A Modified Nodal Integral Method for the Time-Dependent, Incompressible

Navier-Stokes-Energy-Concentration Equations and its Parallel Implementation

BY

FEI WANG

B.S., Tsinghua University, 1992 M.S. Tsinghua University, 1997

M.S., University of California, Los Angeles, 2002

DISSERTATION

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Nuclear Engineering

in the Graduate College of the University of Illinois at Urbana-Champaign, 2003

Urbana, Illinois

ii

© Copyright by Fei Wang, 2003

iii

A Modified Nodal Integral Method for the Time-Dependent, Incompressible Navier-Stokes-Energy-Concentration Equations and its Parallel

Implementation

Fei Wang, Ph.D. Department of Nuclear, Plasma and Radiological Engineering

University of Illinois at Urbana-Champaign, 2003 Rizwan-udiin, Advisor

The nodal integral method can achieve a same accuracy as many conventional

numerical methods using less coarser mesh and less CPU time. In early applications of

the nodal integral method, the nonlinear convection terms were treated as part of the

pseudo source terms. The transverse-averaged continuity equations are used to solve for

two transverse-averaged velocities and two of transverse-averaged momentum equations

are used to solve for transverse-averaged pressures. This leads to a numerical model

asymmetric in spatial directions.

A modified nodal integral method is developed in this dissertation, in which a

Poisson equation is used and the nonlinear convection terms are kept on the left hand side

of the transverse-averaged momentum equations. The numerical model developed has the

following advantages: 1) The Use of Poisson equations leads to a model symmetric in all

spatial directions. 2) The local solution of the transverse averaged velocities has a

component that varies exponentially in space. These exponential terms can capture steep

spatial variation of velocities within each cell, thus, allowing the use of coarse meshes.

3) The appearance of the local Reynolds number in the exponential terms, the scheme

being developed has inherent upwinding.

iv

In this dissertation, the modified nodal integral method is first developed for two-

dimensional, time-dependent, Incompressible Navier-Stokes equations, then extended to

three dimensions. Results from both the two-dimensional and three-dimensional codes

are compared with reference solutions and results obtained using commercial software

Fluent. Comparison of the numerical results proves that the modified nodal integral

method can achieve the same accuracy as other numerical methods using coarse mesh.

A parallel version of the modified nodal integral method is developed.

A modified nodal integral method for Navier-Stokes equations coupled with

energy, specie concentrations is also developed in collaboration with Allen Toreja.

v

ACKNOWLEDGEMENTS

First, I would like to thank my advisor, Professor Rizwan-uddin for his

continuous guidance and encouragement throughout my years of study at University of

Illinois. I would like to thank him also for his support and care in my life and job search.

I would also like to thank Professor Roy Axford, Professor Barclay Jones and Professor

Mark Short for serving on my final defense committee.

I wish to extend special recognition to fund in part by the U.S. Department of

Energy through the University of California under subcontract number B341494. I would

also like to acknowledge support under the Computational Science and Engineering

Fellowship program at University of Illinois at Urbana Champaign.

I would like to thank my parents for the encouragement given to me since my

childhood. Without their support, I could not have come to USA for my Ph.D. studies.

I wish to give special thanks to Allen Toreja for the collaboration in the thesis

work. I would also like to thank my officemates Allen Toreja, Daniel Rock, Doina

Costescu, Quan Zhou for making Room 251 NEL a pleasant working environment.

vi

Table of Contents 1. Introduction .......................................................................................................... 1 1.1. Traditional Numerical Methods ................................................................... 1

1.1.1. Finite Difference Method .................................................................. 2 1.1.2. Finite Volume Method ...................................................................... 2 1.1.3. Finite Element Method ..................................................................... 2

1.2. Numerical methods for the Incompressible Navier-Stokes Equations ........ 3 1.2.1. Conservative Form v.s. Non-conservative Form ...................……. .. 3

1.2.2. Primitive Form v.s. Derived Form ...................................……….. ... 4 1.3. Coarse Mesh Methods .................................................................................. 5 1.4. Nodal Methods ............................................................................................. 7 1.5. Nodal Integral Method ................................................................................. 8 1.5.1. Nodal Integral Method and Transverse Integration Procedure .......... 9 1.5.2. NIM for Nonlinear Equations and Past Applications

to the Navier-Stokes Equations ......................................................... 13 1.6. Modified Nodal Integral Method ................................................................. 14 1.7. Present Work ................................................................................................ 17 2. Modified Nodal Integral Method for the Two-Dimensional, Time-Dependent, Incompressible Navier-Stokes Equations ............................... 19 2.1. Derivation of Poisson Equation for Pressure ............................................... 19 2.2. Transverse Integration Procedure and the Set of ODEs .............................. 21 2.3. Discussion of the Treatment of the Nonlinear Terms .................................. 26 2.4. Transverse-Averaged ODEs ........................................................................ 27 2.5. Local Solutions ............................................................................................ 28 2.6. Set of Discrete Equations in Terms of the Pseudo-Source Terms ............... 31 2.7. Constraint Equations .................................................................................... 33 2.8. Set of Discrete Equations ............................................................................. 37 2.9. Boundary Conditions ................................................................................... 38 3. Application of Modified Nodal Integral Method for Time-Dependent Navier-Stokes Equations – Two Dimensional Case ............................................ 43 3.1. Fully Developed Flow Between Parallel Plates ........................................... 44

3.1.1. Numerical Results .............................................................................. 46 3.2. Developing Flow Between Parallel Plates ................................................... 49 3.3. Classical Lid Driven Cavity Problem .......................................................... 54 3.4. Lid Driven Cavity Problem in a Rectangle with Aspect Ratio = 2 .............. 59 3.5. Modified Lid Driven Cavity Problem .......................................................... 64 3.6. Taylor’s Decaying Vortices ......................................................................... 71 4. Modified Nodal Integral Method for the Three-Dimensional, Time-Dependent, Incompressible Navier-Stokes Equations ............................... 77 4.1. Reformulation and discretization of the N-S Equations ............................. 77

vii

4.2. Transverse Integration Procedure ............................................................... 80 4.3. Local Solutions for the Transverse-Integrated ODEs ................................. 82 4.4. Constraint Equations ................................................................................... 85 4.5. Boundary Conditions .................................................................................. 92 5. Application of Modified Nodal Integral Method for Time-Dependent Navier-Stokes equations – Three Dimensional Case ........................................... 95

5.1. Three-Dimensional Fully Developed Flow in a Rectangular Channel ............................................................................ 95

5.2. Three-Dimensional Developing Flow in a Rectangular Channel ............... 99 5.3. Lid Driven Cavity Flow in a Cube ..............................................................104 5.4. Lid Driven Cavity Flow in a Prism .............................................................110 6. Parallel Implementation of the MNIM for the Navier-Stokes Equations ............117 6.1. Shared Memory v.s. Distributed Memory ..................................................117 6.2. Domain Decomposition ..............................................................................118 6.3. The Ghost Nodes.........................................................................................120 6.4. Load Balancing and Synchronization .........................................................122 6.5. Numerical Results .......................................................................................124 6.6. Conclusion ..................................................................................................126 7. Conclusion ...........................................................................................................127 APPENDIX A. Definition of Coefficients A for Two-Dimensional MNIM ..........................128 B. Definition of Coefficients F for Two-Dimensional MNIM ..........................130 C. Pseudo-Source Terms for Three-Dimensional MNIM ..................................133 D. Definition of Coefficients F for Three-Dimensional MNIM ........................136

E. Modified Nodal Integral Method for Navier-Stokes Equations Coupled with Energy and Concentration Equations ......................................142

E.1. The Boussinesq Approximation ............................................................143 E.2. Thermal Convection ..............................................................................143 E.3. Non-Dimensional Form .........................................................................144 E.4. MNIM for Navier-Stokes Equations Coupled with Energy Equation ..............................................................145 E.5. Development of MNIM for the Energy Equation .................................148 E.5.1. Transverse Integration Procedure ................................................148 E.5.2. Local Solutions and Continuity ...................................................149 E.5.3 Constraint Equations ....................................................................150

viii

E.6. Development of the MNIM for the Specie Concentration Equation .....151 E.7. Numerical Results of the MNIM for the Coupled N-S, Energy and Specie Concentration Equation ..........................................155

References ..................................................................................................................159

ix

List of Acronyms LHS: Left Hand Side

MNIM: Modified Nodal Integral Method

NGFM: Nodal Green’s Function Method

NGTM: Nodal Green’s Tensor Method

NIM: Nodal Integral Method

N-S: Navier-Stokes

ODE: Ordinary Differential Equation

PCBM: Partial Current Balance Method

PDE: Partial Differential Equation

RHS: Right Hand Side

TIP: Transverse Integration Procedure

w.r.t. with respect to

x

List of Tables Table 3.2.1: Numerical comparison with Azmy’s [Azmy1982] results for

developing flow. (1, 1) and (6, 6) are respectively the lower left and top right cells in the domain ..................................... 53

Table 3.5.1: RMS errors and CPU times for Re = 1 (Dirichlet boundary conditions) ............................................................. 66

Table 3.5.2: RMS errors and CPU times for Re = 10 (Dirichlet boundary conditions) ............................................................. 66

Table 3.5.3: RMS errors and CPU times for Re = 20 (Dirichlet boundary conditions) ............................................................. 66

Table 3.5.4: RMS errors and CPU times for Re = 1 (pressure boundary conditions) .............................................................. 70

Table 3.5.5: RMS errors and CPU times for Re = 10 (pressure boundary conditions) .............................................................. 70

Table 3.5.6: RMS errors and CPU times for Re = 20 (pressure boundary conditions) .............................................................. 70

Table 3.6.1: Coefficients of discrete variables in equation (3.18) showing inherent upwinding .................................................................. 76

xi

List of Figures Figure 1.1: Domain discretization for the nodal integral method ............................ 11 (a) Discretization of the spatial domain into x yn n× cells ..................... 11 (b) Space-time cell (i, j, k) and local coordinate system ........................ 11 (c) Details of the local coordinates in cell (i, j) in x-y plane. ................ 11 Figure 2.1: Continuity of transverse averaged pressure ( )xtp y between cell (i, j) and (i, j+1) ............................................................... 30 Figure 2.2: Boundary condition for pressure at the right surface ............................. 40 Figure 3.1.1: Boundary conditions for fully developed flow

between parallel plates ........................................................................... 45 Figure 3.1.2: Flow field for fully developed flow between parallel plates ................. 47 Figure 3.1.3: Comparison of u velocity with exact solution ....................................... 47 Figure 3.1.4: Evolution of centerline velocity for different time steps ....................... 48 Figure 3.2.1: Boundary conditions for developing flow between parallel plates ........ 50 Figure 3.2.2: Flow field for developing flow, Re = 10 ............................................... 52 Figure 3.2.3: Flow field for developing flow, Re = 100 ............................................. 52 Figure 3.3.1: Velocity vectors for classical lid-driven cavity problem for Re = 100 ........................................................................................... 55

(a) Vector length proportional to the velocity magnitude ...................... 55 (b) Uniform vector length ...................................................................... 55

Figure 3.3.2: Velocity profile for classical lid-driven cavity problem for Re = 100. Fine mesh results are from [Ghia 1982] ................................................ 56

(a) u-velocity along the vertical line through geometric center of the cavity .......................................................... 56

(b) v-velocity along the horizontal line through geometric center of the cavity .......................................................... 56 Figure 3.3.3: Velocity vectors for classical lid-driven cavity problem

for Re = 1000 ......................................................................................... 57 (a) Vector length proportional to the velocity magnitude ..................... 57 (b) Uniform vector length ..................................................................... 57

Figure 3.3.4: Velocity profile for classical lid-driven cavity problem for Re = 1000. Fine mesh results are from [Ghia 1982] ................................................ 58 (a) u-velocity along the vertical line through

geometric center of the cavity ......................................................... 58 (b) v-velocity along the horizontal line through geometric center of the cavity ......................................................... 58

Figure 3.4.1: Velocity vectors for lid-driven cavity problem with aspect ratio of 2 for Re = 100. ................................................................................... 60 (a) Vector length proportional to the velocity magnitude ..................... 60 (b) Constant vector length ..................................................................... 60

Figure 3.4.2: U-velocity along the vertical line through geometric center of the cavity for lid-driven cavity problem with aspect ratio of 2 for Re = 100 ........................................................ 61

xii

Figure 3.4.3. Velocity vectors for lid-driven cavity problem for Re = 1000 .............. 62 (a) Vector length proportional to the velocity magnitude ..................... 62 (b) Constant vector length ..................................................................... 62

Figure 3.4.4: Comparison of u-velocity along the vertical line through the geometric center of the cavity for lid-driven cavity problem for Re = 1000 with results obtained using Fluent. Results of the nodal scheme are plotted at the center of the cell and q is the geometric ratio for Non-uniform cell size in x and y directions ...... 63

Figure 3.5.1: Velocity and pressure fields of the modified lid driven cavity problem ........................................................ 67

Figure 3.5.2: Velocity vector plot of the modified lid driven cavity problem ............ 68 Figure 3.6.1: Velocity fields for the Taylor’s decaying

vortices problem at t = 0 ........................................................................ 72 (a) u velocity .......................................................................................... 72 (b) v velocity .......................................................................................... 72

Figure 3.6.1: (c) Pressure field for the Taylor’s decaying vortices problem at t = 0 .............................................................................................. 73 (d) Corresponding velocity vector plot. Coefficients of three neighboring discrete variables at two different locations (A and B) are shown in Table 3.6.1 ................................................. 73

Figure 3.6.2: Numerical and exact solutions of the Taylor’s decaying vortices problem at different times .......................... 75 (a) u velocity .......................................................................................... 75 (b) Pressure ............................................................................................ 75

