr:104 Pr:0.71 Time steps: 1198




Fig. N 1.7


Fig. N 2.7



Fig. N 3.7


Fig. N 4.7




Temperature along horizontal centerline 12X12 Grid structure 32X32 Grid structure

Fig. N 1.8


Fig. N 2.8



Fig. N 3.8


Fig. N 4.8



It is observed from the figure; the Grashoff number increases from 10

3 to 10

4, the computation time and time

steps required for the convergence of the solution increases as compared to the forced and mixed convection

flow for dense grid structure. This is because the temperature difference increases with increasing Grashoff

number. For the less value of Grashoff number in dense grid structure (32X32); the temperature values

along the horizontal center line not close to the published data. This difference may be decreases with the

high dense grid structure.


Chapter-VIII

Scilab CFD codes used in the problem and Algorithm The code is written in non-dimensional form, with Reynolds number (Re=ρU0L/μ) as the governing

parameter for isothermal flow. For convective heat transfer problems, Prandtl number (Pr=ν/α), Grashoff

number (Gr=gβ(TH-TC)L3/ν

2) and Rayleigh Number (Ra=gβ(TH-TC)L

3/να) comes as an additional governing

parameters. However, Gr=0 for forced and RePr=1 for natural convection heat transfer. 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.

Solution Algorithm:

1) Enter the inputs: material properties, geometric parameters (L1 & L2) and maximum number of CVs

in the X and Y directions, B.Cs input and εs.

2) Grid generation: calculate all the geometric parameters of all the CVs.

3) Set ∆ from the stability criteria.

4) Set the initial condition for .

5) Set the boundary condition for

6) Set = for all CVs

7) For =u, calculate fluxes (mass, advection, diffusion) in the X and Y direction at the u-CV faces

using velocity of previous time step and S(j,i)=(Pold(j,i+1))* Y

8) Calculate total advection at all centers Ai,j.

9) Calculate total diffusion at all centers, Di,j.

10) For each ―interior‖ CVs, Predict velocity as = - (-Aj,i+Dj,i+Sj,i)

11) For =v, calculate the fluxes at the v-CV faces using velocity of previous time step with

S(j,i)=(Pold(j,i))-Pold(j+1,i)* X and repeat steps 7-10

12) If max(Divi,j)< ε, then go to step 16 else continue

13) Compute P‘i,j at interior nodes using the mass imbalance Divi,j

14) Pn+1= P

n+ P

‘ for all interior grid points

15) Compute velocity correction using pressure correction, update the predicted star velocity and go to

step 12.

16) The star velocity becomes the velocity for next time step. Solve the energy equation.

17) Go to step 5 continue all steady state.

Scilab codes for Case-I:

// *****************************************************************************

// Codes developed by Vishesh Aggarwal

// Under the supervision of Dr.Atul Sharma, IIT Bombay

// *****************************************************************************

clc;

printf("\n");

printf("*******************************************************************\n");

printf(" LID DRIVEN CAVITY PROBLEM USING 2D STAGGERED GRID NS SOLVER\n");

printf("*******************************************************************\n");

printf("\nGOVERNING PARAMETERS:");

printf("\n\tREYNOLDS NUMBER (Re) BASED ON TOP PLATE VELOCITY\n");

printf("\nBENCHMARK DATA AVAILABLE AT Re = 100, 400, 1000\n");

Re = input("ENTER Re (Must be 100 or 400 or 1000 for benchmarking): ")


// **************************** PROBLEM PARAMETERS *****************************

// NOTE: The parameters are based on non-dimensional governing equation

U = 1; //Top-plate velocity (characteristic velocity scale)

Lx = 1; //Length of domain in x-direction (characteristic length scale)

Ly = 1; //Length of domain in y-direction

dens = 1; //Fluid density

vis = 1/Re; //Fluid viscosity

// ***************************** DEFINE GRID SIZE ******************************

printf("\nENTER THE NO. OF GRID POINTS\n");

// NOTE: The entered value includes the boundary grid points

// This number is based on the pressure cell centre locations

imax = input("IN THE X-DIRECTION : ");

jmax = input("IN THE Y-DIRECTION : ");

dx = Lx/(imax-2); // Grid spacing in x-direction

dy = Ly/(jmax-2); // Grid spacing in y-direction

dV = dx*dy;

// ************ TIME STEP EVALUATION (BASED ON STABILITY CRITERION) ************

// NOTE: Courant–Friedrichs–Lewy (CFL) and Grid Fourier Criterion are used below

// These are only neccessary but not sufficient condition for stability

// since they are obtained from pure convection and pure diffusion, but not for

// the NS equation which is a convection-diffusion equation with a source term

// Furthermore, the maximum velocity needed here to obtained minimum time-step

// is equal to lid velocity.

// If the maximum velocity occurs inside the domain and changes with time,

// then this expression needs to be used after each transient computation.

dt = min(0.5*dx/U, 0.25*((dx*dy)*(dx*dy)/((vis/dens)*(dx*dx + dy*dy))));

// ************************* OTHER CONTROL PARAMETERS **************************

steady_state_criteria = 1e-3; // Used to stop outer time loop

mass_div_criteria = 1e-8; // Used to stop inner mass divergence loop

time_step = 0;

total_time = 0;

// ***************** DEFINING ARRAYS TO HOLD PROBLEM VARIABLES *****************

x = zeros(jmax-1,imax-1);

y = zeros(jmax-1,imax-1);

x_p = zeros(jmax,imax);

y_p = zeros(jmax,imax);

x_u = zeros(jmax,imax-1);

y_u = zeros(jmax,imax-1);

x_v = zeros(jmax-1,imax);

y_v = zeros(jmax-1,imax);

u = zeros(jmax,imax-1);

v = zeros(jmax-1,imax);

p = zeros(jmax,imax);

pc = zeros(jmax,imax);

uold = zeros(jmax,imax-1);

vold = zeros(jmax-1,imax);

ustar = zeros(jmax,imax-1);

vstar = zeros(jmax-1,imax);

Div = zeros(jmax-1,imax-1);

mx1 = zeros(jmax,imax-2);

ax1 = zeros(jmax,imax-2);

dx1 = zeros(jmax,imax-2);

my1 = zeros(jmax,imax-2);

ay1 = zeros(jmax,imax-2);

dy1 = zeros(jmax,imax-2);


mx2 = zeros(jmax-2,imax);

ax2 = zeros(jmax-2,imax);

dx2 = zeros(jmax-2,imax);

my2 = zeros(jmax-2,imax);

ay2 = zeros(jmax-2,imax);

dy2 = zeros(jmax-2,imax);

// ******************** ASSIGNING STAGGERED GRID INFORMATION *******************

// Corner vertices of each p-cell

for i=1:1:imax-1

for j=1:1:jmax-1

x(j,i) = (i-1)*dx;

y(j,i) = (j-1)*dy;

end

end

// Cell center of interior p-cell

for i=2:1:imax-1

for j=2:1:jmax-1

x_p(j,i) = 0.5*(x(j,i) + x(j,i-1));

y_p(j,i) = 0.5*(y(j,i) + y(j-1,i));

end

end

// Cell center of boundary p-cell

for i=2:1:imax-1

x_p(1,i) = 0.5*(x(1,i)+x(1,i-1));

y_p(1,i) = 0;

x_p(jmax,i) = 0.5*(x(jmax-1,i)+x(jmax-1,i-1));

y_p(jmax,i) = Ly;

end

for j=2:1:jmax-1

x_p(j,1) = 0;

y_p(j,1) = 0.5*(y(j,1)+y(j-1,1));

x_p(j,imax) = Lx;

y_p(j,imax) = 0.5*(y(j,imax-1)+y(j-1,imax-1));

end

// Corner p-cells of domain

x_p(1,1) = 0;

y_p(1,1) = 0;

x_p(1,imax) = Lx;

y_p(1,imax) = 0;

x_p(jmax,1) = 0;

y_p(jmax,1) = Ly;

x_p(jmax,imax) = Lx;

y_p(jmax,imax) = Ly;

