Deepak Final Training Report 1-9-2014

Mahatma Gandhi MissionsCollege of Engineering and TechnologyNoida, U.P., India

Report on

Role of Numerical Simulations in Solving Engineering Problems

Carried out at

Indian Institute of Technology, DelhiHauz Khas, New Delhi-110016

From: 10th June - 25th July, 2014

Academic session: 2013 -2014

Submitted by: Submitted to:Name: P. Deepak KumarMechanical Engineering Department,Class: TT-MEMGM COET,Noida, U.P. Roll No: 12095400382

Mahatma Gandhi MissionsCollege of Engineering and TechnologyNoida, U.P., India

Department of Mechanical Engineering

CERTIFICATE

This is to certify that Mr. P. Deepak Kumar of B. Tech. Mechanical Engineering, Class TT-ME and Roll No. 1209540038 has completed his Industrial Training during the academic year 2013-14 from 10th June 2014 to 25th July 2014 at Indian Institute of Technology Delhi, New Delhi, India.

Training Coordinator Head of the Department

Acknowledgements

I would like to express my deep sense of gratitude to my training supervisor Dr. R. K. Pandey, Associate Professor, Department of Mechanical Engineering, I.I.T. Delhi, New Delhi, India, for his guidance, support and encouragement throughout this project work. Moreover, I would like to acknowledge the Department of Mechanical Engineering, I.I.T. Delhi, for providing me all possible help during pursuing of this training. Moreover, I would like to sincerely thank everyone who directly and indirectly helped me in completing this task.

(P. Deepak Kumar)

Date:25 July, 2014Place: New Delhi

Abstract

There are several engineering problems whose performance behaviours are difficult to understand (even in some cases impossible) based on the only experimental/analytical approaches. Thus, in order to understand the behaviours of complex engineering processes/problems, need arises to develop governing equations of the complex processes and thereafter their numerical solutions (called as numerical simulations or computer simulations) for interpretation of the results. It is worth mentioning here that in numerical simulations at a time several input variables easily can be varied for knowing the response/behaviour of the system (s). The numerical techniques such as finite difference method (FDM), finite element method (FEM), finite volume method (FVM) etc. play vital role in discretization of governing equations (PDE) in linear algebraic equations leading to their solutions using Gauss elimination method, Gauss seidel iterative method, tridiagonal matrix algorithm (TDMA) approach etc.

This report incorporates the work carried out in the summer training. Attempts have been made to understand the importance of numerical simulations of engineering problems using finite difference method. How partial differential equation and computational domain are discretized using FDM is understood? Concepts behind the grid independent test and convergence criteria are also understood. Moreover, governing equation (Reynolds equation) for modelling of lubrication in finite journal bearing is derived and numerically solved using FDM. Numerical results have been validated with the published experimental results of prior researchers for developing the confidence in the proposed model. Parametric results for the lubricated finite journal bearing are presented in graphical forms.

Contents Page No.Certificate from Department2Certificate from I.I.T. Delhi3Acknowledgements4Abstract5Contents6List of table8List of figure9CHAPTER-1INTRODUCTION101.1 Background101.2 Numerical Simulation111.2.1 Finite Difference Method (FDM)131.2.2 Finite Element Method (FEM)131.2.3 Spectral Method (SM)131.2.4 Finite Volume Method (FVM)131.3 Scope for the Study14

CHAPTER-2LITERATURE REVIEW ON FINITE DIFFERENCE 15METHOD2.1 Classification of Differential Equation162.1.1 Based on Order162.1.2 Based on Linearity162.1.3 Parabolic Partial Differential Equation172.1.4 Elliptic Partial Differential Equation182.1.5 Hyperbolic Partial Differential Equation192.2 Finite Difference Method202.2.1 Central Difference202.2.2 Forward Difference212.2.3 Backward Difference212.3 Use Forward, Backward & Central Difference222.4 Numerical Errors222.4.1 Round-off Error222.4.2 Truncation Error222.4.3 Discretization Error222.5 Grid Independent Study232.6 Objective of Study23