Figure 4.1: Boundary condition for pressure at the surface x = xmax = x0 ................ 93 Figure 5.1.1: Boundary conditions for 3D fully developed flow

in a rectangular channel ......................................................................... 96 Figure 5.1.2: Velocity profile for 3D fully developed flow in

a rectangular channel at planes y = 0.1, 0.5 and 0.9 .............................. 97 Figure 5.1.3: Velocity profile for 3D fully developed flow in

a rectangular channel at planes z = 0.1, 0.5 and 0.9 .............................. 98 Figure 5.2.1: Boundary conditions for 3D fully developing flow

in a rectangular channel ......................................................................... 100 Figure 5.2.2: Velocity profile for 3D developing flow in a rectangular channel at planes z = 0.1, 0.5 and 0.9 .............................. 101 Figure 5.2.3: Velocity profile for 3D developing flow in a rectangular channel at planes y = 0.1, 0.5 and 0.9 .............................. 102 Figure 5.2.4: Velocity profile for 3D developing flow in a rectangular channel

at plane y = 0.9 (different view angle to show the vector direction) ..... 103 Figure 5.3.1: Configuration of the lid driven cavity problem in a cube ...................... 105 Figure 5.3.2: U-velocity along the vertical centerline for the

3D lid driven cavity cube problem for Re = 100 ................................... 106 Figure 5.3.3. W-velocity along the horizontal centerline for the

3D lid driven cavity cube problem for Re = 100 ................................... 107 Figure 5.3.4: U-velocity along the vertical centerline for the

3D lid driven cavity cube problem for Re = 1000 ................................. 108

xiii

Figure 5.3.5: W-velocity along the horizontal centerline for the 3D lid driven cavity cube problem for Re = 1000 ................................. 109

Figure 5.4.1: Configuration of the lid driven cavity problem in a prism .................... 111 Figure 5.4.2: Center plane velocity vectors for three-dimensional lid-driven

cavity problem in a prism with aspect ratio of 2 for Re = 100. Vector length is proportional to the velocity magnitude ....................... 112

Figure 5.4.3: Comparison of centerline velocity profiles for a prismatic cavity with an aspect ratio of 2 for Reynolds number of 100 .................................................................. 113

Figure 5.4.4: Center plane velocity vectors for three-dimensional lid-driven cavity problem in a prism with aspect ratio of 2 for Re = 1000. Vector length is proportional to the velocity magnitude ....................... 114

Figure 5.4.5: Center plane velocity vectors for three-dimensional lid-driven cavity problem in a prism with aspect ratio of 2 for Re = 1000. Vector length is uniform ........................................................................ 115

Figure 5.4.6: Comparison of centerline velocity profiles for a prismatic cavity with an aspect ratio of 2 for Reynolds number of 1000 ......................... 116

Figure 6.1: Flow chart of parallelization process with domain decomposition ....... 119 Figure 6.2: Domain decomposition and the ghost nodes ......................................... 121 Figure 6.3: Domain decomposition for different number of processors .................. 123

(a) 1 processor (b) 2 processors (c) 3 processors ................... 123 (d) 4 processors (e) 5 processors (f) 6 processors ................... 123

Figure 6.4: Speed-up of parallelized MNIM for lid-driven cavity problem with exact solutions................................................................................ 125

Figure E.1 Coupling of the Navier-Stokes equations and the energy equation ....... 147 Figure E.2 Coupling of the Navier-Stokes equations,

energy and concentration equations ....................................................... 154 Figure E.3: Exact solution and corresponding L1 error surfaces for the

Navier-Stokes-Energy-Concentration equations for the lid driven cavity with energy and specie sources/sinks (mesh size16 x 16) ................................................................................. 157

(a) u velocity. (b) L1 error for u velocity. ............................................. 157 (c) v velocity. (d) L1 error for v velocity .............................................. 157

1

Chapter1

Introduction

Integral, multi-physics simulations of real objects like rockets, airplanes, nuclear reactors,

etc, require fast computers and efficient numerical methods. For example, structural design, fluid

dynamics, and combustion analysis need to be integrated to accurately simulate a rocket, and

coupled neutronics-thermalhydraulics analysis is necessary for the design of efficient and safe

nuclear reactors. Integral simulations of these large objects using traditional numerical methods,

such as finite difference and finite element methods that require a fine mesh for good accuracy,

require large size matrices and long simulation time. Numerical methods that are accurate over

coarse grid or mesh size are hence desirable.

Time-dependent, incompressible Navier-Stokes (N-S) equations are used to simulate fluid

flow by nuclear engineers and others. This research work is aimed at the development of coarse

mesh numerical methods to efficiently solve the three-dimensional, time-dependent,

incompressible N-S equations.

Brief reviews of traditional numerical methods, some issues specific to the N-S equations,

coarse-mesh numerical methods, nodal integral method, and modified nodal integral method are

given in this chapter.

1.1 . Traditional Numerical Methods

A brief review of traditional numerical methods is given in this section.

2

1.1.1. Finite Difference Method

This is the oldest numerical method used to solve Partial Differential Equations (PDEs).

First proposed by Euler in the 18th century, finite difference method is based on Taylor series

expansion or polynomial fitting to approximate the derivatives of the phase variables. Finite

difference method is simple and effective. It is easy to develop high order schemes on regular

grids. The disadvantage is that a large number of grid points are necessary to achieve the desired

level of accuracy [Ferziger 1996].

1.1.2. Finite Volume Method

In the finite volume method, the conservation equations are integrated over control

volumes to obtain a set of algebraic equations. Finite volume method can be used on any types of

grid, thus not relying on the coordinates. It is conservative by construction and easy to program

[Ferziger 1996]. The disadvantage is that it is difficult to develop finite volume methods of order

higher than second in 3D [Ferziger 1996].

1.1.3. Finite Element Method

This approach is based on variational or weighted-residual method. In finite element

method, the original differential equations are first multiplied by a weight function. The weighted

equations are then integrated over an element. A trial function satisfying continuity across

element boundaries is substituted into the weighted integral equations. By minimizing the

residual, a set of non-linear algebraic equations is obtained for the coefficients in the trial

function. The solution thus leads to the best solution from the set of trial functions.

3

The advantage of finite element method is that it can easily accommodate arbitrary

geometries. However the matrices of the linearlized equations are not as well structured as in

other methods [Ferziger 1996].

1.2 . Numerical methods for the Incompressible Navier-Stokes Equations

1.2.1. Conservative Form v.s. Non-conservative Form

Navier-Stokes equations can be written in conservative or non-conservative form.

Although the two forms do not make a difference in pure theoretical fluid dynamics, the choice is

important for the numerical simulation of certain category of fluid flow problems [Anderson

1995]. It has been shown that conservative form of the N-S equations should be used for shock-

capturing method [Anderson 1995]. The conservative form results in smooth and stable shock-

capturing solutions while the non-conservative form leads to unsatisfactory spatial oscillations

(wiggles) upstream and downstream of the shock or the shocks may appear at incorrect locations.

The reason is that there exists a large discontinuity in density ρ (a primary dependent variable) in

non-conservative form across the shock. This discontinuity in turn would compound the numerical

errors associated with the calculation of ρ [Anderson 1995]. In the conservative form, the

dependent variable (the mass flux uρ ) is constant across the shock wave.

On the other hand, shock-fitting method can obtain satisfactory results for either

conservative form or the non-conservative form.

4

1.2.2. Primitive Form v.s. Derived Form

Two-dimensional, incompressible Navier-Stokes equations can be written in primitive-

variable form or derived-variable form (vorticity-stream function form).

In vorticity-stream function form, the mixed elliptic-parabolic, 2-D, incompressible N-S

equations are transferred into one parabolic equation (the vorticity transport equation) and one

elliptic equation (the Poisson equation). These equations are solved in a sequential way or in a

coupled manner. Among numerous others, Dennis et al. [Dennis 1979], Gatski et al. [Gatski

1982], Fasel and Booz [Fasel 1984] and Guj and Stella [Guj 1988] have developed finite

difference type schemes for the vorticity-stream function form.

The derived-variable form is hard to extend to three dimensions. Hence, the primitive-

variable form is commonly used for 2-D and 3-D problems. In primitive form, it is natural to

solve for each velocity component from its corresponding momentum equation. This leaves the

continuity equation for pressure. But there is no pressure term in the continuity equation. Further

more, there is no dominant variable in the incompressible continuity equation. There are two

groups of methods for solving the incompressible N-S equations in primitive variables: coupled

approach and pressure correction approach [Fletcher 1991] [Caughey 1998]. In coupled

approach, an artificial pressure derivative is added to the continuity equation to allow the coupled

hyperbolic system to be advanced in time. Among many others, Steger and Kutler [Steger 1976],

Choi and Merkle [Choi 1985], Kwak et al. [Kwak 1986], Hartwich et al. [Hartwich 1988] have

developed numerical schemes using this approach. In the pressure correction approach, a Poisson

equation is developed for pressure [Harlow 1965][Caughey 1998]. The velocities and the

pressure are de-coupled and solved separately. Examples of numerical schemes for the pressure-

correction category are, marker-and-cell (MAC) [Harlow 1965], SIMPLE and SIMPLER [Caretto

5

1972] [Patankar 1980], the fractional-step method [Chorin 1968], and the primitive-variable

implicit split operator (PISO) method [Issa 1986].

1.3. Coarse Mesh Methods

A number of coarse-mesh numerical schemes, specifically targeted to solve problems over

large computational domains, have been developed over the last three decades [Burns 1975a]

[Azmy 1983] [Lawrence 1986] [Wilson 1987] [Esser 1993a] [Esser 1993b] [Rizwan-uddin 1997]

[Michael 2001] [Rizwan-uddin 2001a] [Wang 2003a]. Characterized by an initial investment of

human effort—now greatly reduced due to the availability of software for algebraic

manipulations—these schemes yield numerical solutions with comparable accuracy in less CPU

time than those obtained with more conventional approaches. Typically, this efficiency is

achieved by using a coarser mesh size than those required by other schemes. Hence, for given

mesh size, a second order coarse-mesh scheme is likely to lead to smaller error than a second

order finite-difference scheme. Applying coarse mesh methods to the N-S equations promises

solution of much larger scale fluid dynamics problems as well as direct numerical simulation

(DNS) of turbulent flow. Coarse-mesh schemes do have some limitations. For example, those

relying on the transverse-integration procedure (explained in section 1.5.1) are restricted to

physical domains with boundaries parallel to one of the axis, i.e., to geometries that can be filled

with brick-like cells. However, as shown by the experience of the nuclear industry, these schemes

provide enough savings in CPU time to justify their development, even if they are applicable to

only a limited set of problems. Moreover, efforts are also underway to relax these restrictions and

hence make the coarse-mesh methods applicable to even larger set of problems [Toreja 1999]

[Toreja 2003].

6

A brief survey of coarse mesh method is given below.

Partial Current Balance Method (PCBM) [Burns 1975 a] [Burns 1975 b], developed for

multi-group neutron diffusion equations, utilizes multidimensional Green’s function with nearest

neighbor coupling to arrive at a set of discrete equations. PCBM results in a large number of

discrete unknowns per cell. Later, because of the simplicity in development, transverse

integration procedure (TIP) became the primary step in the development of new coarse mesh

methods. This procedure leads to numerical schemes with smaller number of discrete unknowns

per cell when compared with PCBM. Built upon the TIP, a Nodal Green’s Function Method

(NGFM) was developed to solve the multi-group neutron diffusion equations [Lawrence 1979]

[Lawrence 1980 a]. The locally defined Green’s functions were first applied to fluid flow

problem in the Nodal Green’s Tensor Method (NGTM) [Horak 1980] [Horak 1985]. Later, the

Nodal Integral Method (NIM) was developed for the steady-state [Azmy 1983] and time-

dependent [Wilson 1988] Navier-Stokes equations. Application of the NIM—also known as nodal

analytical or cell analytic method—to the neutron diffusion problem [Fischer 1981] and fluid

flow problems [Azmy 1983] is mathematically equivalent to the NGTM. Since Green’s function

is not needed in the NIM, it is simpler to develop and implement than NGTM. NIM was applied

to the steady-state Boussinesq equations for natural convection, and to several steady-state

incompressible flow problems [Fischer 1981] [Azmy 1983] [Azmy 1985]. Esser and Witt [Esser

1993b] developed a nodal scheme for the two-dimensional, vorticity-stream function formulation

of the Navier-Stokes equations. This development—that leads to inherent upwinding in the

numerical scheme—however cannot be easily extended to three dimensions. NIM was also

developed and applied to the time-dependent heat conduction problem [Wilson 1988]. Michael et

al developed a second and a third order NIM for the convection-diffusion equation [Michael

7

1994] [Michael 2001], and compared the results with those obtained using the LECUSSO scheme

[Gunther 1992]. They showed that the nodal integral method achieved the same level of accuracy

with significantly less CPU time than the very efficient LECUSSO scheme [Michael 2001].

Nuclear industry has taken full advantage of developments in coarse-mesh methods, and

consequently, they are the workhorse of the nuclear industry’s neutron diffusion and neutron

transport codes [Burns 1975 a] [Burns 1975 b] [Lawrence 1979] [Lawrence 1980a] [Lawrence

1980b] [Fischer 1981]. Other branches of science and engineering have also taken advantage of

similar approaches to develop efficient schemes [Hennart 1986] [Wescott 2001].

1.4. Nodal Methods

Nodal methods [Hennart 1986] are a subset of coarse-mesh methods. A nodal scheme is

developed by approximately satisfying the governing differential equations on finite size brick-

like elements that are obtained by discretizing the space of independent variables. In nodal

method for neutronics, the multi-group neutron diffusion equations are transverse-averaged over

each homogeneous node. Numerical scheme is then developed for the resulting ordinary

differential equations using, for example, the nodal expansion or nodal integral approaches.

Nodal methods are computationally more efficient than finite difference method.