// Cell center of interior u-cell

for i=2:1:imax-2

for j=2:1:jmax-1

x_u(j,i) = x(j,i);

y_u(j,i) = 0.5*(y(j,i)+y(j-1,i));

end

end

// Cell center of boundary u-cell

for i=2:1:imax-2

x_u(1,i) = x(1,i);

y_u(1,i) = 0;


x_u(jmax,i) = x(jmax-1,i);

y_u(jmax,i) = Ly;

end

for j=2:1:jmax-1

x_u(j,1) = 0;

y_u(j,1) = 0.5*(y(j,1)+y(j-1,1));

x_u(j,imax-1) = Lx;

y_u(j,imax-1) = 0.5*(y(j,imax-1)+y(j-1,imax-1));

end

// Corner u-cells of domain

x_u(1,1) = 0;

y_u(1,1) = 0;

x_u(1,imax-1) = Lx;

y_u(1,imax-1) = 0;

x_u(jmax,1) = 0;

y_u(jmax,1) = Ly;

x_u(jmax,imax-1) = Lx;

y_u(jmax,imax-1) = Ly;

// Cell center of interior v-cell

for i=2:1:imax-1

for j=2:1:jmax-2

x_v(j,i) = 0.5*(x(j,i)+x(j,i-1));

y_v(j,i) = y(j,i);

end

end

// Cell center of boundary v-cell

for i=2:1:imax-1

x_v(1,i) = 0.5*(x(1,i)+x(1,i-1));

y_v(1,i) = 0;

x_v(jmax-1,i) = 0.5*(x(jmax-1,i)+x(jmax-1,i-1));

y_v(jmax-1,i) = Ly;

end

for j=2:1:jmax-2

x_v(j,1) = 0;

y_v(j,1) = y(j,1);

x_v(j,imax) = Lx;

y_v(j,imax) = y(j,imax-1);

end

// Corner v-cells of domain

x_v(1,1) = 0;

y_v(1,1) = 0;

x_v(1,imax) = Lx;

y_v(1,imax) = 0;

x_v(jmax-1,1) = 0;

y_v(jmax-1,1) = Ly;

x_v(jmax-1,imax) = Lx;

y_v(jmax-1,imax) = Ly;

// ************************ APPLYING INITIAL CONDITIONS ************************

for i=1:1:imax


for j=1:1:jmax

u(j,i) = 0;

v(j,i) = 0;

ustar(j,i) = 0;

vstar(j,i) = 0;

p(j,i) = 0;

end

end

// ******************** FUNCTION: APPLY BOUNDARY CONDITION *********************

// NOTE: Boundary condition application is encapsulated in a function

// It allows ease in modification of boundary conditions based on problem setup

// It can be called within the main loop repeatedly if the problem demands

function [u, v, ustar, vstar, p]=APPLY_BC(u, v, ustar, vstar, p)

funcprot(0);

// Bottom Boundary

for i=1:1:imax

u(1,i) = 0;

v(1,i) = 0;

ustar(1,i) = 0;

vstar(1,i) = 0;

p(1,i) = p(2,i);

end

// Top Boundary

for i=1:1:imax

u(jmax,i) = U;

v(jmax-1,i) = 0;

ustar(jmax,i) = U;

vstar(jmax-1,i) = 0;

p(jmax,i) = p(jmax-1,i);

end

// Left Boundary

for j=1:1:jmax

u(j,1) = 0;

v(j,1) = 0;

ustar(j,1) = 0;

vstar(j,1) = 0;

p(j,1) = p(j,2);

end

// Right Boundary

for j=1:1:jmax

u(j,imax-1) = 0;

v(j,imax) = 0;

ustar(j,imax-1) = 0;

vstar(j,imax) = 0;

p(j,imax) = p(j,imax-1);

end

endfunction

// ************* FUNCTION: PRESSURE CORRECTION INITIAL CONDITION ***************

function [pc]=APPLYIC_PCORR(pc)

funcprot(0);

for j=1:1:jmax

for i=1:1:imax

pc(j,i) = 0;

end

end

endfunction

// ************* FUNCTION: PRESSURE CORRECTION BOUNDARY CONDITION **************

function [pc]=APPLYBC_PCORR(pc)


for j=1:1:jmax

pc(j,1) = pc(j,2);

pc(j,imax) = pc(j,imax-1);

end

for i=1:1:imax

pc(1,i) = pc(2,i);

pc(jmax,i) = pc(jmax-1,i);

end

endfunction

// ********************* MAIN TIME LOOPING BEGINS HERE *************************

unsteadiness = 1e6;

while unsteadiness > steady_state_criteria

// Apply boundary conditions

[u,v,ustar,vstar,p] = APPLY_BC(u,v,ustar,vstar,p);

// Store old time level data

uold = u;

vold = v;

//******************************************************************

// Predict new time level velocities

// Fluxes across u-velocity cell faces

for j=2:1:jmax-1

for i=1:1:imax-2

mx1(j,i) = dens*0.5*(u(j,i)+u(j,i+1));

ax1(j,i) = max(mx1(j,i),0)*u(j,i) - max(-mx1(j,i),0)*u(j,i+1);

dx1(j,i) = vis*(u(j,i+1)-u(j,i))/(x_u(j,i+1)-x_u(j,i));

end

end

for j=1:1:jmax-1

for i=2:1:imax-2

my1(j,i) = dens*0.5*(v(j,i)+v(j,i+1));

ay1(j,i) = max(my1(j,i),0)*u(j,i) - max(-my1(j,i),0)*u(j+1,i);

dy1(j,i) = vis*(u(j+1,i)-u(j,i))/(y_u(j+1,i)-y_u(j,i));

end

end

// Fluxes across v-velocity cell faces

for j=2:1:jmax-2

for i=1:1:imax-1

mx2(j,i) = dens*0.5*(u(j,i)+u(j+1,i));

ax2(j,i) = max(mx2(j,i),0)*v(j,i) - max(-mx2(j,i),0)*v(j,i+1);

dx2(j,i) = vis*(v(j,i+1)-v(j,i))/(x_v(j,i+1)-x_v(j,i));

end

end

for j=1:1:jmax-2

for i=2:1:imax-1

my2(j,i) = dens*0.5*(v(j,i)+v(j+1,i));

ay2(j,i) = max(my2(j,i),0)*v(j,i) - max(-my2(j,i),0)*v(j+1,i);

dy2(j,i) = vis*(v(j+1,i)-v(j,i))/(y_v(j+1,i)-y_v(j,i));

end

end

// Predict cell center velocities

for j=2:1:jmax-1

for i=2:1:imax-2

Au = (ax1(j,i)-ax1(j,i-1))*dy + (ay1(j,i)-ay1(j-1,i))*dx;

Du = (dx1(j,i)-dx1(j,i-1))*dy + (dy1(j,i)-dy1(j-1,i))*dx;

Su = (p(j,i)-p(j,i+1))*dy;

ustar(j,i) = u(j,i) + (dt/(dens*dV))*(Du-Au+Su);


end

end

for j=2:1:jmax-2

for i=2:1:imax-1

Av = (ax2(j,i)-ax2(j,i-1))*dy + (ay2(j,i)-ay2(j-1,i))*dx;

Dv = (dx2(j,i)-dx2(j,i-1))*dy + (dy2(j,i)-dy2(j-1,i))*dx;

Sv = (p(j,i)-p(j+1,i))*dx;

vstar(j,i) = v(j,i) + (dt/(dens*dV))*(Dv-Av+Sv);

end

end

//******************************************************************

// Divergence term (mass error) evaluation per cell

RMS_Div = 1e6;

[pc] = APPLYIC_PCORR(pc);

count = 0;

while (RMS_Div > mass_div_criteria)