CHAPTER-3GOVERNING EQUATIONS AND 24COMPUTATIONAL PROCEDURE3.1 Continuity Equation243.2 Navier Stokes Equation253.3 Reynolds Equation283.4 Miscellaneous Relations313.4.1 Film Thickness Relation313.4.2 Load Relation313.4.3 Coefficient of Friction323.4.4 Power Loss323.4.5 Leakage333.5 Discretization of Reynolds Equation using FDM33CHAPTER-4RESULTS AND DISCUSSIONS354.1 Grid Independent Test354.2 Convergence Criteria364.3 Validation of Numerical Model364.4 Pressure Distribution in Lubricating Film374.5 Power Loss and Side Leakage46

CHAPTER-5CONCLUSIONS 47

References48

List of Tables

Table No.TopicPage No.

1.1Comparison of analytical, numerical & experimental method12

4.1Demonstration for grid independent test [N=1000rpm, L=D=0.04m, Cr=53.33m, =0.1 Pa-s, =0.5]35

4.2Convergence test at the grid (101,51) [N=1000rpm, L=D=0.04m, Cr=53.33m, =0.1 Pa-s, =0.5]36

4.3Comparison of numerical results with the experimental results [N=1000rpm, L=D=0.04m, Cr=53.33m, =0.1 Pa-s, =0.5]37

4.4Results for Power loss and side leakage46

List of Figures

Figure No.TopicPage No.

2.1Boundary Condition for parabolic differential equations17

2.2Boundary Condition for elliptic differential equations18

2.3Boundary Condition for hyperbolic differential equations19

3.1Infinitesimally small fluid element24

3.2Schematic diagram of infinitesimal element with forces25

3.3Journal bearing geometry with coordinate system29

4.1(a)3-D pressure profile in lubricating film [=0.3, RPM=500, =0.1 Pa-s, L=D=0.04m, Cr=53.33m]38

4.1(b)Pressure Distribution in circumferential direction [=0.3, RPM=500, =0.1 Pa-s, L=D=0.04m, Cr=53.33m]38

4.1(c)Pressure distribution in axial direction [=0.3, RPM=500, =0.1 Pa-s, L=D=0.04m, Cr=53.33m]39

4.1(d)Pressure Contours in 2-D [=0.3, RPM=500, =0.1 Pa-s, L=D=0.04m, Cr=53.33m]39













CHAPTER-1

INTRODUCTION

This chapter presents background related to the proposed project. It highlights the importance of numerical simulations in the field of engineering. Moreover, at the end of the chapter, the scope of the study is also provided.

1.1 Background

There are several engineering problems whose performance behaviours are difficult to understand (even in some cases impossible) based on the only experimental/analytical approaches. Thus, in order to understand the behaviours of complex engineering processes, need arises to develop the governing equations of the processes and thereafter their numerical solutions (called as numerical simulations or computer simulations) for interpretation of the results.

The numerical simulation has vital benefits as described below:

1. Many input parameters can be changed simultaneously and its effect can be observed on the engineering processes through numerical simulation. It must be noted here that it is very difficult and an expensive affair for setting the experimental facilities in comparison to doing computer simulation.2. Numerical simulation is much cheap and convenient for individual.3. Numerical modelling is versatile. A large number of problems of various levels can be simulated/ modelled on the computer.4. Computer simulation helps in understanding and improving the models and physical condition of the problem.

It is worth noting here that in some conditions numerical simulation is ideal substitute for the experiments such as assessing the effect of nuclear explosion, spreading of fire in a building etc.

1.2 Numerical Simulation

Numerical simulation of any engineering system/problem on computer involves the mathematical representation of related system/problem using set of equations and solving it. Numerical simulation is also popularly known as computer simulation. But there is a minute difference between these two. In computer simulation, the problem is dealt graphically either using governing equations or without using it. However, in numerical simulation the problem is dealt by solving the governing equations.