FLARE model developed in 1964 is a representative of the first generation of nodal

method [Delp 1964] [Lawrence 1986]. Since then, nodal methods have been the preferred method

to solve the multi-group neutron diffusion equations by the nuclear industry [Joo 1997] [Shatilla

1997] [Iwamoto 1998] [Jiang 1998]. A good, but somewhat dated, review of nodal methods

developed in the nuclear industry is given by Lawrence [Lawrence 1986]. Nodal methods, as a

general class of computational schemes, are discussed by Hennart [Hennart 1986]. A comparison

8

of nodal schemes and exact finite difference schemes has appeared recently [Rizwan-uddin

2001a].

In the early development of nodal schemes the brick-like elements were referred to as

nodes — hence the schemes were called nodal. Nodes in nodal methods are however similar to

the elements of the finite element approach, i.e. they are finite volumes — and not points — in the

space of independent variables. This is often a source of confusion since “node” is already used in

the finite difference and finite volume methods to refer to a “point” in space. To avoid this

confusion we will refer to the finite size brick-like volume in the space-time domain as a cell.

(Consequently, nodal integral approach has also been called the “cell-analytic” approach

[Elnawawy 1990].) As in the space-time finite element method (FEM), time in the nodal

approach may be treated in the same manner as any spatial direction.

As mentioned above, nodal schemes have been developed for the Navier-Stokes equations.

Though highly innovative, those early applications did not take full advantage of the potential that

the nodal approach offers. Consequently, these schemes for the Navier-Stokes equations can be

further improved. To lay down the groundwork for the scheme developed in chapter 2, pertinent

features of the nodal integral scheme are outlined in the next section. Past applications to the

Navier-Stokes equations are also discussed, leading to suggestions for improvements.

1.5. Nodal Integral Method

Steps essential to the NIM are outlined briefly in section 1.5.1. Issues relevant to the

treatment of the nonlinear terms, and specifically those relevant to the Navier-Stokes equations,

are separately discussed in section 1.5.2, leading to the modified scheme developed in chapter 2.

9

1.5.1. Nodal Integral Method and Transverse Integration Procedure

In general, development of a nodal integral method can be split into the following four

steps:

a) After discretizing the space-time domain into brick-like cells, each PDE is reduced to a set

of ODEs by applying the transverse integration procedure (TIP) over a cell. The

dependent variables in these ODEs are referred to as transverse-averaged variables.

b) These ODEs are split into homogeneous and inhomogeneous (also called, pseudo-source)

terms. After making certain assumptions about the homogeneous and inhomogeneous

terms, the ODEs are solved analytically for local solutions within each cell using the

discrete values of the transverse-averaged variables at the cell surfaces as boundary

conditions. The transverse-averaged variables evaluated at the cell surfaces are the discrete

variables of the nodal scheme.

c) Continuity of these transverse-averaged variables (and their derivatives for second order

ODEs) is imposed on cell boundaries to obtain a set of discrete equations.

d) Constraint conditions are next used to eliminate the coefficients of expansion of the

pseudo-source terms (identified in step (b)) to obtain a set of discrete equations with

number equal to the number of discrete unknowns per cell.

Steps (a) and (b) are further explained below.

In the TIP, after discretizing the space-time domain of independent variables (X, Y, T) into

finite size computational cells of size ( x y tΔ ×Δ ×Δ ), cell specific local coordinates (x, y, t), with

origin at the center of the cell, are introduced. Hence, with 2 , 2 2x a y b and t τΔ = Δ = Δ = , the cell

is given by , ,a x a b y b tτ τ− ≤ ≤ + − ≤ ≤ + − ≤ ≤ + . See Figure 1.1. Each governing PDE is then

integrated locally over the space-time cell over all independent variables except one, leading to an

10

ODE. Repeating this process with different combinations of independent variables leads to a set

of ODEs for each PDE. The ODEs are for transverse-averaged variables, such as , , ( )xti j ku y , which

is defined as the u velocity, u(x, y, t), transverse-averaged locally over the cell in x and t

directions, i.e.

, , , ,1( ) ( , , )

4

axt

i j k i j ka

u y u x y t dx dtab

τ

τ− −≡ ∫ ∫ . (1.1)

Here, the over-bar and the symbols that follow (xt

) indicate the independent variables over

which the local averaging has been carried out. The subscripts i, j and k respectively identify the

cell in x, y and t directions. The discrete unknowns of the nodal approach are the transverse-

averaged variables evaluated at the cell-surfaces. In other words, the discrete unknowns are the

unknowns (u(x, y, t), v(x, y, t) and p(x, y, t) for the Navier-Stokes equations) averaged over

surfaces of the space-time cells. For example, one of the discrete unknown is the transverse-

averaged variable , , ( )xti j ku y evaluated at y = b, i.e., , , , ,( )xt xt

i j k i j ku y b u= ≡ . (See Figure 1.1.)

While step (a) is common to almost all nodal methods, step (b) is crucial in understanding

the difference between different nodal integral approaches, and is further elaborated here. In

general, ODEs obtained after the TIP do not have analytical solution. The basic idea behind NIM

is to analytically solve in each cell as much of the transverse-integrated ODEs as possible [Azmy

1983] [Rizwan-uddin 1997] for a homogeneous solution, and obtain (approximate) particular

solutions corresponding to the remaining terms. Hence, each ODE is split into two parts: a group

of terms that are retained on the LHS, and remaining terms that are written on the RHS. Splitting

the ODEs into terms retained on the LHS and those kept on the RHS is not arbitrary. In general,

only the terms of the ODEs that are linear in the dependent variable to be solved using that ODE,

are retained on the LHS. The nonlinear terms, as well as linear terms that involve other dependent

11

Y

X

(i, j) (i-1, j) (i+1, j)

(i, j+1)

(i, j-1)

(1, ny)

(1, 1)

(nx, ny)

(nx, 1)

(a)

-ai

x

y

+ai

+bj

-bj

(0,0) ,yt

i ju

, 1xt

i ju −

,xt

i ju( )xtu y

( )ytu x

1,yt

i ju −

y

t x

, , 1(back)xyi j ku −

, 1, (bottom)xti j ku −

1, , (left)yti j ku −

, , (front)xyi j ku

, , (right)yti j ku

, , (top surface)xti j ku

(b) (c)

Figure 1.1: Domain discretization for the nodal integral method. (a) Discretization of the spatial domain into x yn n× cells. (b) Space-time cell (i, j, k) and local coordinate system.

(c) Details of the local coordinates in cell (i, j) in x-y plane.

12

variables, are lumped together on the RHS of the equation as inhomogeneous terms (traditionally

called the pseudo-source term). Solutions to these ODEs are then written as the sum of

homogeneous and particular solutions. The homogeneous part of the solution of these ODEs then

consists of polynomial, trigonometric, exponential or other functions. Since this (homogeneous)

component of the solution is obtained by analytically solving a part of the transverse-averaged

ODEs, it is likely to capture characteristics that are directly relevant to the problem. The

homogeneous solution can thus be considered to be a “finite set of natural basis functions”

specific to the problem—or at least to a part of the problem. This feature makes nodal integral

method distinct from other numerical methods—such as Fourier, collocation and spectral etc—in

which “basis functions” independent of the problem at hand are usually employed. (It is for this

reason that nodal integral method is also known as nodal analytical method.) Particular solutions,

corresponding to the terms that are lumped on the RHS in the pseudo-source term, are obtained

after expanding the pseudo-source terms in a set of complete basis functions and truncating at a

desired level. Hence, the terms lumped in the pseudo-source terms (and the physical process that

these terms represent) are less accurately captured by the numerical scheme than those that

contribute to the homogeneous part of the solution. Consequently, it is desirable to retain as many

terms on the LHS in the transverse-averaged ODEs as possible [Rizwan-uddin 1997].

In step (c), the general solution within each cell—consisting of the homogeneous and

particular parts—is used to obtain the set of discretized equations. The coefficients of expansion

of the pseudo-source terms, which appear in the particular solutions, are initially unknown. They

are eliminated in step (d), leading to a set of discrete algebraic equations.

13

1.5.2. NIM for Nonlinear Equations and Past Applications to the Navier-Stokes Equations

In early applications of the NIM, the nonlinear terms were treated as part of the pseudo-

source terms [Azmy 1983] [Wilson 1988]. For example, in the NIM developed for the time-

dependent, two-dimensional Navier-Stokes (N-S) equations, the nonlinear convection terms as

well as the pressure gradient term were lumped into the pseudo-source term [Wilson 1988]. In

addition, dogged by the absence of pressure in the continuity equation, normal stress, instead of

pressure, was used as an independent variable. Consequently, the standard continuity and the

momentum equations for the two-dimensional, time-dependent, incompressible flow, after the

transverse integration were transformed into the following set of ODEs [Azmy 1983] [Wilson

1988]:

3( )yt

ytdu xdx

ϕ= (continuity equation integrated over y and t) (1.2)

3( )xt

xtdv ydy

ϕ= (continuity equation integrated over x and t) (1.3)

1( )xy

xydu tdt

ϕ= (x momentum equation integrated over x and y) (1.4)

2

12

( )xtxtd u y

dyϕ= (x momentum equation integrated over x and t) (1.5)

4( )yt

ytxd xdx

τ ϕ= (x momentum equation integrated over y and t) (1.6)

2( )xy

xydv tdt

ϕ= (y momentum equation integrated over x and y) (1.7)

2

22

( )ytytd v x

dxϕ= (y momentum equation integrated over y and t) (1.8)

4

( )xty xtd ydy

τϕ= (y momentum equation integrated over x and t) (1.9)

14

where the normal stresses are defined as

xuPx

τ μ ∂≡ −

∂ (1.10)

yvPy

τ μ ∂≡ −

∂. (1.11)

Terms not explicit on the left hand side of the transverse-averaged equations [(1.2) – (1.9)] were

lumped in the pseudo-source (ϕ) terms on the right hand side.

These ODEs were solved to obtain cell-interior solutions for the transverse-averaged

variables. For example, equation (1.5) led to a quadratic (local) variation in y for the transverse-

averaged u velocity, ( )xtu y ; and equation (1.2) led to a linear variation in x for the transverse-

averaged u velocity, ( )ytu x . The formulation consequently led to asymmetries in the local

solutions of transverse-averaged u (and v) velocities in the x and y directions. Moreover, lumping

all the convection terms, when solving the momentum equation, into the RHS, also meant that the

homogeneous part of the analytical solution captured only the diffusion process—and not

convection.

Hence, there are three desirable features in any new coarse-mesh nodal numerical scheme

for the N-S equations: 1) local analytical solution that are more representative of the N-S

equations and the physical processes they represent; 2) formulation in terms of only the primitive

variables; and 3) a numerical scheme that is symmetric in all spatial directions. A recipe to

incorporate these features in a modified nodal integral scheme is given in the next section.

1.6. Modified Nodal Integral Method

Motivated by the desire to “exactly” solve more of the ODEs, i.e., to obtain homogeneous

solution to a larger fraction of the ODEs, a Modified Nodal Integral Method (MNIM) was

15

proposed by Rizwan-uddin [Rizwan-uddin 1997]. The method proposed was successfully applied

to Burgers’ equation [Rizwan-uddin 1997], and led to lower CPU time when compared with the

conventional NIM. The new approach was further modified later and applied to the 2D Burgers’

equations [Wescott 2001]. The ideas introduced are similar to the concept of “delayed

coefficients” in which part of the nonlinear convection term is evaluated in terms of the u and v

velocities at the previous time step [Anderson 1995]. Thus, in the MNIM for the 2D Burgers’

equations, one of the nonlinear convection terms, in its approximated form, is retained on the left

hand side of the ODE, and the homogeneous part of the solution is written for the diffusion as

well as the convection term. That is, for the N-S equations, instead of equation (1.2) for ( )ytu x

and equation (1.5) for ( )xtu y — which would respectively lead to linear and quadratic local

transverse-averaged velocities — two ODEs are respectively obtained by locally transverse-

averaging the x-momentum equation over x and t, and over y and t. These equations are of the

form

2

12

( ) ( )t ttd u du

d d

η ηημ μν σ ϕ

μ μ− = (1.12)

where η = y, x, μ = x, y, and σ = u0, v0. u0 and v0 are the cell-averaged u and v velocities at the

current or previous time step. Consequently, a larger part of the transverse-integrated ODEs is

analytically solved for ( )ytu x and ( )xtu y leading to local cell-interior solutions of the constant +

linear + exponential form

3 1 4( ) .t tvu C e Cσμ

η ημ ϕ μ= + + (1.13)

These local, cell-interior solutions capture the effect of diffusion as well as convection, and are

more representative of the physics than the linear or quadratic local variations for the transverse-

averaged velocities used in earlier development of the NIM. Solution of the 1-D [Rizwan-uddin

16

1997] and 2-D [Wescott 2001] Burgers’ equations with the modified nodal approach—in which

the convection term is retained on the LHS and contributes to the homogeneous part of the

analytical solution—showed that the resulting analytical solution for the cell interior variation is

capable of capturing steep variations within large size spatial cells. Moreover, the numerical

scheme that results also has inherent upwinding.

Another feature of the NIM is that, because part of the ODEs is analytically solved within

each node, local solutions within each node are also available. This is different from the

traditional and more popular finite difference method, in which solution is only available at the

grid points. This feature makes it easier for multi-grid implementation of the nodal methods.

Specifically, because approximate expressions for the variable’s space-time distribution within the

node is available, better restriction operators to project results from fine mesh to coarse mesh and

better prolongation operator (from coarse mesh to fine mesh) can be devised.

A nodal scheme for the Navier-Stokes equations only in terms of primitive variables can

be developed by using the Poisson equation for pressure [Wang 2000]. Solving the Poisson

equation for pressure leaves the two momentum equations to be solved for the velocities.