// NOTE: It may be needed to restrict the maximum no. of iterations

// besides checking convergence for some flow problems

// Further, applying boundary conditions for USTAR and VSTAR

// within this loop is also useful for channel flow problems

RMS_Div = 0;

for j=2:1:jmax-1

for i=2:1:imax-1

Div(j,i) = (ustar(j,i)-ustar(j,i-1))*dens*dy + (vstar(j,i)-vstar(j-1,i))*dens*dx;

if (RMS_Div<abs(Div(j,i))) then

RMS_Div = abs(Div(j,i));

end

end

end

// Corrector step

for j=2:1:jmax-1

for i=2:1:imax-1

aW = dens*dt*dy/(x_p(j,i)-x_p(j,i-1));

aE = dens*dt*dy/(x_p(j,i+1)-x_p(j,i));

aS = dens*dt*dx/(y_p(j,i)-y_p(j-1,i));

aN = dens*dt*dx/(y_p(j+1,i)-y_p(j,i));

aP = aW+aE+aS+aN;

pc(j,i) = (aE*pc(j,i+1) + aW*pc(j,i-1) + aN*pc(j+1,i) + aS*pc(j-1,i) - Div(j,i))/aP;

end

end

[pc] = APPLYBC_PCORR(pc);

for j=2:1:jmax-1

for i=2:1:imax-2

ustar(j,i) = ustar(j,i) + (dt/(dens*dV))*(pc(j,i)-pc(j,i+1))*dy;

end

end

for j=2:1:jmax-2

for i=2:1:imax-1

vstar(j,i) = vstar(j,i) + (dt/(dens*dV))*(pc(j,i)-pc(j+1,i))*dx;

end

end

count = count+1;

end

printf("\tIterations in the mass convergence loop = %d\n", count)

//******************************************************************

// Update new time level pressure


for j=2:1:jmax-1

for i=2:1:imax-1

p(j,i) = p(j,i) + pc(j,i);

end

end

// Update new time level velocities

u = ustar;

v = vstar;

//******************************************************************

// Check for convergence of the solution

RMS1=0;

for i=1:1:imax-1

for j=1:1:jmax

RMS1 = RMS1 + (u(j,i)-uold(j,i))*(u(j,i)-uold(j,i));

end

end

RMS1 = sqrt(RMS1/((imax-1)*jmax));

RMS2=0;

for i=1:1:imax

for j=1:1:jmax-1

RMS2 = RMS2 + (v(j,i)-vold(j,i))*(v(j,i)-vold(j,i));

end

end

RMS2 = sqrt(RMS2/(imax*(jmax-1)));

RMS_RESIDUE = max(RMS1,RMS2);

unsteadiness = RMS_RESIDUE;

time_step = time_step+1;

total_time = total_time+dt;

printf('Time step = %4d, Error = %5.3e\n',time_step,unsteadiness);

end

//*************************** Output ****************************

// U-VELOCITY

xu = zeros(imax-1);

yu = zeros(jmax);

for i=1:1:imax-1

xu(i) = x_u(1,i);

end

for j=1:1:jmax

yu(j) = y_u(j,1);

end

U_TRAN = zeros(imax-1,jmax);

for j=1:1:jmax

for i=1:1:imax-1

U_TRAN(i,j) = u(j,i);

end

end

xset('window',1);

clf(1);

UMIN = min(u);

UMAX = max(u);

colorbar(UMIN,UMAX);

title('U-VELOCITY CONTOUR PLOT OVER THE DOMAIN', 'color', 'black', 'fontsize',3);

Sgrayplot(xu,yu,U_TRAN,strf="041");

xset("colormap",jetcolormap(32));

// V-VELOCITY


xv = zeros(imax);

yv = zeros(jmax-1);

for i=1:1:imax

xv(i) = x_v(1,i);

end

for j=1:1:jmax-1

yv(j) = y_v(j,1);

end

V_TRAN = zeros(imax,jmax-1);

for j=1:1:jmax-1

for i=1:1:imax

V_TRAN(i,j) = v(j,i);

end

end

xset('window',2);

clf(2);

VMIN = min(v);

VMAX = max(v);

colorbar(VMIN,VMAX);

title('V-VELOCITY CONTOUR PLOT OVER THE DOMAIN', 'color', 'black', 'fontsize',3);

Sgrayplot(xv,yv,V_TRAN,strf="041");

xset("colormap", jetcolormap(32));

// CENTRELINE PLOTS

xset('window',3);

clf(3);

uc = zeros(jmax);

if (modulo(imax,2)==0) then

for j=1:1:jmax

uc(j) = u(j,imax/2);

end

else

for j=1:1:jmax

uc(j) = (u(j,(imax-1)/2)+u(j,(imax-1)/2+1))/2;

end

end

// GHIA ET AL. (1982) JOURNAL OF COMPUTATIONAL PHYSICS VOL. 48, pp.387-411 @ Re = 100

ypA = [1 0.9766 0.9688 0.9609 0.9531 0.8516 0.7344 0.6172 0.5 0.4531 0.2813 0.1719 0.1016 0.0703 0.0625 0.0547 0];

ucA100 = [1 0.84123 0.78871 0.73722 0.68717 0.23151 0.00332 -0.13641 -0.20581 -0.2109 -0.15662 -0.1015 -0.06434 -0.04775

-0.04192 -0.03717 0];

ucA400 = [1 0.75837 0.68439 0.61756 0.55892 0.29093 0.16256 0.02135 -0.11477 -0.17119 -0.32726 -0.24299 -0.14612 -

0.10338 -0.09266 -0.08186 0];

ucA1000 = [1 0.65928 0.57492 0.51117 0.46604 0.33304 0.18719 0.05702 -0.0608 -0.10648 -0.27805 -0.38289 -0.2973 -0.2222 -

0.20196 -0.18109 0];

title('VARIATION OF U-VELOCITY ALONG THE VERTICAL CENTRELINE', 'color', 'black', 'fontsize',3);

xlabel("U-VELOCITY");

ylabel("Y-DISTANCE");

plot2d(uc,yu,axesflag=1);

if (Re==100) then

plot2d(ucA100,ypA,leg="Benchmarks Results (Ghia et al. (1982))");

e=gce();

e1=e.children(1);e1.line_mode="off";e1.mark_style=0;e1.mark_size_unit="point";e1.mark_size=4;

end

if (Re==400) then


e=gce();


end


if (Re==1000) then


e=gce();


end

xset('window',4);

clf(4);

vc = zeros(imax);

if (modulo(jmax,2)==0) then

for i=1:1:imax

vc(i) = v(jmax/2,i);

end

else

for i=1:1:imax

vc(i) = (v((jmax-1)/2,i)+v((jmax-1)/2+1,i))/2;

end

end


vcA100 = [0 -0.05906 -0.07391 -0.08864 -0.10313 -0.16914 -0.22445 -0.24533 0.05454 0.17527 0.17507 0.16077 0.12317

0.1089 0.100091 0.09233 0];

vcA400 = [0 -0.12146 -0.15663 -0.19254 -0.22847 -0.23827 -0.44993 -0.38598 0.05186 0.30174 0.30203 0.28124 0.22965

0.2092 0.19713 0.1836 0];

vcA1000 = [0 -0.21388 -0.27669 -0.33714 -0.39188 -0.5155 -0.42665 -0.31966 0.02526 0.32235 0.33075 0.37095 0.32627

0.30353 0.29012 0.27485 0];

xpA = [1 0.9688 0.9609 0.9531 0.9453 0.9063 0.8594 0.8047 0.5 0.2344 0.2266 0.1563 0.0938 0.0781 0.0703 0.0625 0];

title('VARIATION OF V-VELOCITY ALONG THE HORIZONTAL CENTRELINE', 'color', 'black', 'fontsize',3);

xlabel("X-DISTANCE");

ylabel("V-VELOCITY");

plot2d(xv,vc,axesflag=1);

if (Re==100) then

plot2d(xpA,vcA100,leg="Benchmarks Results (Ghia et al. (1982))");

e=gce();


end

if (Re==400) then


e=gce();


end

if (Re==1000) then


e=gce();


end

// STREAM FUNCTION

SI = zeros(jmax,imax-1);