The governing equations of many engineering problems happen to be so complex that these cannot be solved using classical mathematics approach i.e. analytically. Thus, need arises to solve such equations using numerical techniques. Two examples are provided below to explain the difference between the analytical and numerical methods for solutions:

Example-1: Solve the following equation for x-

(1)Using the formula provided in the classical mathematics, Eq. 1 is solved for x as follows:

=== -1, 2

Thus, the solution achieved of Eq.1 is called the solution through the analytical/exact method.

Example-2: Solve the following equation for x-

(2)

Equation (2) is a transcendental equation, which does not have exact/analytical solution. This type of equation is solved using iterative (numerical) method. Thus, the solution of Eq. 2 starts as follows:

Let

(3)

For the values of x= -2 and 0, the sign of y(x) is noticed positive and negative, respectively. For the error value of 10-3, the solution of equation (3) yields x -1.49.

This method of achieving the solution for Eq.2 is called the solution through numerical approach. A comparison between the analytical, numerical, and experimental approaches have been presented in Table 1.1 for the ready reference of readers.

Table 1.1 Comparison of analytical, numerical, and experimental methods

MethodAdvantagesDisadvantages

Analyticalsimple usually in formula form Result is difficult to compute. Usually applicable to linear problems Applicable to simple physics and geometry

NumericalNo restriction to linearity, applicable to complex problems, low cost and high speed of computation Not very accurate Boundary condition problems

ExperimentalRealistic in nature Apparatus is required Measurement problem Scaling difficulties Probe errors High cost

Many numerical methods exist for discretization of partial differential governing equations. However, the following methods are widely used for numerical simulation of engineering problems:

1.2.1 Finite Difference Method (FDM)

This method is based on the approximating the derivatives in the partial differential equation using truncated Taylor series. This method is easy for the newcomers in the numerical simulation area due to its straight forwardness and simplicity. However, this method is not preferred for solving the engineering problems with high degree of physical complexity.

1.2.2 Finite Element Method (FEM)

This method yields solution at discrete elements by assuming that the governing partial differential equation is applicable to the continuum within each element. It is based on integral minimization principle and provides regional approximation to the governing equations. This method is normally used in solid mechanics and fluid mechanics. It easily solves complex problems as compared to Finite Difference Method. The only disadvantage is that complicated matrix operations are needed to be solved.

1.2.3 Spectral Method (SM)

This method is more accurate than first and second order Finite Difference Method and can be combined with standard Finite Difference Method. In this method, the approximation is based on expansions of independent variable into finite series of smooth functions. It is more complex in comparison to standard Finite Difference Method. Applying complex boundary condition is difficult in this method.

1.2.4 Finite Volume Method (FVM)In this method the calculation domain is divided into a number of non-overlapping control volumes such that there is one control volume surrounding each grid point. The differential equation is integrated over each control volume and the result contains the unknown for a group of grid point. The advantage of this method is that it is physically sound and the disadvantage is that it is not straight forward like Finite Difference Method.

1.3 Scope of the Study

To understand the importance of numerical simulation in solving the complex engineering problems, it is easier to use finite difference method (FDM) by a newcomer in the field of numerical simulation. Through passes of time, some recent improvements have taken place in FDM, which makes FDM much superior and can be applied to irregular boundary problems too. Thus, there is scope to study the engineering problem (lubrication in hydrodynamic journal bearing) numerically using the FDM.

************

CHAPTER-2

LITERATURE REVIEW ON FINITE DIFFERENCE METHOD

This chapter presents literature review related to finite difference method. Moreover, at the end of this chapter objective of the present study is also provided.