Consequently, this also eliminates the asymmetries between different spatial directions. (The

asymmetry between ( )ytu x and ( )xtu y in the original development [Azmy 1983] resulted from

the fact that continuity equation in its primitive form was used to solve for ( )ytu x , while the x-

momentum equation was solved to determine ( )xtu y .) Michael and Dorning [Michael 2000a, b]

have developed a nodal scheme for the Navier-Stokes equations in primitive variables recently.

This scheme is similar in its treatment of the transverse-averaged velocities to the scheme

developed below. However, motivated by the desire to develop scheme that could be back-fitted

in some existing production level codes, approximations were introduced to develop discrete

17

equations for a single, cell-averaged pressure. The approach in the current work does not rely on

similar approximations, and two discrete equations are retained for the two transverse-averaged

pressures for each cell.

1.7. Present Work

Modified Nodal Integral Method for the two dimensional, time-dependent, incompressible,

isothermal Navier-Stokes equations is developed in Chapter 2. Rather than using the conventional

continuity equation [Fischer 1981] [Azmy 1983], or the vorticity-stream function formulation

[Esser 1993b] (which is difficult to extend to three dimensions), the momentum equations are

retained in primitive variables, and the conventional continuity equation is replaced by a Poisson

equation written in terms of pressure. In the classical application of the NIM [Fischer 1981]

[Azmy 1983], asymmetries exist in the local solution of u and v velocities in the x and y

directions. Use of Poisson equation for pressure eliminates the asymmetries between different

spatial directions in the MNIM scheme.

In chapter 3, the scheme is used to solve several test problems: fully developed flow and

developing flow between parallel plates, lid driven cavity problem with exact solution, classical

lid-driven cavity problem in a square and in a rectangle with aspect ratio of two, and Taylor-Green

flow problem. Numerical results for these problems obtained using MNIM are presented and

compared with exact or reference solutions.

The modified nodal integral method is expanded to three-dimensions in chapter 4 and

numerical results for three-dimensional problems obtained using MNIM are presented in

chapter 5.

18

In chapter 6, the MNIM is parallelized using domain decomposition technique. Because

of its scalability, MPI is chosen to implement the parallelization. Speed-up results are presented

for the lid driven cavity problem with exact solutions.

Conclusion and suggestions for future are given in chapter 7.

The MNIM is coupled with the energy and specie concentration equations in appendix E.

(This part is carried out in collaboration with Allen Toreja.)

19

Chapter 2

Modified Nodal Integral Method for the Two-Dimensional, Time-Dependent, Incompressible Navier-Stokes Equations

Two-dimensional, time-dependent, incompressible, isothermal Navier-Stokes equations

in primitive variable form are

v 0uX Y

∂ ∂∂ ∂

+ = (2.1)

2 2

2 2

1v ( , , ) 0Xu u u u u pu v b X Y TT X Y X Y X

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂

⎡ ⎤+ + − + + + =⎢ ⎥

⎣ ⎦ (2.2)

2 2

2 2

v v v v v 1v ( , , ) 0Ypu v b X Y T

T X Y X Y ρ Y∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂

⎡ ⎤+ + − + + + =⎢ ⎥

⎣ ⎦ (2.3)

where, ( , , )Xb X Y T and ( , , )Yb X Y T represent volumetric sources such as gravity. Notice that the

capital X, Y and T are used for global coordinate variables, while the lower case x, y and t are

reserved for local coordinate variables introduced later in this chapter.

A modified nodal integral method is developed in this chapter to numerically solve

equations (2.1-2.3). It is natural to solve for each velocity component from its corresponding

momentum equation. This leaves the continuity equation for pressure. But there is no pressure

term in the continuity equation. To deal with this problem, a Poisson equation is developed for

pressure by combining the two momentum equations [Harlow 1965] [Tannehill 1997]. This

equation, when coupled with the continuity equation, can be used to solve for pressure.

2.1. Derivation of Poisson Equation for Pressure

Differentiating equation (2.2) with respect to (w.r.t.) X and equation (2.3) w.r.t. Y yield

20

2 2 2

2 2 2

1v 0Xbu u u u u pu v vT X X X X Y X X X Y X X

∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ρ ∂ ∂

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞+ + − − + + =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(2.4)

and

2 2 2

2 2 2

v v v v v 1v 0Ybpu v vT Y Y X Y Y Y X Y Y Y Y

∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ρ ∂ ∂

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞+ + − − + + =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

. (2.5)

Adding equations (2.4) and (2.5) yields

2 2

2 2

D D Dv vT X Y

∂ ∂ ∂∂ ∂ ∂

− −

2

2

1v Xbu u puX X X Y X X

∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ρ ∂ ∂

⎛ ⎞ ⎛ ⎞+ + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2

2

v v 1v 0YbpuY X Y Y Y Y

∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ρ ∂ ∂

⎛ ⎞ ⎛ ⎞+ + + + =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

. (2.6)

where, the dilatation term D is given by

vuDX Y

∂ ∂∂ ∂

≡ + . (2.7)

After expanding the derivatives, equation (2.6) is written as

2 22 2

2 2

v v2 X Yb bp p u uX Y X Y X Y X Y

∂ ∂∂ ∂ ∂ ∂ ∂ ∂ρ ρ ρ ρ ρ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎛ ⎞ ⎛ ⎞+ = − − − − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2

2 2vD D D D Du v vT X Y X Y

∂ ∂ ∂ ∂ ∂ρ∂ ∂ ∂ ∂ ∂⎡ ⎤

− + + − −⎢ ⎥⎣ ⎦

(2.8)

Note that equation (2.8) is derived from the two scalar momentum equations, and must be

combined with the continuity equation before it is used to solve for pressure. Since the continuity

equation is simply 0D = , setting the square bracket on the RHS of equation (2.8) to zero leads

to an equation that can be used to solve for pressure. However, several authors have pointed out

that setting D in equation (2.8) identically to zero may lead to an unstable numerical scheme

21

[Ghia 1977] [Tannehill 1997]. Hence, while solving the Poisson equation for pressure, retention

of, for example, the temporal derivative of the local dilatation is considered essential for the

convergence of a numerical scheme. Moreover, a discretization of the dilatation term D

consistent with the continuity equation is believed to be important to ensure the convergence of

the numerical scheme.

An alternative formulation of the pressure equation (2.8) is obtained by realizing that

2 2 22v v v v2 2u u u uD

X Y X Y X Y X Y∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ = + − = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

. (2.9)

Hence, equation (2.8) can also be written as

2 2

2 2

v v2 2 X Yb bp p u uX Y X Y Y X X Y

∂ ∂∂ ∂ ∂ ∂ ∂ ∂ρ ρ ρ ρ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

+ = − − −

2 22

2 2vD D D D Du v v DT X Y X Y

∂ ∂ ∂ ∂ ∂ρ∂ ∂ ∂ ∂ ∂⎡ ⎤

− + + − − +⎢ ⎥⎣ ⎦

. (2.10)

A nodal method is developed in the following sections to numerically solve the

incompressible, time-dependent N-S equations using equations (2.2), (2.3) and (2.8). Equations

(2.2) and (2.3) are reproduced below for easy reference:

2 2

2 2

1v 0Xu u u u u pu v v bT X Y X Y X

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂

+ + − − + + = (2.11)

2 2

2 2

v v v v v 1v 0Ypu v v b

T X Y X Y ρ Y∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂

+ + − − + + = . (2.12)

2.2. Transverse Integration Procedure and the Set of ODEs

In the nodal method, the space-time domain (X, Y, T) is first discretized into cells (i, j, k)

of size (2 2 2 )i j ka b τ× × with cell-centered local coordinates ( , ,i i j ja x a b y b− ≤ ≤ − ≤ ≤

k ktτ τ− ≤ ≤ ). Figure 1.1 in chapter 1 shows the discretized spatial domain, a space-time cell, and

22

the local coordinates in a cell with origin located at the center of the cell. As a prelude to the

development of the numerical scheme, the pressure and momentum equations are re-written in

terms of the local coordinate system in the cell (i, j, k) in the following form:

222 2

2 2

v v2 yx bbp p u ux y x y x y x y

∂∂∂ ∂ ∂ ∂ ∂ ∂ρ ρ ρ ρ ρ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎛ ⎞⎛ ⎞+ = − − − − −⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2

2 2vD D D D Du v vt x y x y

∂ ∂ ∂ ∂ ∂ρ∂ ∂ ∂ ∂ ∂

⎡ ⎤− + + − −⎢ ⎥

⎣ ⎦ (2.13)

2 2

p 2 2

1v ( , , ) ( ) (v v )p x p pu u u u u p u uu v b x y t u ut x y x y x x y

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂ ∂ ∂

⎡ ⎤+ + − + = − − − − − −⎢ ⎥

⎣ ⎦ (2.14)

2 2

p p2 2

v v v v v 1 v vv ( , , ) ( ) (v v )p y ppu v b x y t u u

t x y x y ρ y x y∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎡ ⎤+ + − + = − − − − − −⎢ ⎥

⎣ ⎦ (2.15)

where

vuDx y

∂ ∂∂ ∂

≡ + , (2.16)

and pu and v p are respectively the cell-averaged u and v velocities at the previous time step.

Convection terms based on cell-averaged velocities at the previous time step have been added to

both sides of equations (2.11) and (2.12) to obtain equations (2.14) and (2.15). The reason for

writing the momentum equations in this form was alluded to in the previous section (delayed

coefficients), and it will become further obvious in the next sections. By applying the local

transverse integration procedure to equations (2.13), (2.14) and (2.15), eight transverse-

integrated ordinary differential equations are obtained below.

Applying the transverse-integration operator 14

k i

k i

a

ai k

dxdta

τ

ττ − −∫ ∫ to equations (2.13), (2.14)

and (2.15) respectively yields

23

2

12

( ) ( )xt

xtd p y S ydy

= (2.17)

2

22

( ) ( )v ( )xt xt

xtp

du y d u yv S ydy dy

− = (2.18)

2

32

v ( ) v ( )v ( )xt xt

xtp

d y d yv S ydy dy

− = , (2.19)

where, the cell-specific subscripts (i, j, k) on independent variables have been omitted, and

, , , ,1( ) ( , , ) , , v,

4k i

k i

axt

i j k i j kai k

y x y t dx dt u pa

τ

τφ φ φ

τ − −≡ =∫ ∫ . (2.20)

, , ( )yti j k xφ and , , ( )xy

i j k tφ are similarly defined. Terms not explicit in equations (2.17) – (2.19) are

lumped into the right hand as pseudo-source terms:

222

2

1 2 2

2 2

v v21( )

4v

k i

k i

yxa

xt

ai k

bbp u ux x y x y x y

S y dxdta D D D D Du v v

t x y x y

τ

τ

∂∂∂ ∂ ∂ ∂ ∂ρ ρ ρ ρ ρ∂ ∂ ∂ ∂ ∂ ∂ ∂

τ ∂ ∂ ∂ ∂ ∂ρ∂ ∂ ∂ ∂ ∂

− −

⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟+ + + + +⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠≡ − ∫ ∫ ⎜ ⎟⎛ ⎞⎜ ⎟+ + + − −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(2.21)

2

2 2

1 1( ) (v v )4

k i

k i

axt

p xai k

u u u u pS y dxdt u v ba t x y x x

τ

τ

∂ ∂ ∂ ∂ ∂τ ∂ ∂ ∂ ∂ ρ ∂− −

⎛ ⎞≡ − + + − − + +∫ ∫ ⎜ ⎟

⎝ ⎠ (2.22)

and

2

3 p 2

1 v v v v 1( ) (v v )4

k i

k i

axt

yai k

pS y dxdt u v ba t x y x ρ y

τ

τ

∂ ∂ ∂ ∂ ∂τ ∂ ∂ ∂ ∂ ∂− −

⎛ ⎞≡ − + + − − + +∫ ∫ ⎜ ⎟

⎝ ⎠. (2.23)

There are no approximations introduced up to this stage of the development.

The inhomogeneous pseudo-source terms in equations (2.21) – (2.23) are then expanded

in Legendre polynomials. Truncation of these expansions at specific order determines the order

of the numerical scheme. Here, the expansion is truncated at the zeroth order, which is consistent

with the goal of a second order scheme [Azmy 1983]. [In general, truncating at higher order, in

24

conjunction with other consistent approximations, leads to numerical scheme of order higher

than second [Elnawawy 1990] [Michael 1994].] The above process yields

2

12

( )xtxtd p y S

dy= (2.24)

2

22

( ) ( )vxt xt

xtp

du y d u yv Sdy dy

− = (2.25)

and

2

32

v ( ) v ( )vxt xt

xtp

d y d yv Sdy dy

− = . (2.26)

Note that it is only the absence of the argument that differentiates 1 ( )xtS y in equation (2.17) from

1xtS in equation (2.24). Latter is the zeroth order Legendre expansion of the former.