SI_TRAN = zeros(imax-1,jmax);

for j=1:1:jmax

SI(j,1) = 0;

SI(j,imax-1) = 0;

end

for i=1:1:imax-1

SI(1,i) = 0;

SI(jmax,i) = 0;

end


for j=2:1:jmax-1

for i=2:1:imax-2

SI(j,i) = u(j-1,i)*(y(j,i)-y(j-1,i)) + SI(j-1,i);

end

end

for j=1:1:jmax

for i=1:1:imax-1

SI_TRAN(i,j) = SI(j,i);

end

end

xset('window',5);

clf(5);

SI_MIN = min(abs(SI));

SI_MAX = max(abs(SI));

colorbar(SI_MIN,SI_MAX);

title('STREAM FUNCTION PLOT', 'color', 'black', 'fontsize',3);

Sgrayplot(xu,yu,SI_TRAN,strf="041");


Scilab codes for Case-II:

// *****************************************************************************



// *****************************************************************************

clc;

printf("\n");

printf("*******************************************************************\n");

printf("FORCED/MIXED CONVECTION IN A LID DRIVEN CAVITY ON 2D STAGGERED GRID\n");

printf("*******************************************************************\n");

printf("\n*******************************************************************");

printf("\n LID-DRIVEN CAVITY WITH ISOTHERMAL WALLS IS CONSIDERED HERE....");

printf("\n THE TOP WALL IS CONSIDERED HOTTER,");

printf("\n WHILE THE REMAINING THREE WALLS ARE TAKEN TO BE COOLER");

printf("\n*******************************************************************");

printf("\n\nGOVERNING PARAMETERS:");

printf("\n\t- REYNOLDS NUMBER (Re) BASED ON TOP PLATE VELOCITY");

printf("\n\t- GRASHOFF NUMBER (Gr) BASED ON DIFFERENCE IN TOP AND BOTTOM WALL TEMPERATURES");

printf("\n\t- PRANDTL NUMBER (Pr) OF THE FLUID");

printf("\n\nBENCHMARK DATA AVAILABLE AT Re = 100, Pr = 1, Gr = 0 and (+/-)1E6\n");

Re = input("ENTER THE REYNOLDS NUMBER (Must be 100 for benchmarking): ");

Pr = input("ENTER THE PRANDTL NUMBER (Must be 1 for benchmarking): ");

Gr = input("ENTER THE GRASHOFF NUMBER (Must be 0 or +1E6 or -1E6 for benchmarking): ");

Ri = Gr/(Re*Re);

printf("RICHARDSON NUMBER : %2.1f\n",Ri);







vis = 1/Re; //Fluid viscosity

Cp = 1; //Specific heat

dif = 1/(Re*Pr); //Thermal diffusivity

TW1 = 1; //Top wall temperature

TW2 = 0; //Bottom wall temperature

// ***************************** DEFINE GRID SIZE ******************************









dV = dx*dy;

// ************ TIME STEP EVALUATION (BASED ON STABILITY CRITERION) ************





// Furthermore, the maximum velocity needed here to obtained minimum time-step

// is equal to lid velocity.

// If the maximum velocity occurs inside the domain and changes with time,

// then this expression needs to be used after each transient computation.

dt = min(0.5*dx/U, 0.25*((dx*dy)*(dx*dy)/((vis/dens)*(dx*dx + dy*dy))),0.25*((dx*dy)*(dx*dy)/(dif*(dx*dx + dy*dy))));

if Gr<0 then

// NOTE: A -ve Gr demands a stringent time step criterion; hence taking even more conservative time step

dt = 0.25*dt;

end




time_step = 0;

total_time = 0;













T = zeros(jmax,imax);




Told = zeros(jmax,imax);

















mxT = zeros(jmax-1,imax-1);

axT = zeros(jmax-1,imax-1);

dxT = zeros(jmax-1,imax-1);

myT = zeros(jmax-1,imax-1);

ayT = zeros(jmax-1,imax-1);

dyT = zeros(jmax-1,imax-1);



for i=1:1:imax-1

for j=1:1:jmax-1

x(j,i) = (i-1)*dx;

y(j,i) = (j-1)*dy;

end

end


for i=2:1:imax-1

for j=2:1:jmax-1

x_p(j,i) = 0.5*(x(j,i) + x(j,i-1));

y_p(j,i) = 0.5*(y(j,i) + y(j-1,i));

end

end


for i=2:1:imax-1

x_p(1,i) = 0.5*(x(1,i)+x(1,i-1));

y_p(1,i) = 0;


y_p(jmax,i) = Ly;

end

for j=2:1:jmax-1

x_p(j,1) = 0;

y_p(j,1) = 0.5*(y(j,1)+y(j-1,1));

x_p(j,imax) = Lx;


end


x_p(1,1) = 0;

y_p(1,1) = 0;

x_p(1,imax) = Lx;

y_p(1,imax) = 0;

x_p(jmax,1) = 0;

y_p(jmax,1) = Ly;




for i=2:1:imax-2

for j=2:1:jmax-1

x_u(j,i) = x(j,i);

y_u(j,i) = 0.5*(y(j,i)+y(j-1,i));

end

end



for i=2:1:imax-2

x_u(1,i) = x(1,i);

y_u(1,i) = 0;


y_u(jmax,i) = Ly;

end

for j=2:1:jmax-1

x_u(j,1) = 0;

y_u(j,1) = 0.5*(y(j,1)+y(j-1,1));

x_u(j,imax-1) = Lx;


end


x_u(1,1) = 0;

y_u(1,1) = 0;

x_u(1,imax-1) = Lx;

y_u(1,imax-1) = 0;

x_u(jmax,1) = 0;

y_u(jmax,1) = Ly;




for i=2:1:imax-1

for j=2:1:jmax-2

x_v(j,i) = 0.5*(x(j,i)+x(j,i-1));

y_v(j,i) = y(j,i);

end

end


for i=2:1:imax-1

x_v(1,i) = 0.5*(x(1,i)+x(1,i-1));

y_v(1,i) = 0;


y_v(jmax-1,i) = Ly;

end

for j=2:1:jmax-2

x_v(j,1) = 0;

y_v(j,1) = y(j,1);

x_v(j,imax) = Lx;


end


x_v(1,1) = 0;

y_v(1,1) = 0;

x_v(1,imax) = Lx;

y_v(1,imax) = 0;

x_v(jmax-1,1) = 0;

y_v(jmax-1,1) = Ly;





for i=1:1:imax

for j=1:1:jmax

u(j,i) = 0;

v(j,i) = 0;

ustar(j,i) = 0;

vstar(j,i) = 0;

p(j,i) = 0;

T(j,i) = 0;

end

end





function [u, v, ustar, vstar, p, T]=APPLY_BC(u, v, ustar, vstar, p, T)

funcprot(0);

// Bottom Boundary

for i=1:1:imax

u(1,i) = 0;

v(1,i) = 0;

ustar(1,i) = 0;

vstar(1,i) = 0;

p(1,i) = p(2,i);

T(1,i) = TW2;

end

// Top Boundary

for i=1:1:imax

u(jmax,i) = U;

v(jmax-1,i) = 0;

ustar(jmax,i) = U;



T(jmax,i) = TW1;

end

// Left Boundary

for j=1:1:jmax

u(j,1) = 0;

v(j,1) = 0;

ustar(j,1) = 0;

vstar(j,1) = 0;

p(j,1) = p(j,2);

T(j,1) = TW2;

end

// Right Boundary

for j=1:1:jmax

u(j,imax-1) = 0;

v(j,imax) = 0;


vstar(j,imax) = 0;