Finite difference method has played significant role in solving the various engineering problems. Peaceman and Rachford [1] have nicely described the numerical solutions of parabolic and elliptic differential equations. One of the major obstacles to the solution of the multi-dimensional compressible Navier-Stokes equations is the large amount of computer time needed in achieving the solution. Most previous finite difference based methods involved the explicit difference schemes in the unsteady form of the governing equations, which resulted in one or more stability restrictions. These stability restrictions lower computational efficiency by imposing a smaller time step than would otherwise be desirable. Thus, a key disadvantage of explicit methods is the stability limits. Therefore, maximum time step is fixed by the spatial mesh size rather than the physical time dependence or the desired temporal accuracy. However, in contrast to explicit methods, implicit methods tend to be stable for large time steps and hence offer the prospect of substantial increase in the computational efficiency. In an effort to exploit the favourable stability of implicit methods, an alternating direction differencing technique was developed by the authors [2]. In this method, the governing equations are replaced by either a Crank-Nicolson method or backward time difference approximation. Terms involving nonlinearities at the implicit time level are linearised by Taylor expansion about the known time level, and spatial difference approximations are introduced. This technique leads to systems of one-dimensional coupled linear difference equations which can be solved efficiently by standard elimination methods.

2.1 Classification of Differential Equation

In literature, the partial differential equations are classification as follows:

2.1.1 Based on Order

The highest order partial derivative in an equation is the order of that equation. The example of first order equation is given as:

(2.1)The example of second order equation is given as:

(2.2)

Where, Dependent variable or field variable, Independent variables

2.1.2 Based on Linearity

The partial differential equation is classified into linear and non-linear equation on the basis of linearity. In linear equations, coefficients are constant or functions of independent variables.

(2.3)In non-linear equations, the coefficients are functions of the dependent variable, which are written as below:

(2.4)

(2.5)

Apart from above classification, partial differential equation is classified into three categories depending on the sign of expression (refer eqn. (2.5)).

2.1.3 Parabolic Partial Differential Equation

Partial differential equation is parabolic if it satisfies the following condition:

(2.6)

Heat conduction equation (2.6) fall under parabolic differential equation category.

Heat conduction equation:

(2.7)

In parabolic partial differential equations, the solution advances outward from the known initial values as illustrated in Fig. 2.1.

tOpen End

Prescribed Boundary Condition

Solution marches outward with time from initial conditionPrescribed Boundary Condition

x

Initial boundary condition is specified here

Fig. 2.1 Boundary condition for parabolic differential equation

2.1.4 Elliptic Partial Differential Equation

A partial differential equation is said elliptic when following relation holds:

(2.8)

The following equations fall under this category:

Laplace equation:

(2.9)Poissons equation:

(2.10)

The field variables in elliptic partial differential equation satisfy a closed domain and the boundary conditions on the closed boundary of the domain as shown in Fig. 2.2.

Solution domain of the function (x, y).

Boundary conditions prescribed on entire boundary, usually as a value of (x, y) or the normal derivative.

Fig. 2.2 Boundary condition for elliptic differential equation

2.1.5 Hyperbolic Partial Differential Equation

Any partial differential equation can be hyperbolic partial differential equation only when it satisfies the following equation:

(2.11)

An example of hyperbolic partial differential equation is wave equation.

Wave equation:

(2.12)

The solution domain happens to be open ended for parabolic partial differential equation). Two initial boundary conditions are required to start the solution as indicated in Fig. 2.3.

tOpen End

Boundary Condition

Boundary Condition

x

Initial Condition

Fig. 2.3 Boundary condition for elliptic differential equation

2.2 Finite Difference Method

Finite difference method originates from Taylor series expansion assuming the function to be smooth i.e. continuous and differentiable.

2.2.1 Central Difference

The Taylor series expansion is given as:

(2.13)

(2.14)

Subtracting eqn. (2.14) from eqn. (2.13) yields:

(2.15)

If h

Deepak Final Training Report 1-9-2014

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