Similarly, applying the transverse-integration operator 14

jk

k j

b

bj k

dydtb

τ

ττ − −∫ ∫ , to equations

(2.13), (2.14) and (2.15), and approximating the pseudo-source terms by constants, result in

2

12

( )ytytd p x S

dx= (2.27)

2

22

( ) ( )yt ytyt

pdu x d u xu v S

dx dx− = (2.28)

and

2

32

v ( ) v ( )yt ytyt

pd x d xu v S

dx dx− = , (2.29)

where, the definitions of the pseudo-source terms prior to truncation are

222

2

1 2 2

2 2

v v21( )

4v

jk

k j

yxb

yt

bj k

bbp u uy x y x y x y

S x dydtb D D D D Du v v

t x y x y

τ

τ

∂∂∂ ∂ ∂ ∂ ∂ρ ρ ρ ρ ρ∂ ∂ ∂ ∂ ∂ ∂ ∂

τ ∂ ∂ ∂ ∂ ∂ρ∂ ∂ ∂ ∂ ∂

− −

⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟+ + + + +⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠≡ − ∫ ∫ ⎜ ⎟⎛ ⎞⎜ ⎟+ + + − −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(2.30)

25

2

2 2

1 1( ) v ( )4

jk

k j

byt

p xbj k

u u u u pS x dydt u u v bb t y x y x

τ

τ

∂ ∂ ∂ ∂ ∂τ ∂ ∂ ∂ ∂ ρ ∂− −

⎛ ⎞≡ − + + − − + +∫ ∫ ⎜ ⎟

⎝ ⎠ (2.31)

and

2

3 2

1 v v v v 1( ) v ( )4

jk

k j

byt

p ybj k

pS x dydt u u v bb t y y y ρ y

τ

τ

∂ ∂ ∂ ∂ ∂τ ∂ ∂ ∂ ∂ ∂− −

⎛ ⎞≡ − + + − − + +∫ ∫ ⎜ ⎟

⎝ ⎠. (2.32)

Next, applying the operator, 14

ji

i j

ba

a bi j

dxdya b − −

∫ ∫ , to equations (2.14) and (2.15), and expanding and

truncating the pseudo-source terms yields,

2( )xy

xydu t Sdt

= (2.33)

3v ( )xy

xyd t Sdt

= (2.34)

where, the pre-truncated pseudo-source terms— 2 ( )xyS t and 3 ( )xyS t —are given by

2 2

2 2 2

1 1( ) v4

ji

i j

baxy

xa bi j

u u u u pS t dxdy u v v ba b x y x y x

∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ρ ∂− −

⎛ ⎞≡ − + − − + +∫ ∫ ⎜ ⎟

⎝ ⎠ (2.35)

2 2

3 2 2

1 v v v v 1( ) v4

ji

i j

baxy

ya bi j

pS t dxdy u v v ba b x y x y ρ y

∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂− −

⎛ ⎞≡ − + − − + +∫ ∫ ⎜ ⎟

⎝ ⎠. (2.36)

Note that due to the absence of a time derivative, the pressure equation leads to only two ODEs.

Equations (2.24) – (2.29), (2.33) and (2.34) form the set of eight ODEs that will be solved to

develop the set of discrete equations.

The reason behind the introduction of the convection term based on the (known) cell-

averaged velocity at the previous time step should now be obvious. These terms are linear, and

hence allow the convection term – albeit a linear one – to contribute to the homogeneous solution

of the transverse averaged momentum equations. A brief discussion of the treatment of the

nonlinear term in nodal analytical schemes is given in the next section.

26

2.3. Discussion of the Treatment of the Nonlinear Terms

First nodal integral scheme for the Navier-Stokes equations [Azmy 1983] was developed

with only the diffusion terms contributing to the homogeneous solutions of the transverse-

averaged differential equations. Hence, except for the diffusion term, all other terms in the

momentum equations were lumped in the pseudo-source terms. See, for example, equations (1.5)

and (1.8). Cognizant of the advantages in obtaining homogeneous solution of the transverse-

integrated momentum equations that locally capture the diffusion as well as the convection

process, it is desirable, when the transverse averaged equations are split for the homogenous and

particular components of the solution, to retain the convection terms on the LHS. However,

convection terms, being nonlinear, do not lend themselves easily to analytical solutions. Hence,

following the procedure for the convection-diffusion equation—in which the velocity field is

assumed known, and therefore the convection term can be retained on the LHS [Elnawawy

1990]—a modified nodal scheme for the 1-D Burgers’ equation was developed by approximating

the nonlinear convection term u ∂u/∂x by u0 ∂u/∂x, where the velocity u0 is the (unknown) cell-

averaged u velocity at the current time step. The non-linearity was resolved through an iterative

process. However, this approach was computationally expensive since the unknown cell-

averaged velocities, u0 and v0, appear as argument of exponential functions that must be

repeatedly evaluated during the iteration process.

To avoid this computational overhead, the scheme was further modified, and also applied

to the 2-D Burgers’ equation [Wescott 2001]. To reduce the computational burden, convection

terms based on cell-averaged velocities at the previous time step are added to both sides of the

transverse-integrated momentum equations [Wescott 2001]. For example, the term vp ∂u/∂y is

added to both sides of the u momentum equation before it is transverse-integrated in the x and t

27

directions, where v p is the cell-averaged v velocity at the previous time step. The nonlinear

term, v u/ y dx dt∂ ∂∫∫ , is moved to the right hand side and lumped into a modified pseudo-

source term. This procedure is followed for the Navier-Stokes equations in the previous section.

It led to equations (2.25), (2.26), (2.28) and (2.29), which can be solved analytically within each

cell. Solutions of these equations for the cell-interior variations of the velocity are of constant +

linear + exponential form. Clearly, this functional dependence can more accurately capture a

wider range of cell-interior variations than the quadratic variation that results when all the

convection terms are lumped into the pseudo-source term. The coefficients in the resulting

scheme depend on the velocities pu and v p , and thus the scheme possesses inherent upwinding,

though the upwinding is based on velocities at the previous time step. Thus, by introducing the

cell-averaged velocities at the previous time step, the exponentials need to be evaluated only

once for each time step rather than once every iteration, which significantly reduces the

computational burden [Wescott 2001].

2.4. Transverse-Averaged ODEs

The final set of eight transverse-integrated ordinary differential equations is

2

12

( )xtxtd p y S

dy= (2.37)

2

12

( )ytytd p x S

dx= (2.38)

2

22

( ) ( )vxt xt

xtp

du y d u yv Sdy dy

− = (2.39)

2

32

v ( ) v ( )vxt xt

xtp

d y d yv Sdy dy

− = (2.40)

28

2

22

( ) ( )yt ytyt

pdu x d u xu v S

dx dx− = (2.41)

2

32

v ( ) v ( )yt ytyt

pd x d xu v S

dx dx− = (2.42)

2( )xy

xydu t Sdt

= (2.43)

and

3v ( )xy

xyd t Sdt

= , (2.44)

where , ,i i j ja x a b y b− ≤ ≤ − ≤ ≤ and k ktτ τ− ≤ ≤ ; cell-specific subscripts (i, j, k) have been

omitted; and the right hand sides represent the truncated expansions of the pseudo-source terms.

Complete symmetry exists between u and v velocities, and between x and y directions in this

formulation.

2.5. Local Solutions

These ODEs are solved locally within each cell. The local solution of the ODEs for

transverse-integrated pressure is quadratic. For example, the solution of equation (2.37) is

1 2( ) 1 22

xtxt Sp y y C y C= + + . (2.45)

A similar solution can be written for ( )ytp x . The local solution for ( )xtu y is of the form

v

3 2 4( )p y

xt xtvu y C e S y C= + + , (2.46)

and solutions for the other transverse-integrated velocities ( )ytu x , v ( )xt y and v ( )yt x are similar.

The solutions for ( )xyu t and v ( )xy t are linear in time. For example,

29

2 5( )xy xyu t S t C= + . (2.47)

Recognizing that the discrete unknowns associated with the cell (i, j, k) will be the

surface-averaged variables on cell surfaces, the constants Ci (i = 1, 2, …) in the above solutions

are eliminated in favor of these discrete unknowns by imposing boundary conditions, or initial

conditions, on cell surfaces normal to the independent variable. For example, boundary

conditions for equation (2.45) are

, , , , , , , 1,( ) , ( )xt xt xt xti j k j i j k i j k j i j kp y b p p y b p −= + = = − = . (2.48)

See Figure 2.1. The resulting expressions for , , ,( ), ( ), v ( )xt xt xti j i j i jp y u y y , , ( )xy

i ju t and ,v ( )xyi j t are

( )1 , 2 2, , , 1 , , 1

1 1( ) ( ) ( )2 2 2

xti jxt xt xt xt xt

i j j i j i j i j i jj

Sp y y b p p y p p

b − −= − + − + + (2.49)

1,2 v ,

,

, ,

,

Re v2 , , , , 1 ,

, 2 ,Re v,,

Re v Re v2 , , , , 1 ,

Rev,

( 2 v v ) 1( )vv ( 1 )

(1 ) v vv ( 1 )

i j yp i jv

i j

i j i j

i j

xt xt xtj i j i j p i j i j p i jxt xt

i j i jp i jp i j

xt xt xtj i j i j p i j i j p i j

p i j

e b S u uu y e S y

e

b S e u u ee

−

−

− + −= +

− +

+ − ++

− +

(2.50)

1,2 v ,

,

, ,

,

Re v3 , , , , 1 ,

, 3 ,Re v,,

Re v Re v3 , , , , 1 ,

Re v,

( 2 v v v v ) 1v ( )vv ( 1 )

(1 ) v v v vv ( 1 )

i j yp i jv

i j

i j i j

i j

xt xt xtj i j i j p i j i j p i jxt xt

i j i jp i jp i j

xt xt xtj i j i j p i j i j p i j

p i j

e b Sy e S y

e

b S e ee

−

−

− + −= +

− +

+ − ++

− +

(2.51)

2 1( ) ( )xy xy xyku t S t uτ −= + + , (2.52)

and

3 1v ( ) ( ) vxy xy xykt S t τ −= + + (2.53)

30

Figure 2.1: Continuity of transverse averaged pressure ( )xtp y between cell (i, j) and (i, j+1).

x

y

(0, 0) (i, j+1)

x

y

(0, 0) (i, j)

y = -bj

y = +bj

y = -bj+1

y = +bj+1

kjijxt byp

,1,1)(+

+−=

kjijxt byp

,,)( =

xtkjikjij

xtkjij

xt pbypbyp ,,,1,1,,)()( ≡−===

++

31

where the local Reynolds number in the y direction is defined as

,,

2 vRe v j p i j

i j

bv≡ (2.54)

and the subscript k for current time step variables has been omitted. Solution for , ( )yti jp x ,

, ( )yti ju x and ,v ( )yt

i j x can be obtained similarly. Like ,Re vi j , ,Re i ju is defined as

,,

2Re i p i j

i j

a uu v≡ . (2.55)

This completes steps (a) and (b) discussed in Sec. 1.5.1. Local solutions obtained above are used

in the next section to derive the set of discrete equations (step c).

2.6. Set of Discrete Equations in Terms of the Pseudo-Source Terms

A set of discrete equations is obtained by imposing continuity of each variable at cell

interfaces (continuity of derivative for second order equations). For the first order ODEs for

, ,xy

i j ku and , ,v xyi j k , the algebraic equations are obtained by simply evaluating the local solutions for

( )xyu t and v ( )xy t —equations (2.52) and (2.53)—at t τ= . For the second order ODEs,

continuity of the transverse averaged variable is (automatically) imposed by simply using the

same notation to identify the discrete variable at the interface between the two neighboring cells.

For example, for transverse averaged pressure between cells (i, j, k) and (i, j+1, k) this means,

, , , 1, 1 , ,, , , 1,( ) ( )xt xt xt

i j k j i j k j i j ki j k i j kp y b p y b p+ + +

= = = − ≡ . (2.56)

See Figure 2.1 for details. Then, imposing the continuity of the derivative at the cell interfaces

yields a three-point scheme. For example, for transverse averaged pressure between cells (i, j, k)

and (i, j+1, k), this means

32

, , , 1,1

, , , 1,

( ) ( )xt xti j k i j k

j j

i j k i j k

dp dpy b y b

dy dy+

+

+

= = = − . (2.57)

Equation (2.57) leads to the discrete equation for pressure , ,xti j kp . Repeating the same process for

the other variables, a total of six coupled, algebraic equations per cell for , , , , , , , ,, v , , ,xt xt xt yti j k i j k i j k i j ku p u

, , , ,v ,yt yti j k i j kp are derived in terms of the pseudo-source terms, S’s.

Eight discrete algebraic equations thus obtained (six mentioned above, and two discussed

earlier for , ,xy

i j ku and , ,v xyi j k ) are

1, , 1 , 1 1 , 1 1 , 1

1 1

( ) 1 1 02 2 2

j j xt xt xt xt xti j i j i j j i j j i j

j j j j

b bp p p b S b S

b b b b+

− + + ++ +

+− − + + = (2.58)

1, 1, 1, 1 , 1 1 1,

1 1

( ) 1 1 02 2 2

yt yt yt yt yti ii j i j i j i i j i i j

i i i i

a a p p p a S a Sa a a a

+− + + +

+ +

+− − + + = (2.59)

21 2 , 22 2 , 1 23 , 1 23 24 , 24 , 1( ) 0xt xt xt xt xti j i j i j i j i jA S A S A u A A u A u+ − ++ + − + + = (2.60)

21 3 , 22 3 , 1 23 , 1 23 24 , 24 , 1v ( ) v v 0xt xt xt xt xti j i j i j i j i jA S A S A A A A+ − ++ + − + + = (2.61)

51 2 , 52 2 1, 53 1, 53 54 , 54 1,( ) 0yt yt yt yt yti j i j i j i j i jA S A S A u A A u A u+ − ++ + − + + = (2.62)

51 3 , 52 3 1, 53 1, 53 54 , 54 1,v ( ) v v 0yt yt yt yt yti j i j i j i j i jA S A S A A A A+ − ++ + − + + = (2.63)

, , , 1 2 ,2 0xy xy xyi j i j k i ju u Sτ−− − = (2.64)

, , , 1 3 ,v v 2 0xy xy xyi j i j k i jSτ−− − = , (2.65)

where, once again, the subscript k for current time step variables has been omitted, k-1 denotes

the previous time-step values, and 21A , 22A , ... , and 51A , 52A , ... , are coefficients which are

functions of ,, , ,Rei j i ja b v u and ,Re vi j . For example,

33

,,

, ,

RevRe v,

21 23Rev Rev,

v2 1 ;v(1 ) (1 )

i ji j

i j i j

p i jj

p i j

eb eA A

v e v e≡ + ≡

− −. (2.66)

A list of definitions for all the coefficients A can be found in appendix A.