T(j,imax) = TW2;

end

endfunction



funcprot(0);

for j=1:1:jmax


for i=1:1:imax

pc(j,i) = 0;

end

end

endfunction

// ************* FUNCTION: PRESSURE CORRECTION BOUNDARY CONDITION **************


for j=1:1:jmax

pc(j,1) = pc(j,2);


end

for i=1:1:imax

pc(1,i) = pc(2,i);


end

endfunction


unsteadiness = 1e6;



[u,v,ustar,vstar,p,T] = APPLY_BC(u,v,ustar,vstar,p,T);


uold = u;

vold = v;

Told = T;

//******************************************************************

// Solving temperature equation

// Fluxes in x-direction for T

for j=2:1:jmax-1

for i=1:1:imax-1

mxT(j,i) = dens*Cp*u(j,i);

axT(j,i) = max(mxT(j,i),0)*T(j,i) - max(-mxT(j,i),0)*T(j,i+1);

dxT(j,i) = dif*(T(j,i+1)-T(j,i))/(x_p(j,i+1)-x_p(j,i));

end

end

// Fluxes in y-direction for T

for j=1:1:jmax-1

for i=2:1:imax-1

myT(j,i) = dens*Cp*v(j,i);

ayT(j,i) = max(myT(j,i),0)*T(j,i) - max(-myT(j,i),0)*T(j+1,i);

dyT(j,i) = dif*(T(j+1,i)-T(j,i))/(y_p(j+1,i)-y_p(j,i));

end

end

// Get new time level temperatures

for j=2:1:jmax-1

for i=2:1:imax-1

DTp = (dxT(j,i)-dxT(j,i-1))*dy + (dyT(j,i)-dyT(j-1,i))*dx;

ATp = (axT(j,i)-axT(j,i-1))*dy + (ayT(j,i)-ayT(j-1,i))*dx;

T(j,i) = T(j,i) + 1.25*dt*(DTp - ATp)/dV;

end

end

//******************************************************************



for j=2:1:jmax-1

for i=1:1:imax-2


mx1(j,i) = dens*0.5*(u(j,i)+u(j,i+1));



end

end

for j=1:1:jmax-1

for i=2:1:imax-2

my1(j,i) = dens*0.5*(v(j,i)+v(j,i+1));



end

end


for j=2:1:jmax-2

for i=1:1:imax-1

mx2(j,i) = dens*0.5*(u(j,i)+u(j+1,i));



end

end

for j=1:1:jmax-2

for i=2:1:imax-1

my2(j,i) = dens*0.5*(v(j,i)+v(j+1,i));



end

end


for j=2:1:jmax-1

for i=2:1:imax-2



Su = (p(j,i)-p(j,i+1))*dy;


end

end

for j=2:1:jmax-2

for i=2:1:imax-1



Sv = (p(j,i)-p(j+1,i))*dx + Ri*0.5*(T(j,i)+T(j+1,i))*dV;


end

end

//******************************************************************


RMS_Div = 1e6;


count = 0;






RMS_Div = 0;

for j=2:1:jmax-1

for i=2:1:imax-1





end

end

end

// Corrector step

for j=2:1:jmax-1

for i=2:1:imax-1





aP = aW+aE+aS+aN;


end

end


for j=2:1:jmax-1

for i=2:1:imax-2


end

end

for j=2:1:jmax-2

for i=2:1:imax-1


end

end

count = count+1;

end


//******************************************************************


for j=2:1:jmax-1

for i=2:1:imax-1

p(j,i) = p(j,i) + pc(j,i);

end

end


u = ustar;

v = vstar;

//******************************************************************


RMS1=0;

for i=1:1:imax-1

for j=1:1:jmax


end

end


RMS2=0;

for i=1:1:imax

for j=1:1:jmax-1


end

end



RMS3=0;

for i=1:1:imax

for j=1:1:jmax

RMS3 = RMS3 + (T(j,i)-Told(j,i))*(T(j,i)-Told(j,i));

end

end

RMS3 = sqrt(RMS3/(imax*jmax));

RMS_RESIDUE = max(RMS1,RMS2,RMS3);





end

//*************************** Output ****************************

// U-VELOCITY

xu = zeros(imax-1);

yu = zeros(jmax);

for i=1:1:imax-1

xu(i) = x_u(1,i);

end

for j=1:1:jmax

yu(j) = y_u(j,1);

end


for j=1:1:jmax

for i=1:1:imax-1


end

end

xset('window',1);

clf(1);

UMIN = min(u);

UMAX = max(u);





// V-VELOCITY

xv = zeros(imax);

yv = zeros(jmax-1);

for i=1:1:imax

xv(i) = x_v(1,i);

end

for j=1:1:jmax-1

yv(j) = y_v(j,1);

end


for j=1:1:jmax-1

for i=1:1:imax


end

end

xset('window',2);

clf(2);


VMIN = min(v);

VMAX = max(v);





// CENTRELINE PLOTS

xset('window',3);

clf(3);

uc = zeros(jmax);


for j=1:1:jmax


end

else

for j=1:1:jmax


end

end


ypA = [1 0.9766 0.9688 0.9609 0.9531 0.8516 0.7344 0.6172 0.5 0.4531 0.2813 0.1719 0.1016 0.0703 0.0625 0.0547 0];

ucA100 = [1 0.84123 0.78871 0.73722 0.68717 0.23151 0.00332 -0.13641 -0.20581 -0.2109 -0.15662 -0.1015 -0.06434 -0.04775

-0.04192 -0.03717 0];

title('VARIATION OF U-VELOCITY ALONG THE VERTICAL CENTRELINE', 'color', 'black', 'fontsize',3);




if ((Re==100)&(Gr==0)) then


e=gce();


end

xset('window',4);

clf(4);

vc = zeros(imax);


for i=1:1:imax


end

else

for i=1:1:imax


end

end


vcA100 = [0 -0.05906 -0.07391 -0.08864 -0.10313 -0.16914 -0.22445 -0.24533 0.05454 0.17527 0.17507 0.16077 0.12317

0.1089 0.100091 0.09233 0];

xpA = [1 0.9688 0.9609 0.9531 0.9453 0.9063 0.8594 0.8047 0.5 0.2344 0.2266 0.1563 0.0938 0.0781 0.0703 0.0625 0];

title('VARIATION OF V-VELOCITY ALONG THE HORIZONTAL CENTRELINE', 'color', 'black', 'fontsize',3);




if ((Re==100)&(Gr==0)) then


e=gce();



end

// STREAM FUNCTION



for j=1:1:jmax

SI(j,1) = 0;

SI(j,imax-1) = 0;

end

for i=1:1:imax-1

SI(1,i) = 0;

SI(jmax,i) = 0;

end

for j=2:1:jmax-1

for i=2:1:imax-2


end

end

for j=1:1:jmax

for i=1:1:imax-1


end

end

xset('window',5);

clf(5);







// TEMPERATURE PLOT

T_TRAN = zeros(imax,jmax);

for j=1:1:jmax

for i=1:1:imax

T_TRAN(i,j) = T(j,i);

end

end

xT = zeros(imax);

yT = zeros(jmax);

for i=1:1:imax

xT(i) = x_p(1,i);

end

for j=1:1:jmax

yT(j) = y_p(j,1);

end

xset('window',6);

clf(6);

T_MIN = min(abs(T));

T_MAX = max(abs(T));

colorbar(T_MIN,T_MAX);

title('TEMPERATURE CONTOUR PLOT', 'color', 'black', 'fontsize',3);

Sgrayplot(xT,yT,T_TRAN,strf="041");


// VERTICAL CENTRELINE PLOT


xset('window',7);

clf(7);