Three characteristics of the numerical scheme being developed can be identified at this

stage. First, the local solution of the transverse averaged velocities has a component that varies

exponentially in space. These exponential terms can capture steep spatial variation of velocities

within each cell, thus, allowing the use of coarse meshes. Second, because of the appearance of

the local Reynolds number in the exponential terms, the scheme being developed has inherent

upwinding [Wescott 2001]. Third, local Reynolds number based only on cell-averaged velocities

at the previous time step appear as argument of the exponential terms. Hence, these terms can be

evaluated at the beginning of each time step outside the iteration loop, which significantly

reduces the computation time.

2.7. Constraint Equations

The eight discrete algebraic equations (2.58 – 2.65) per cell, given in the last section, are

in terms of sixteen unknowns: , , , ,, v , , ,xt xt xt yti j i j i j i ju p u , , , ,v , , , vyt yt xy xy

i j i j i j i jp u , 1 , 1 ,,xt yti j i jS S , 2 , 2 , 2 ,, , ,xt yt xy

i j i j i jS S S

3 , 3 , 3 ,, ,xt yt xyi j i j i jS S S . Thus, eight more equations are needed to close the set of equations (step d).

Following the nodal approach [Azmy 1983], these are developed next using eight constraint

equations. Three constraint equations are obtained by ensuring that the continuity and the

momentum equations are satisfied over each cell in an integral sense. Applying the cell-

averaging operator, 18

jk i

k j i

b a

b ai j k

dxdydta b

τ

ττ − − −∫ ∫ ∫ , on equations (2.13), (2.14) and (2.15) respectively

yields

34

1 1 1 0xt ytS S f+ + = (2.67)

2 2 2 2 0xt yt xyS S S f+ + + = (2.68)

and

3 3 3 3 0xt yt xyS S S f+ + + = , (2.69)

where

22

, 1, , 1, , , 1 , , 11

, , 1 , 1,

v v v v2

2 2 2 2

2 2 2

yt yt yt yt xt xt xt xti j i j i j i j i j i j i j i j

i i j j

xt xt yt ytyi j yi j xi j xi j

j i

u u u uf

a a b b

b b b b Db a

ρ ρ ρ

ρ ρ ρτ

− − − −

− −

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞− − − −≡ + + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞− −+ +⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

(2.70)

2 0 , 1, 0 , , 1

, 1,

1 1( ) ( ) (v v ) ( )2 2

1 1 ( )2

yt yt xt xtp i j i j p i j i j

i j

yt yt xyti j i j x

i

f u u u u u ua b

p p baρ

− −

−

≡ − − + − −

+ − + (2.71)

and

3 0 , 1, 0 , , 11 1( ) (v v ) (v v ) (v v )

2 2yt yt xt xt

p i j i j p i j i ji j

f u ua b− −≡ − − + − − , , 1

1 1 ( )2

xt xt xyti j i j y

j

p p bbρ −+ − + (2.72)

and the following approximation

1 ( , , ) ( , , )8

1 1( , , ) ( , , )8 8

jk i

jk i

j jk i k i

j jk i k i

b a

i j k b a

b ba a

i j k i j kb ba a

x y t x y t dxdydta b

x y t dxdydt x y t dxdydta b a b

τ

τ

τ τ

τ τ

φ ψτ

φ ψτ τ

−− −

− −− − − −

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

≈∫

∫ ∫

∫ ∫

∫ ∫ ∫ ∫ (2.73)

has been used to arrive at equations (2.70) – (2.72). Approximating the average of the product by

product of the averages, as above, is known to be second order accurate [Azmy 1983]. For

example, ( , , )v( , , ) u x y tx y ty

∂∂

is locally averaged over x and t as

35

01 ( , , ) ( ) ( )v( , , ) v ( ) v

4k i

k i

xt xtaxt

ai k

u x y t du y du yx y t dxdt ya y dy dy

τ

τ

∂τ ∂− −

≈ ≈∫ ∫ . (2.74)

where, 0v is the current time step cell-averaged velocity. To simplify and simultaneously retain

the stability of the numerical scheme, in equation (2.70), we replace the square bracket on the

RHS of equation (2.13) with (D/2τ) evaluated at the current time step, where τ is half time step.

The other five constraint equations are obtained by imposing the condition that the cell-averaged

variables be unique, independent of the order of integration, i.e.

1 1( ) ( )2 2

j k

j k

bxty xt xy xyt

j kb

u u y dy u t dt ub

τ

ττ

− −

≡ = ≡∫ ∫ (2.75)

1 1( ) ( )2 2

i k

i k

aytx yt xy xyt

i ka

u u x dx u t dt ua

τ

ττ

− −

≡ = ≡∫ ∫ (2.76)

1 1v v ( ) v ( ) v2 2

j k

j k

bxty xt xy xyt

j kb

y dy t dtb

τ

ττ

− −

≡ = ≡∫ ∫ (2.77)

1 1v v ( ) v ( ) v2 2

i k

i k

aytx yt xy xyt

i ka

x dx t dta

τ

ττ

− −

≡ = ≡∫ ∫ (2.78)

1 1( ) ( )2 2

ji

i j

baytx yt xt xty

i ja b

p p x dx p y dy pa b

− −

≡ = ≡∫ ∫ . (2.79)

These constraint conditions are simplified after substituting the local solutions given by

equations (2.49 – 2.53), and the corresponding expressions for the other dependent variables. For

example, equation (2.75) yields

, ,

, ,

Re v Re v

2 , 2 , , , 1 , 1Re v Re v2, , ,

1 1 1v v Re v1 1

i j i j

i j i j

jxt xy xy xti j i j i j k i j

p i j p i j i j

b e eS S u ue e

τ − −

⎛ ⎞ ⎛ ⎞+− − − + − +⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟− − +⎝ ⎠⎝ ⎠

36

,

,

Re v

, Re v,

11 0Re v1

i j

i j

xti j

i j

eue

⎛ ⎞⎛ ⎞− − =⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟− +⎝ ⎠⎝ ⎠

. (2.80)

This can be rewritten as

91 2 , 2 , , , 1 92 , 1 92 ,(1 ) 0xt xy xy xt xti j i j i j k i j i jA S S u A u A uτ − −− − + + − = (2.81)

where the definitions of 91A and 92A are obvious from the comparison of equations (2.80) and

(2.81). A list of definitions for all the coefficients A can be found in appendix A. Similarly,

equations (2.76), (2.77) and (2.78) lead to the following set of equations:

1 2 , 2 , , , 1 2 1, 2 ,(1 ) 0yt xy xy yt yta i j i j i j k a i j a i jA S S u A u A uτ − −− − + + − = (2.82)

91 3 , 3 , , , 1 92 , 1 92 ,v v (1 )v 0xt xy xy xt xti j i j i j k i j i jA S S A Aτ − −− − + + − = (2.83)

1 3 , 3 , , , 1 2 1, 2 ,v v (1 )v 0yt xy xy yt yta i j i j i j k a i j a i jA S S A Aτ − −− − + + − = , (2.84)

where 1aA , 2aA have definitions similar to 91A and 92A . The uniqueness of pressure, equation

(2.79), has the following form

2 2

, , 1 , 1, 1 , 1 ,1 1 1 1 02 2 2 2 3 3

jxt xt yt yt xt ytii j i j i j i j i j i j

b ap p p p S S− −+ − − − + = . (2.85)

Thus, a total of sixteen algebraic equations are derived for sixteen unknowns for each

cell: eight from the continuity of transverse-integrated variables and their derivatives [equations

(2.58) – (2.65)]; three from the cell-averaged conservation equations [equations (2.67) – (2.69)];

and five from the uniqueness conditions [equations (2.81) – (2.85)]. The pseudo-source terms

are eliminated next from this set, leaving only eight physically relevant unknowns and eight

equations per cell.

37

2.8. Set of Discrete Equations

The final set of eight, discrete, algebraic equations are

17 , 11 , 1 12 , 1 13 , 1,

14 , 1 1, 1 15 1 , 16 1 , 1

( )

( )

xt xt xt yt yti j i j i j i j i j

yt yti j i j i j i j

F p F p F p F p p

F p p F f F f− + −

+ − + +

= + + +

+ + + + (2.86)

27 , 21 1, 22 1, 23 , , 1 24 1, 1, 1

25 1 , 26 1 1,

( ) ( )yt yt yt xt xt xt xti j i j i j i j i j i j i j

i j i j

F p F p F p F p p F p p

F f F f− + − + + −

+

= + + + + +

+ + (2.87)

37 , 31 , 1 32 , 1 33 , , , 1 34 , 1 , 1, 1( ) ( )xt xt xt xy xy xy xyi j i j i j i j i j k i j i j kF u F u F u F u u F u u− + − + + −= + + + + + (2.88)

37 , 31 , 1 32 , 1 33 , , , 1 34 , 1 , 1, 1v v v (v v ) (v v )xt xt xt xy xy xy xyi j i j i j i j i j k i j i j kF F F F F− + − + + −= + + + + + (2.89)

57 , 51 1, 52 1, 53 , , , 1 54 1, 1, , 1( ) ( )yt yt yt xy xy xy xyi j i j i j i j i j k i j i j kF u F u F u F u u F u u− + − + + −= + + + + + (2.90)

57 , 51 1, 52 1, 53 , , , 1 54 1, 1, , 1v v v (v v ) (v v )yt yt yt xy xy xy xyi j i j i j i j i j k i j i j kF F F F F− + − + + −= + + + + + (2.91)

77 , 71 , 72 , 1 73 , , 1 74 , 75 1, 2 ,xy xt xt xy yt yt

i j i j i j i j k i j i j i jF u F u F u F u F u F u f− − −= + + + + + (2.92)

77 , 71 , 72 , 1 73 , , 1 74 , 75 1, 3 ,v v v v v vxy xt xt xy yt yti j i j i j i j k i j i j i jF F F F F F f− − −= + + + + + , (2.93)

where F’s are coefficients that are functions of , ,, , , ,Re ,Re vi j i j i ja b v uτ . [Ai,j is also a function

of , ,, , , ,Re ,Re vi j i j i ja b v uτ .] For example,

11 2 2

312 2( )

j

j i j

bF

b a b= +

+ (2.94)

23 9231 21

91

A AF AA

= − (2.95)

and

7391 1

1 1 1 12 a

FA A τ

⎛ ⎞= + −⎜ ⎟

⎝ ⎠. (2.96)

38

A list of definitions for all the coefficients F can be found in appendix B. The discrete unknowns

for the cell (i, j, k) are the variables averaged on the cell surfaces:

, , , ,, v , , ,xt xt xt yti j i j i j i ju p u , , , ,v , , , vyt yt xy xy

i j i j i j i jp u .

2.9. Boundary Conditions

Though the discrete unknowns in the scheme developed in the previous section are the

dependent variables averaged over the surfaces of the space-time cell (i, j, k) in X-Y-T space,

boundary conditions for surface averaged velocities are relatively straightforward. No slip

boundary conditions are imposed on solid surfaces. In addition, Dirichlet condition can also be

specified, for example, on inlet surfaces. Nodal scheme developed in the previous section leads

to a collocated discretization. Hence, along with boundary conditions on u and v velocities,

pressure boundary conditions are also needed.

Boundary conditions for pressure on no-slip surfaces are derived using the x- and y

momentum equations [Ghia 1977] [Gresho 1987] [Ferziger 1996] [Tannehill 1997]. For

example, on vertical no-slip surfaces, u = v = 0, ut

∂∂

= 2

2 0uy

∂∂

= , and thus, the u-momentum

equation, averaged locally over y and t, becomes [Gresho 1987]

2

2

1 ( ) ( ) 0yt yt

ytx

dp x d u xv bdx dxρ

− + = . (2.97)

One straightforward approach to derive the discrete form of this boundary condition is to satisfy

equation (2.97) on the surface of the boundary cell. This can be achieved by substituting in

equation (2.97) the expressions derived in section 2.5 for the local solution of transverse-

integrated pressure ( )ytp x

39

1 2( ) 1 22

ytyt Sp x x B x B= + + (2.98)

and transverse-integrated velocity ( )ytu x

u

3 2 4( )p x

yt ytvu x B e S x B= + + (2.99)

Thus, algebraic equations for transverse-averaged pressure on vertical boundaries can be easily

obtained. However, local solutions for ( )ytp x and ( )ytu x are second order accurate.

Consequently, the second derivative of ( )ytu x only has zeroth order accuracy, and the discrete

form of the boundary condition will also be only zeroth order accurate. Hence, to derive a second

order accurate boundary condition for pressure, consistent with the second order accuracy of the

scheme, a fourth order accurate finite difference expression for ytu (x) on the boundaries is used.

For a stationary right vertical surface the boundary condition is developed as follows. Let the

discrete variable ytu on the right surface of a boundary cell (i, j), and on the right surfaces of

cells (i-1, j), (i-2, j) and (i-3, j) be represented by 0 ,( )yt yti ju x u= , 0 1 1,( )yt yt

i ju x h u −− = ,

0 1 2 2,( )yt yti ju x h h u −− − = and 0 1 2 3 3,( )yt yt

i ju x h h h u −− − − = (see Figure 2.2). A second order

accurate discrete approximation for the second derivative at x = x0 is

20 1 2 3 1 2 3

, 1,21 1 2 1 2 3 1 2 2 3

21 2 3 1 22, 3,

2 1 2 3 3 2 3 1 2 3

( ) 2( 3 2 ) 2 (2 2 )( )( ) ( )

2 (2 ) 2 (2 ) ( )( ) ( )( )

ytyt yt

i j i j

yt yti j i j

d u x x h h h h h hu udx h h h h h h h h h h

h h h h hu u O hh h h h h h h h h h

−

− −

= + + + += − +

+ + + +

+ + +− +

+ + + +

(2.100)

The large template for the second derivative can be reduced by imposing additional conditions.

For example, the continuity equation

v 0ux y

∂ ∂∂ ∂

+ = (2.101)

40

h4 h3 h2 h1

(i-3, j) (i-2, j) (i-1, j) (i, j)

ytu = 3,yt

i ju − 2,yt

i ju − 1,yt

i ju − ,yt

i ju

x = (x0 – h1 – h2 – h3) ( x0 – h1 – h2) ( x0 – h1) x0

Figure 2.2: Boundary condition for pressure at the right surface.

41

when imposed along a stationary, vertical, no-slip wall ( v 0x

∂∂

= ), requires

0ux

∂∂

= . (2.102)

Transverse integrating equation (2.102) locally over y and t yields

0ytdu

dx= . (2.103)

Using a third order accurate finite difference expression for the first derivative, equation (2.103)

becomes

0 1 2 1 2 3, 1,

1 1 2 1 2 3 1 2 2 3

31 1 2 3 1 1 22, 3,

2 1 2 3 3 2 3 1 2 3

( ) ( )( ) 1 1 1( )( )

( ) ( ) ( ) 0( ) ( )( )

ytyt yt

i j i j

yt yti j i j

du x x h h h h hu udx h h h h h h h h h h

h h h h h h hu u O hh h h h h h h h h h

−

− −

= + + += + + −

+ + + +

+ + ++ − + =

+ + + +

(2.104)

Equation (2.104) is used to eliminate the discrete variable farthest from the surface ( 3,yt

i ju − ) from

the four-point finite difference expression (equation (2.100)), resulting in a second order

accurate, three-point scheme for the second derivative at the wall,2

02

( )ytd u x xdx

= ,

2 2 220 1 1 2 2

,2 2 2 21 1 2

21 2 11, 2,2 2

1 2 2 1 2

( ) 2( 3 3 ) ( )

2( ) 2 ( ) ( )

ytytyt

i jwall

yt yti j i j

d u x x h h h hd u udx dx h h h

h h hu u O hh h h h h− −

= + += = − +

+

+− +

+

. (2.105)

A second order accurate scheme for the first derivative of pressure at the wall has the

following form

0

, 1, , 2 1, 1 2 2, 1 2

1 2 1 2( )

- ( )( )

( )

yt yt yt yt ytyti j i j i j i j i j

wall x x

p p p h p h h p hdp O hdx h h h h

− − −

=

− + += + +

+ (2.106)

42

[However, the results obtained using the second order scheme and those obtained using the cell

interior expression for ( )ytp x to evaluate the derivativeyt

wall

dpdx

, yield similar numerical results.]

These expressions for 2

2

ytd udx

and ytdp

dx are substituted in equation (2.97) to obtain the discrete

form of the pressure boundary condition for ,yti jp at the right wall. Pressure boundary conditions

for the other walls are similarly derived.

This completes the development of the set of discrete equations for the time-dependent,

incompressible Navier-Stokes equations. This set of equations has been implemented in a

Fortran code, and tested on several two-dimensional, steady-state and time-dependent fluid flow

problems. Steady-state problems are solved by marching in time. Several iterative approaches

have been tested to solve the final set of algebraic equations at each time step. Results presented

in the next chapter are based on a Gauss-Seidel iterative procedure in conjunction with a

SIMPLE-like algorithm that couples the field variables. That is, for fixed pressure field,

velocities are evaluated from bottom left to the top right of the domain, row by row. Next,

keeping velocities fixed, discrete pressure values are evaluated (row by row) from lower left of

the domain to the top right of the domain. For given velocity field, around fifteen pressure-

sweeps yield near optimum convergence. (For pressure updates, ADI scheme was tested but was

found to be less efficient.) As is the case with many other iterative approaches, for most

examples studied here only a single sweep to update the velocity, for given pressure field, was

found to be sufficient. Numerical results are reported in the next section.

43

Chapter 3

Application of Modified Nodal Integral Method for Time-Dependent Navier-Stokes Equations – Two Dimensional Case

Numerical scheme and the boundary conditions developed in chapter two were coded in

Fortran. The code runs on a SUN Ultra II Workstation as well as on a PC running LINUX

operating system. All CPU times reported in this section are for simulations carried out on the

PC. CPU times for the 1.5 GHz PC were lower by up to a factor of eight when compared with

those for the SUN Ultra II workstation.

The MNIM developed for the Navier-Stokes equations has been applied to the following

steady state problems: fully-developed flow and developing flow between parallel plates, lid

driven cavity problem with exact solutions, classic lid driven cavity problem and lid driven

cavity problem with aspect ratio of two. The time dependent code is used to solve the steady

state problems by marching in time. The code is tested on a two-dimensional, time-dependent

Taylor’s decaying vortices problem.

Three of these problems, the fully developed flow between parallel plates, modified lid

driven cavity problem, and Taylor’s decaying vortices problem have exact analytical solutions,

Exact solutions allow easy debugging of the code as well as numerical estimate of the order of

the numerical scheme. All problems have been used extensively by researchers to test numerical

schemes developed for the Navier-Stokes equations.

44

3.1. Fully Developed Flow Between Parallel Plates

The 2D fully developed flow between two parallel plates is one of the most simple flow

problems with exact solution. The boundary conditions for this problem are shown in Figure

3.1.1. The fully developed feature is captured by setting the derivative with respect to x equal to

zero ( 0=∂∂x

) at the inlet and exit planes. Wall boundary conditions described in Chapter 2 are

used for top and bottom surfaces. Constant pressure is enforced at the inlet and exit planes.

The well-known exact solution for this problem is

2 2 2

02 2( ) ( )(1 ) (1 )2L dp y yu y u

dx L Lμ= − − = − (3.1)

Re is based on half of the channel width L and the maximum velocity 0u

vLu0Re = (3.2)

where

)(2

)0(2

0 dxdpLuu −==

μ. (3.3)

45

Figure 3.1.1: Boundary conditions for fully developed flow between parallel plates.

2

2

0v 0

1 0

xt

xt

xt xtxt

y

u

dp d uv bdy dyρ

=

=

− + + =

2

2

0v 0

1 0

xt

xt

xt xtxt

y

u

dp d uv bdy dyρ

=

=

− + + =

,

0

v 0

0

yt

yt

yti j

dudx

ddx

p

=

=

=, 0

0

v 0

yt

yt

yti j

dudx

ddx

p p

=

=

=

x

y

Ly =

Ly −=

46

3.1.1. Numerical Results

Numerical results for the Poiseuille flow are obtained for the following parameter values,

0.5, ( 0) 4.0, ( 1) 0, 0.005, 1L p x p x μ ρ= = = = = = = (3.4)

The exact centerline velocity is

2

0 (0) ( ) 1002L dpu u

dxμ= = − = , (3.5)

and thus, the Reynolds number is

40Re 10u Lv

= = . (3.6)

The velocity profile obtained using MNIM is shown in Figure 3.1.2. Numerical results agree

very well with exact solution (Figure 3.1.3). Since a time marching scheme with u(x, y, t = 0) =

1.0, v(x, y, t = 0) = 0 is used to solve a steady-state problem, numerical solutions approach the

exact solution asymptotically as time evolves. The evolutions of the centerline u-velocity for two

different time-step sizes are shown in Figure 3.1.4. When time (the product of time step number

and time step size) is used as the x-axis, the evolutions of centerline u-velocity for the two

different time step sizes are very close to each other, though the time step size has been doubled.

However, numerical experiments showed that the time-step cannot be arbitrarily large when

solving steady state problems. The upper bound on time step size for convergence is a function

of mesh size, Reynolds number and initial condition.

47

x

y

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.1.2: Flow field for fully developed flow between parallel plates.

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-20

0

20

40

60

80

100

120

ubxt

y

Numerical Solution Exact Solution

Figure 3.1.3: Comparison of u velocity with exact solution.

48

0 10 20 30 40 50

50

60

70

80

90

100

Ubx

t at c

ente

r lin

e

Time = (time step # ) x (time step size)

Time step size = 1.0 Time step size = 2.0

Figure 3.1.4: Evolution of center-line velocity for different time steps.

49

3.2. Developing Flow Between Parallel Plates

Developing flow between parallel plates (or inlet flow problem) is another steady state

fluid flow problem. In this problem, velocity at the inlet plane is uniform, while pressure is

specified at the exit. The length is chosen to be twice the entrance length so that the flow is fully

developed at the exit [Schlicting 1968]. The flow field develops from uniform along y direction

to a parabolic distribution at the exit, where the fully developed boundary conditions are

enforced. Boundary conditions for developing flow problem are shown in Figure 3.2.1.

Inlet boundary condition for pressure is derived from the u-momentum equation (2.2)

2 2

2 2

1v ( , , ) 0xu u u u u pu v b x y tt x y x y x

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂

⎡ ⎤+ + − + + + =⎢ ⎥

⎣ ⎦ (3.7)

applied to the set of cells whose left surface coincides with the inlet plane. Velocity v is zero at

the inlet, thus v 0y

∂∂

= . Using continuity equation v 0ux y

∂ ∂∂ ∂

+ = , it is obvious that 0ux

∂∂

= . For a

constant u-velocity inlet boundary condition, the u-momentum equation transverse-averaged

over y and t is reduced to the following form

2

2

1 0yt yt

ytx

dp d uv bdx dxρ

− + = , (3.8)

which is the same as wall boundary condition (2.97). Particular attention is needed when the inlet

velocity is not uniform and the viscosity term 2

2

uvy

∂∂

− cannot be eliminated in the u-momentum

equation.

50

Figure 3.2.1: Boundary conditions for developing flow between parallel plates.

2

2

0v 01 v 0

xt

xt

xt xtxt

y

u

dp dv bdy dyρ

=

=

− + =

2

2

0v 01 v 0

xt

xt

xt xtxt

y

u

dp dv bdy dyρ

=

=

− + =

,

0

v 0

0

yt

yt

yti j

d udx

ddx

p

=

=

=x

y

Ly =

Ly −=

2

2

v 01 0

ytin

yt

yt ytyt

x

u u

dp d uv bdx dxρ

=

=

− + =

51

Velocity profile for 2Re 10inu Lv

= = ( 0.01inu = , 5L = , 0.01v = ) in a domain with size 10 10×

is shown in Figure 3.2.2. A detailed comparison of velocities with those reported by Azmy

[Azmy 1982] is listed in Table 3.2.1. The results obtained using MNIM agree very well with

earlier results [Azmy 1982]. Velocity profile for Reynolds number of 100 is shown in

Figure 3.2.3.

52

x

y

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Figure 3.2.2: Flow field for developing flow, Re = 10.

x

y

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

Figure 3.2.3: Flow field for developing flow, Re = 100.

53

Table 3.2.1: Numerical comparison with Azmy’s [Azmy1982] results for developing flow. (1, 1) and (6, 6) are respectively the lower left and top right cells in the domain.

Node # MNIM NIM (Azmy [1982])

Ubxy Vbxy Ubxy Vbxy 1, 1 0.00847 0.00158 0.00863 0.00137 1, 2 0.01088 0.00205 0.01093 0.00180 1, 3 0.01042 0.00021 0.01043 0.00043 1, 4 0.01042 -0.00021 0.01043 -0.00043 1, 5 0.01088 -0.00205 0.01093 -0.00180 1, 6 0.00847 -0.00158 0.00863 -0.00137 2, 1 0.0065 0.00092 0.00631 0.00096 2, 2 0.01158 0.0021 0.01184 0.00194 2, 3 0.01194 0.00076 0.01185 0.00098 2, 4 0.01194 -0.00076 0.01185 -0.00098 2, 5 0.01158 -0.0021 0.01184 -0.00194 2, 6 0.0065 -0.00092 0.00631 -0.00096 3, 1 0.00495 0.00037 0.00502 0.00033 3, 2 0.01146 0.00115 0.01159 0.00090 3, 3 0.01338 0.00062 0.01340 0.00057 3, 4 0.01338 -0.00062 0.01340 -0.00057 3, 5 0.01146 -0.00115 0.01159 -0.00090 3, 6 0.00495 -0.00037 0.00502 -0.00057 4, 1 0.00465 0.00004 0.00458 0.00010 4, 2 0.01122 0.00035 0.01126 0.00029 4, 3 0.01409 0.00025 0.01416 0.00019 4, 4 0.01409 -0.00025 0.01416 -0.00019 4, 5 0.01122 -0.00035 0.01126 -0.00029 4, 6 0.00465 -0.00004 0.00458 -0.00010 5, 1 0.00439 0.00005 0.00445 0.00002 5, 2 0.0111 0.00015 0.01115 0.00007 5, 3 0.01436 0.00009 0.01440 0.00005 5, 4 0.01436 -0.00009 0.01440 -0.00005 5, 5 0.0111 -0.00015 0.01115 -0.00007 5, 6 0.00439 -0.00005 0.00445 -0.00002 6, 1 0.00442 -0.00003 0.00442 0.00000 6, 2 0.01107 -0.00003 0.01107 0.00001 6, 3 0.01442 0 0.01442 0.00000 6, 4 0.01442 0 0.01442 0.00000 6, 5 0.01107 0.00003 0.01107 0.00001 6, 6 0.00442 0.00003 0.00442 -0.00000

54

3.3. Classical Lid Driven Cavity Problem

In this well-known problem [Pan 1967], the flow in a square cavity of dimension one on

each side is driven by the moving lid. The other three surfaces are at zero velocity. Numerical

results obtained over a very fine mesh [Ghia 1982] have been widely used for comparison.

The velocity profile for Re =100 is plotted in Figure 3.3.1. Figure 3.3.1 (a) is plotted with

vector length proportional to the magnitude of velocities. Figure 3.3.1 (b) is plotted with uniform

vector length in order to show the flow directions for small velocity locations. Figures 3.3.2

shows the u and v velocities along the vertical and horizontal lines passing through the center of

the box for a Reynolds number of 100. Fine mesh results from [Ghia 1982] are also shown.

Results presented in Figures 3.3.2 are obtained using 4 4× , 8 8× and 16 16× non-uniform

meshes. The geometric factor used was 1.3. Even the results on a very coarse 4 4× mesh, for

this relatively low Reynolds number problem, agree fairly well with the fine-mesh data.