Tc = zeros(jmax);


for j=1:1:jmax

Tc(j) = (T(j,imax/2)+T(j,imax/2+1))/2;

end

else

for j=1:1:jmax

Tc(j) = T(j,(imax+1)/2);

end

end

title('VARIATION OF TEMPERATURE ALONG THE VERTICAL CENTRELINE', 'color', 'black', 'fontsize',3);

xlabel("TEMPERATURE");


plot2d(Tc,yT,axesflag=1);

if (Gr == 1E6)&(Re == 100)&(Pr == 1) then

// Torrance et al. (1972) JOURNAL OF FLUID MECHANICS VOL. 51, pp.221-231

ypp = [0 0.2176 0.4484 0.6731 0.8371 0.9484 1];

Tcp = [0 0.01 0.03 0.1 0.3 0.7 1];

plot2d(Tcp,ypp,leg="Benchmarks Results (Torrance et al. (1972))");

e=gce();


end

if (Gr == 0)&(Re == 100)&(Pr == 1) then


ypp = [0 0.1172 0.3438 0.7872 0.9268 0.9795 1];

Tcp = [0 0.1 0.3 0.5 0.7 0.9 1];


e=gce();


end

if (Gr == -1E6)&(Re == 100)&(Pr == 1) then


ypp = [0 0.01826 0.04868 0.4847 0.8925 0.9534 1];

Tcp = [0 0.1 0.3 0.4 0.5 0.7 1];


e=gce();


end

Scilab codes for Case-III and IV:

// *****************************************************************************



// *****************************************************************************

clc;

printf("\n");


printf("*******************************************************************\n");

printf(" NATURAL CONVECTION IN A SQUARE ENCLOSURE ON 2D STAGGERED GRID\n");

printf("*******************************************************************\n");

printf("\n*******************************************************************");

printf("\n ENCLOSURE IS DIFFERENTIALLY HEATED");

printf("\n THE TOP AND BOTTOM WALLS ARE INSULATED,");

printf("\n SIDE WALLS ARE ISOTHERMAL, WITH LEFT WALL BEING HOTTER");

printf("\n*******************************************************************");

printf("\n\nGOVERNING PARAMETERS:");

printf("\n\t- PRANDTL NUMBER (Pr) OF THE FLUID");

printf("\n\t- RAYLEIGH NUMBER (Ra) OF THE FLUID");

printf("\n\nBENCHMARK DATA AVAILABLE AT Pr = 0.71 & Ra = 1E3 or 1E4 or 1E5 or 1E6\n");

Pr = input("ENTER THE PRANDTL NUMBER (Must be 0.71 for benchmarking) : ");

Ra = input("ENTER THE RAYLEIGH NUMBER (Must be 1E3 or 1E4 or 1E5 or 1E6 for

benchmarking): ");







vis = Pr; //Fluid viscosity

Cp = 1; //Specific heat

dif = 1; //Thermal diffusivity

TW1 = 1; //Top wall temperature

TW2 = 0; //Bottom wall temperature

// ***************************** DEFINE GRID SIZE ******************************








dV = dx*dy;

// Initial time step value

// NOTE: Adaptive time stepping is used here to speedup convergence

dt = 0.001;




time_step = 0;

total_time = 0;













T = zeros(jmax,imax);





Told = zeros(jmax,imax);
















mxT = zeros(jmax-1,imax-1);

axT = zeros(jmax-1,imax-1);

dxT = zeros(jmax-1,imax-1);

myT = zeros(jmax-1,imax-1);

ayT = zeros(jmax-1,imax-1);

dyT = zeros(jmax-1,imax-1);



for i=1:1:imax-1

for j=1:1:jmax-1

x(j,i) = (i-1)*dx;

y(j,i) = (j-1)*dy;

end

end


for i=2:1:imax-1

for j=2:1:jmax-1

x_p(j,i) = 0.5*(x(j,i) + x(j,i-1));

y_p(j,i) = 0.5*(y(j,i) + y(j-1,i));

end

end


for i=2:1:imax-1

x_p(1,i) = 0.5*(x(1,i)+x(1,i-1));

y_p(1,i) = 0;


y_p(jmax,i) = Ly;

end

for j=2:1:jmax-1

x_p(j,1) = 0;

y_p(j,1) = 0.5*(y(j,1)+y(j-1,1));

x_p(j,imax) = Lx;


end


x_p(1,1) = 0;

y_p(1,1) = 0;


x_p(1,imax) = Lx;

y_p(1,imax) = 0;

x_p(jmax,1) = 0;

y_p(jmax,1) = Ly;




for i=2:1:imax-2

for j=2:1:jmax-1

x_u(j,i) = x(j,i);

y_u(j,i) = 0.5*(y(j,i)+y(j-1,i));

end

end


for i=2:1:imax-2

x_u(1,i) = x(1,i);

y_u(1,i) = 0;


y_u(jmax,i) = Ly;

end

for j=2:1:jmax-1

x_u(j,1) = 0;

y_u(j,1) = 0.5*(y(j,1)+y(j-1,1));

x_u(j,imax-1) = Lx;


end


x_u(1,1) = 0;

y_u(1,1) = 0;

x_u(1,imax-1) = Lx;

y_u(1,imax-1) = 0;

x_u(jmax,1) = 0;

y_u(jmax,1) = Ly;




for i=2:1:imax-1

for j=2:1:jmax-2

x_v(j,i) = 0.5*(x(j,i)+x(j,i-1));

y_v(j,i) = y(j,i);

end

end


for i=2:1:imax-1

x_v(1,i) = 0.5*(x(1,i)+x(1,i-1));

y_v(1,i) = 0;


y_v(jmax-1,i) = Ly;

end

for j=2:1:jmax-2

x_v(j,1) = 0;


y_v(j,1) = y(j,1);

x_v(j,imax) = Lx;


end


x_v(1,1) = 0;

y_v(1,1) = 0;

x_v(1,imax) = Lx;

y_v(1,imax) = 0;

x_v(jmax-1,1) = 0;

y_v(jmax-1,1) = Ly;




for i=1:1:imax

for j=1:1:jmax

u(j,i) = 0;

v(j,i) = 0;

ustar(j,i) = 0;

vstar(j,i) = 0;

p(j,i) = 0;

T(j,i) = 0;

end

end





function [u, v, ustar, vstar, p, T]=APPLY_BC(u, v, ustar, vstar, p, T)

funcprot(0);

// Bottom Boundary

for i=1:1:imax

u(1,i) = 0;

v(1,i) = 0;

ustar(1,i) = 0;

vstar(1,i) = 0;

p(1,i) = p(2,i);

T(1,i) = T(2,i);

end

// Top Boundary

for i=1:1:imax

u(jmax,i) = 0;

v(jmax-1,i) = 0;

ustar(jmax,i) = 0;



T(jmax,i) = T(jmax-1,i);

end

// Left Boundary

for j=1:1:jmax

u(j,1) = 0;

v(j,1) = 0;

ustar(j,1) = 0;

vstar(j,1) = 0;

p(j,1) = p(j,2);


T(j,1) = TW1;

end

// Right Boundary

for j=1:1:jmax

u(j,imax-1) = 0;

v(j,imax) = 0;


vstar(j,imax) = 0;


T(j,imax) = TW2;

end

endfunction



funcprot(0);

for j=1:1:jmax

for i=1:1:imax

pc(j,i) = 0;

end

end

endfunction

// ************* FUNCTION: PRESSURE CORRECTION BOUNDARY CONDITION

**************


for j=1:1:jmax

pc(j,1) = pc(j,2);


end

for i=1:1:imax

pc(1,i) = pc(2,i);


end

endfunction


unsteadiness = 1e6;



[u,v,ustar,vstar,p,T] = APPLY_BC(u,v,ustar,vstar,p,T);


uold = u;

vold = v;

Told = T;

//******************************************************************

// Solving temperature equation

// Fluxes in x-direction for T

for j=2:1:jmax-1

for i=1:1:imax-1

mxT(j,i) = dens*Cp*u(j,i);

axT(j,i) = max(mxT(j,i),0)*T(j,i) - max(-mxT(j,i),0)*T(j,i+1);

dxT(j,i) = dif*(T(j,i+1)-T(j,i))/(x_p(j,i+1)-x_p(j,i));

end

end

// Fluxes in y-direction for T

for j=1:1:jmax-1

for i=2:1:imax-1

myT(j,i) = dens*Cp*v(j,i);


ayT(j,i) = max(myT(j,i),0)*T(j,i) - max(-myT(j,i),0)*T(j+1,i);

dyT(j,i) = dif*(T(j+1,i)-T(j,i))/(y_p(j+1,i)-y_p(j,i));

end

end

// Get new time level temperatures

for j=2:1:jmax-1

for i=2:1:imax-1

DTp = (dxT(j,i)-dxT(j,i-1))*dy + (dyT(j,i)-dyT(j-1,i))*dx;

ATp = (axT(j,i)-axT(j,i-1))*dy + (ayT(j,i)-ayT(j-1,i))*dx;

T(j,i) = T(j,i) + dt*(DTp - ATp)/dV;

end

end

//******************************************************************



for j=2:1:jmax-1

for i=1:1:imax-2

mx1(j,i) = dens*0.5*(u(j,i)+u(j,i+1));



end

end

for j=1:1:jmax-1

for i=2:1:imax-2

my1(j,i) = dens*0.5*(v(j,i)+v(j,i+1));



end

end


for j=2:1:jmax-2

for i=1:1:imax-1

mx2(j,i) = dens*0.5*(u(j,i)+u(j+1,i));



end

end

for j=1:1:jmax-2

for i=2:1:imax-1

my2(j,i) = dens*0.5*(v(j,i)+v(j+1,i));



end

end


for j=2:1:jmax-1

for i=2:1:imax-2



Su = (p(j,i)-p(j,i+1))*dy;


end

end

for j=2:1:jmax-2

for i=2:1:imax-1



Sv = (p(j,i)-p(j+1,i))*dx + Ra*Pr*0.5*(T(j,i)+T(j+1,i))*dV;



end

end

//******************************************************************


RMS_Div = 1e6;


count = 0;






RMS_Div = 0;

for j=2:1:jmax-1

for i=2:1:imax-1




end

end

end

// Corrector step

for j=2:1:jmax-1

for i=2:1:imax-1





aP = aW+aE+aS+aN;


end

end


for j=2:1:jmax-1

for i=2:1:imax-2


end

end

for j=2:1:jmax-2

for i=2:1:imax-1


end

end

count = count+1;

end


//******************************************************************


for j=2:1:jmax-1

for i=2:1:imax-1

p(j,i) = p(j,i) + pc(j,i);

end

end


u = ustar;

v = vstar;


//******************************************************************


RMS1=0;

for i=1:1:imax-1

for j=1:1:jmax


end

end


RMS2=0;

for i=1:1:imax

for j=1:1:jmax-1


end

end


RMS3=0;

for i=1:1:imax

for j=1:1:jmax

RMS3 = RMS3 + (T(j,i)-Told(j,i))*(T(j,i)-Told(j,i));

end

end

RMS3 = sqrt(RMS3/(imax*jmax));

RMS_RESIDUE = max(RMS1,RMS2,RMS3);





// TIME STEP EVALUATION (BASED ON STABILITY CRITERION)





// Thus, the minimum of the two is further reduced by 50%.

// If the solution diverges, the 0.5 value below needs further reduction.

UMAX = max(abs(u));

VMAX = max(abs(v));

dt_advection = 1/((UMAX/dx)+(VMAX/dy));

dt_diffusion_vel = (0.5/Pr)*((dx*dy)*(dx*dy))/(dx*dx+dy*dy);

dt_diffusion_temp = 0.5*((dx*dy)*(dx*dy))/(dx*dx+dy*dy);

dt = 0.5*min(dt_advection, dt_diffusion_vel,dt_diffusion_temp);

end

//*************************** Output ****************************

// U-VELOCITY

xu = zeros(imax-1);

yu = zeros(jmax);

for i=1:1:imax-1

xu(i) = x_u(1,i);

end

for j=1:1:jmax

yu(j) = y_u(j,1);

end


for j=1:1:jmax

for i=1:1:imax-1



end

end

xset('window',1);

clf(1);

UMIN = min(u);

UMAX = max(u);





// V-VELOCITY

xv = zeros(imax);

yv = zeros(jmax-1);

for i=1:1:imax

xv(i) = x_v(1,i);

end

for j=1:1:jmax-1

yv(j) = y_v(j,1);

end


for j=1:1:jmax-1

for i=1:1:imax


end

end

xset('window',2);

clf(2);

VMIN = min(v);

VMAX = max(v);





// CENTRELINE PLOTS

xset('window',3);

clf(3);

uc = zeros(jmax);


for j=1:1:jmax


end

else

for j=1:1:jmax


end

end

title('VARIATION OF U-VELOCITY ALONG THE VERTICAL CENTRELINE', 'color', 'black',

'fontsize',3);




xset('window',4);

clf(4);

vc = zeros(imax);



for i=1:1:imax


end

else

for i=1:1:imax


end

end

title('VARIATION OF V-VELOCITY ALONG THE HORIZONTAL CENTRELINE', 'color', 'black',

'fontsize',3);




// DENG AND TANG (2002) INT. J. HEAT MASS TRANSFER VOL. 45, pp.2373-2385

xpA3 = [0 0.1021 0.2028 0.3031 0.4001 0.5001 0.5998 0.7026 0.8027 0.9008 1];

vcA3 = [0 3.1988 3.6673 2.8549 1.5034 0.0169 -1.5371 -2.9226 -3.70122 -3.1652 0];

xpA4 = [0 0.1005 0.2013 0.3020 0.4004 0.4988 0.6020 0.7004 0.7987 0.9018 1];

vcA4 = [0 19.4787 15.7820 8.3886 3.4123 0 -3.5545 -8.5308 -15.6398 -19.1943 0];

xpA5 = [0 0.0628 0.1043 0.2031 0.3033 0.4016 0.4977 0.6008 0.7014 0.8015 0.9002 0.9326 1];

vcA5 = [0.1882 69.3615 55.7130 12.8960 -0.4185 -1.2494 0.1909 1.2477 0.4150 -14.0342 -59.4990 -

69.3580 0.5665];

xpA6 = [0 0.0079 0.0375 0.0958 0.1548 0.1928 0.2524 0.4381 0.6405 0.7334 0.8048 0.8429 0.8878

0.9182 0.9625 0.9870 0.9974 1];

vcA6 = [0 100.0 225.18 74.1007 0.7194 -5.0360 0.7194 2.1583 -0.7194 2.1583 6.4748 0.7194 -42.4460 -

111.5110 -225.1800 -133.0940 -33.8129 0];

if (Pr==0.71)&(Ra==1e4) then

plot2d(xpA3,vcA3,leg="Benchmarks Results (Deng anf Tang (2002))");

e=gce();


end

if (Pr==0.71)&(Ra==1e4) then


e=gce();


end

if (Pr==0.71)&(Ra==1e4) then


e=gce();


end

if (Pr==0.71)&(Ra==1e4) then


e=gce();


end

// STREAM FUNCTION



for j=1:1:jmax

SI(j,1) = 0;

SI(j,imax-1) = 0;

end

for i=1:1:imax-1

SI(1,i) = 0;


SI(jmax,i) = 0;

end

for j=2:1:jmax-1

for i=2:1:imax-2


end

end

for j=1:1:jmax

for i=1:1:imax-1


end

end

xset('window',5);

clf(5);







// TEMPERATURE PLOT

T_TRAN = zeros(imax,jmax);

for j=1:1:jmax

for i=1:1:imax

T_TRAN(i,j) = T(j,i);

end

end

xT = zeros(imax);

yT = zeros(jmax);

for i=1:1:imax

xT(i) = x_p(1,i);

end

for j=1:1:jmax

yT(j) = y_p(j,1);

end

xset('window',6);

clf(6);

T_MIN = min(abs(T));

T_MAX = max(abs(T));

colorbar(T_MIN,T_MAX);

title('TEMPERATURE CONTOUR PLOT', 'color', 'black', 'fontsize',3);

Sgrayplot(xT,yT,T_TRAN,strf="041");


// VERTICAL CENTRELINE PLOT

xset('window',7);

clf(7);

Tc = zeros(jmax);


for j=1:1:jmax

Tc(j) = (T(j,imax/2)+T(j,imax/2+1))/2;

end

else

for j=1:1:jmax

Tc(j) = T(j,(imax+1)/2);

end

end


title('VARIATION OF TEMPERATURE ALONG THE VERTICAL CENTRELINE', 'color', 'black',

'fontsize',3);

xlabel("TEMPERATURE");


plot2d(Tc,yT,axesflag=1);


ypA5 = [0 0.10489 0.28205 0.40558 0.50116 0.59672 0.71327 0.89508 1];

TcA5 = [0.18 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.82];

ypA6 = [0 0.1586 0.2908 0.3894 0.5 0.6058 0.7067 0.8389 1];

TcA6 = [0.151 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.848];

if (Pr==0.71)&(Ra==1e4) then

plot2d(TcA5,ypA5,leg="Benchmarks Results (Deng anf Tang (2002))");

e=gce();


end

if (Pr==0.71)&(Ra==1e4) then

plot2d(TcA6,ypA6,leg="Benchmarks Results (Deng anf Tang (2002))");

e=gce();


end

// HORIZONTAL CENTRELINE PLOT

xset('window',8);

clf(8);

TcH = zeros(imax);


for i=1:1:imax

TcH(i) = (T(jmax/2,i)+T(jmax/2+1,i))/2;

end

else

for i=1:1:imax

TcH(i) = T((imax+1)/2,i);

end

end

title('VARIATION OF TEMPERATURE ALONG THE HORIZONTAL CENTRELINE', 'color', 'black',

'fontsize',3);


ylabel("TEMPERATURE");

plot2d(xT,TcH,axesflag=1);


xpA3 = [0 0.09028 0.1782 0.2778 0.3819 0.5 0.6181 0.7245 0.8217 0.9120 1];

TcA3 = [1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0];

xpA4 = [0 0.04323 0.08529 0.13202 0.1951 0.3469 0.4661 0.6671 0.8096 0.8703 0.9171 0.9591 1];

TcA4 = [1 0.9 0.8 0.7 0.6 0.5 0.5 0.5 0.4 0.3 0.2 0.1 0];

if (Pr==0.71)&(Ra==1e4) then

plot2d(xpA3,TcA3,leg="Benchmarks Results (Deng anf Tang (2002))");

e=gce();


end

if (Pr==0.71)&(Ra==1e4) then

plot2d(xpA4,TcA4,leg="Benchmarks Results (Deng anf Tang (2002))");

e=gce();


end


Chapter-IX

Conclusion The higher velocity is required to top lid to shift verticity towards top left corner of the cavity; as it is

observed from isothermal and forced convection fluid flow. The higher temperature difference i.e higher

value of Grashoff number will forms two verticities in case of mixed convection case with increasing

Grashoff number. The lower viscous with higher velocity fluid i.e lower Prandl number with higher

Reynolds number required to create primary and secondary verticities with different characteristics and the

verticities will decreases with low viscous fluid i.e high Prandl number with low Reynolds number. Only

buoyancy force and horizontal temperature difference are sufficient to create centralized verticity in the

cavity.

Higher temperature difference between the walls of the cavity (higher Grashoff number)and higher velocity

of the fluid particles (higher Reynolds number) will produce sharp changes in the vertical velocity

component along the horizontal center line of the cavity. As the Grashoff increases from zero in the

isothermal case to higher values (105,10

6 and 2X10

6) in case of mixed convection fluid flow; the number of

verticities increases from two to three for lower Reynolds number. As the Grashoff number increases; the

verticities are disappearing. The fluid flow by purely buoyancy force will creates almost equal magnitude

different velocity verticities at left and right wall of the cavity.

The sharp change in the direction of the fluid particles is at short distance from the left side surface of the

cavity for higher value of the Grashoff number; as the number increases, the distance at which sudden

change in the direction of the fluid will decrease velocity of the fluid along x-direction (u). More

temperature difference is required to quick turn of the fluid particle along the horizontal direction of the

cavity. The only buoyancy force in the cavity (natural convection case) makes the fluid particles to take two

sharp turns at top and bottom surface along the vertical center line of the cavity.

As the temperature difference between the cavity walls increases with increasing Grashoff number i.e

increasing buoyancy effect than the inertia effect; the top lid temperature effect will decreases and becomes

uniform at higher value of the Grashoff number. If only buoyancy effect (natural convection fluid flow)

present with top and bottom walls of the cavity are insulated; the inverted S-shape temperature contours are

formed and they are moving towards top and bottom of the cavity.

In natural convection fluid flow; the computation time required for the dense grid structure is very high, it is

about two days for 100 Reynolds number with 32X32 grid structure. This is because the temperature

difference increases with increasing Grashoff number and only buoyancy force is dominant the fluid flow

with very negligible inertia of the fluid flow as compared to the forced and mixed fluid flow cases. For the

less value of Grashoff number in dense grid structure (32X32); the temperature values along the horizontal

center line not close to the published data. This difference may be decreases with the high dense grid

structure.


Chapter-X

References ORIGINAL PAPER

1] ‗Numerical prediction of eddy structure in a shear-driven cavity‘ by T. P. Chiang, W. H. Sheu; Published

by Computational Mechanics 20 (1997) 379-396 Springer-Verlag 1997.

2] A text book ‗Fundamentals of finite element method for heat and fluid flow‘ by Ronald W.Lewish,

Perumal Nithirasu and Kankanhalli N.Seetaramu; Published by Jhon Wiley & Sons Ltd.

3] ‗Parallel edge-based solution of viscoplastic flows with the SUPG/PSPG formulation‘ by Renato N. Elias

Marcos A. D. Martins Alvaro L. G. A. Coutinho ; Published by Comput. Mech. (2006) 38: 365–381

DOI 10.1007/s00466-005-0012-y

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; Published by Computational Mechanics 20 (1997) 541±550 Ó Springer-Verlag 1997

5] ‗A variational multi scale stabilized formulation for the incompressible Navier–Stokes equations‘ by Arif

Masud · Ramon Calderer ; Published by Comput Mech (2009) 44:145–160 DOI 10.1007/s00466-008-0362-

3

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Leung, C.-M. Woo ; Published by Computational Mechanics 32 (2003) 185–198 _ Springer-Verlag 2003

DOI 10.1007/s00466-003-0474-8

7] ‗Interpolation functions in the immersed boundary and finite element methods‘ by Xingshi Wang · Lucy

T. Zhang ; Published by Comput Mech (2010) 45:321–334 DOI 10.1007/s00466-009-0449-5

8] ‗Nonlinear fluid–structure interaction problem. Part I: implicit partitioned algorithm, nonlinear stability

proof and validation examples‘ by Christophe Kassiotis · Adnan Ibrahimbegovic ·Rainer Niekamp· Hermann

G. Matthies ; Published by Comput Mech (2011) 47:305–323 DOI 10.1007/s00466-010-0545-6

9] ‗A dynamic mesh refinement technique for Lattice Boltzmann simulations on octree-like grids‘ by Philipp

Neumann · Tobias Neckel ; Published by Comput Mech DOI 10.1007/s00466-012-0721-y

10] ‗A numerical study of secondary flow and large eddies in a driven cavity‘ by Y. H. Yau, A. Badarudin

and P. A. Rubini ; Published by Journal of Mechanical Science and Technology 26 (1) (2012) 93~102

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0913-y

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RECTANGULAR CAVITY WITH A MOVING WALL‘ by K. N. Volkov; Published by Journal of

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http://www.springerlink.com/content/1738-494x

Lid Driven Cavity CFD Simulation Report by S N Topannavar

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