For a Reynolds number of 1000, the velocity vector profile is plotted in Figure 3.3.3.

Figure 3.3.4 (a) shows the u velocity component along the vertical line passing through the

center of the box. Fine-mesh results of [Ghia 1982] are also plotted. Figure 3.3.4(b) shows the

v velocity component along the horizontal line passing through the center of the box. For nodal

method, ytu and v xt values are plotted at the center of the cell. Results are presented for 12 12× ,

16 16× and 20 20× non-uniform mesh. Non-uniform meshes for this problem were generated

using a geometric factor of 1.4 from the center of the cavity toward the wall. Even for as coarse

as 12 12× mesh, results match fairly well with the reference data. Results obtained on the 16 16×

mesh compare very well with those reported in [Ghia 1982] (30 30× mesh) obtained using a

variable explicit/implicit method for unstructured meshes.

55

x

y

0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a)

x

y

0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

Figure 3.3.1: Velocity vectors for classical lid-driven cavity problem for Re = 100 (a) Vector length proportional to the velocity magnitude. (b) Uniform vector length.

56

0.0 0.2 0.4 0.6 0.8 1.0-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0Re=100

u

y

Ghia [1982] mesh 4X4 mesh 8x8 mesh 16x16

(a)

0.0 0.2 0.4 0.6 0.8 1.0-0.4

-0.2

0.0

0.2

0.4

0.6Re=100

v

x

Ghia [1982] mesh 8x8 mesh 16x16

(b)

Figure 3.3.2: Velocity profile for classical lid-driven cavity problem for Re = 100. Fine mesh results are from [Ghia 1982]. (a). u-velocity along the vertical line through geometric center of the cavity (b) v-velocity along the horizontal line through geometric center of the cavity.

57

x

y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a)

x

y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

Figure 3.3.3: Velocity vectors for classical lid-driven cavity problem for Re = 1000. (a) Vector length proportional to the velocity magnitude. (b) Uniform vector length.

58

0.0 0.2 0.4 0.6 0.8 1.0-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0Re=1000

u

y

Ghia [1982] mesh 12x12 mesh 16x16 mesh 20x20

(a)

0.0 0.2 0.4 0.6 0.8 1.0-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6Re=1000

v

x

Ghia [1982] mesh 12x12 mesh 16x16 mesh 20x20

(b) Figure 3.3.4: Velocity profile for classical lid-driven cavity problem for Re = 1000. Fine mesh results are from [Ghia 1982]. (a). u-velocity along the vertical line through geometric center of the cavity. (b) v-velocity along the horizontal line through geometric center of the cavity.

59

3.4. Lid Driven Cavity Problem in a Rectangle with Aspect Ratio = 2

In this problem the dimension in the y direction is twice the size of the cavity in the x

direction. The flow is divided into two main regions. A strong vortex is formed in the upper half

of the domain, while a weak vortex in the opposite direction is formed in the lower half of the

domain. The flow structure in the upper half of the domain is similar to those obtained in a

square cavity.

The velocity vector profile for Re = 100 is plotted in Figure 3.4.1. Figure 3.4.1 (a) shows

vector length proportional to the magnitude of velocities. Figure 3.4.1 (b) is plotted with uniform

vector length in order to clearly show the flow direction at locations where the velocity is small.

The u-velocity along the vertical centerline is plotted in Figure 3.4.2.

The velocity vector profile for Re = 1000 is plotted in Figure 3.4.3. The u-velocity along

vertical centerline obtained using MNIM is compared with results obtained using commercial

CFD software Fluent. Very good agreement is achieved for Re = 1000. Also shown is the

convergence of mesh refinement from a 30 60× mesh to a 40 80× mesh for MNIM.

60

x

y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

1.25

1.5

1.75

2

x

y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

1.25

1.5

1.75

2

(a) (b)

Figure 3.4.1: Velocity vectors for lid-driven cavity problem with aspect ratio of 2 for Re = 100. (a) Vector length proportional to the velocity magnitude. (b) Constant vector length.

61

0.0 0.5 1.0 1.5 2.0-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

ubxt

y

Figure 3.4.2: u-velocity along the vertical line through geometric center of the cavity for lid-driven cavity problem with aspect ratio of 2 for Re = 100.

62

x

y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

1.25

1.5

1.75

2

x

y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

1.25

1.5

1.75

2

(a) (b)

Figure 3.4.3. Velocity vectors for lid-driven cavity problem for Re = 1000. (a) Vector length proportional to the velocity magnitude. (b) Constant vector length.

63

0.0 0.5 1.0 1.5 2.0

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

u

y

MNIM 40x80 q=1.3x1.05 Fluent 80x160 q=1.2x1.05 MNIM 30x60 q=1.3x1.05

Figure 3.4.4: Comparison of u-velocity along the vertical line through the geometric center of the cavity for lid-driven cavity problem for Re = 1000 with results obtained using Fluent. Results of the nodal scheme are plotted at the center of the cell and q is the geometric ratio for Non-uniform cell size in x and y directions.

64

3.5. Modified Lid Driven Cavity Problem

A variation of the classical lid driven cavity problem has been proposed by Shin et al

[Shin 1989]. This problem—here referred to as the modified lid driven cavity problem—has an

exact analytical solution. The modifications include a lid velocity that varies along the lid, i.e.,

ulid = u(x), and space-dependent body forces within the cavity. The fact that the lid velocity is

equal to zero at the two corners eliminates the singularity that exists at those two points in the

classical lid driven cavity problem. This problem was solved by Shin et al to compare nine

numerical schemes developed for the Navier-Stokes equations [Shin 1989]. The exact solution of

the modified lid driven cavity problem (0 ≤ x ≤ 1 and 0 ≤ y ≤ 1) is given by [Shin 1989]

2 3 4 3( , ) 8( 2 )( 2 4 )u x y x x x y y= − + − + (3.9)

2 3 2 4v( , ) 8(2 6 4 )( )x y x x x y y= − − + − + (3.10)

and

( )

3 4 52 3 3

4 85 6 7 3 2 2 2 4

8( , ,Re) 24( ) (2 6 4 )( 2 4 )Re 3 2 5

64 2 3 2 ( 2 4 ) ( 2 12 )( )2 2

x x xp x y y x x x y y

x xx x x y y y y y

⎛ ⎞= − + + − + − + +⎜ ⎟

⎝ ⎠⎛ ⎞

− + − + − − + + − + − +⎜ ⎟⎝ ⎠

(3.11)

where the non-uniformly distributed body forces are given by

( )

3 4 5

2 3 2 2 4

2 3 4 5 6 3 2 4

4 85 6 7 2 3 2 4

24( )8( , , Re) 3 2 5Re

2(2 6 4 )( 2 12 ) ( 12 24 )( )

( 2 8 14 12 4 )( 2 4 )( )64

( 2 3 2 ) ( 2 12 )( 2 4 ) 24 ( )2 2

x

x x xb x y

x x x y x y y

x x x x x y y y yx xx x x y y y y y y

⎛ ⎞− + +⎜ ⎟= − +⎜ ⎟⎜ ⎟− + − + + − + − +⎝ ⎠

⎛ ⎞− − + − + − − + − + +⎜ ⎟⎜ ⎟− + − + − − + − + + − +⎜ ⎟⎝ ⎠

(3.12)

0yb = . (3.13)

The lid velocity is given by

65

2 3 4( ) ( , 1) 16( 2 ).lidu x u x y x x x= = = − + (3.14)

The velocity and pressure fields, u(x, y), v(x, y) and p(x, y), over 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1,

are shown in Figure 3.5.1. A vector plot of the velocity field for Re = 1 is shown in Figure 3.5.2.

This steady-state problem is also solved by starting from an arbitrary initial condition

(zero uniform velocity, and zero uniform pressure) and marching in time till steady state is

reached. The results for Reynolds number Re = 1 , 10 and 20 are given in Tables 3.5.1-3.5.3.

These results were obtained with Dirichlet boundary conditions for all variables, including

pressure, on all surfaces. RMS errors in xyu , xyv , xtp and ytp are reported for different mesh

sizes. CPU times are also reported. For Re = 1, even for as coarse as 5 5× uniform mesh, the

numerical scheme developed here yields a small RMS error of 0.006 and 0.001 for xyu and xyv ;

and it takes only 1.2 s of CPU time on the PC. The CPU time is low despite the fact that very

simple Gauss-Seidel sweeps are used repeatedly at each time step till convergence. For larger

problems, significant savings in CPU time can be achieved by incorporating more efficient

solvers.

66

Table 3.5.1: RMS errors and CPU times for Re = 1 (Dirichlet boundary conditions).

Mesh xyu xyv xtp ytp CPU time (s)

5 x 5 0.5816 x 10-2 0.1058 x 10-2 0.1849 x 10-1 0.2211 x 10-1 1.2

10 x 10 0.1313 x 10-2 0.2595 x 10-3 0.5448 x 10-2 0.5539 x 10-2 2.2

20 x 20 0.2779 x 10-3 0.6216 x 10-4 0.1703 x 10-2 0.1709 x 10-2 31.9

Table 3.5.2: RMS errors and CPU times for Re = 10 (Dirichlet boundary conditions).

Mesh xyu xyv xtp ytp CPU time (s)

5 x 5 0.4077 x 10-2 0.3715 x 10-2 0.8003 x 10-2 0.8373 x 10-2 1.3

10 x 10 0.9918 x 10-3 0.7849 x 10-3 0.1585 x 10-2 0.1628 x 10-2 3.3

20 x 20 0.2511 x 10-3 0.1879 x 10-3 0.3740 x 10-3 0.3752 x 10-3 37.5

Table 3.5.3: RMS errors and CPU times for Re = 20 (Dirichlet boundary conditions).

Mesh xyu xyv xtp ytp CPU time (s)

5 x 5 0.5398 x 10-2 0.6616 x 10-2 0.7904 x 10-2 0.8506 x 10-2 2.4

10 x 10 0.1173 x 10-2 0.1333 x 10-2 0.1623 x 10-2 0.1736 x 10-2 6.4

20 x 20 0.2946 x 10-3 0.3119 x 10-3 0.3758 x 10-3 0.3771 x 10-3 85.1

67

(a) (b)

(c)

(c)

Figure 3.5.1: Velocity and pressure fields of the modified lid driven cavity problem.

(a) u velocity (b) v velocity (c) Pressure

00.25

0.5

0.75

1

x

0

0.25

0.5

0.75

1

y

-0.20

0.2

0.4

u

00.25

0.5

0.75

1

x

00.25

0.5

0.75

1

x

0

0.25

0.5

0.75

1

y

-0.4-0.2

00.2

0.4

v

00.25

0.5

0.75

1

x

0

0.25

0.5

0.75

1

x

0

0.25

0.5

0.75

1

y

0

2

4

6

p

0

0.25

0.5

0.75

1

x

68

Figure 3.5.2: Velocity vector plot of the modified lid driven cavity problem.

69

The near second order accuracy of the scheme can be seen from the tables, confirming

the (at least) second order nature of the approximations introduced in the development. As the

Reynolds number is increased to 10, the RMS error for velocity and pressure (except for xyv ) in

general decrease, in some case by as much as a factor of 5. However, the error in cell-averaged v

velocity increases. The error in xyu , xtp and ytp remain roughly the same as the Reynolds

number is increased to 20, while the error in xyv increases by as much as a factor of almost 2.

The problem was then solved using the pressure boundary conditions developed in

section 2.9. No-slip boundary conditions were imposed for velocities on all surfaces. The

problem was solved for Reynolds number of 1, 10 and 20, on 5 5× , 10 10× and 20 20× uniform

meshes. See Tables 3.5.4-3.5.6. The RMS errors for Re = 1 are in general higher than the

corresponding RMS errors found with Dirichlet boundary conditions for pressure. However,

RMS errors in xyu are lower by as much as a factor of 2. The RMS errors for velocities are in the

range of 0.22 × 10-2 to 0.95 × 10-4. As the Reynolds number is increased to 10 and then 20,

RMS error in velocities either remain approximately constant or increase, while RMS errors in

pressure decrease significantly. It should be noted that several schemes based on central finite

difference approach, tested and reported in [Shin 1989], failed to converge to the correct

solution for Re > 10, and those that did, converged to grossly inaccurate solutions [Shin 1989].

Moreover, RMS errors in the numerical results obtained here are lower than the errors in eight of

the nine schemes tested in [Shin 1989]. Only the results obtained using the 4/4 non-staggered

(HO) scheme are comparable with those obtained using the nodal scheme.

70

Table 3.5.4: RMS errors and CPU times for Re = 1 (pressure boundary conditions).

Mesh xyu xyv xtp ytp CPU time (s)

5 x 5 0.2184 x 10-2 0.2180 x 10-2 0.7851 x 10-1 0.9017 x 10-1 0.8

10 x 10 0.5278 x 10-3 0.4302 x 10-3 0.1757 x 10-1 0.1918 x 10-1 7.56

20 x 20 0.1251 x 10-3 0.9451 x 10-4 0.4193 x 10-2 0.4404 x 10-2 133.3

Table 3.5.5: RMS errors and CPU times for Re = 10 (pressure boundary conditions).

Mesh xyu xyv xtp ytp CPU time (s)

5 x 5 0.2229 x 10-2 0.2709 x 10-2 0.1074 x 10-1 0.1039 x 10-1 6.2

10 x 10 0.5421 x 10-3 0.5075 x 10-3 0.2473 x 10-2 0.2456 x 10-2 30.3

20 x 20 0.1400 x 10-3 0.1171 x 10-3 0.6152 x 10-3 0.6045 x 10-3 375.1

. Table 3.5.6: RMS errors and CPU times for Re = 20 (pressure boundary conditions).

Mesh xyu xyv xtp ytp CPU time (s)

5 x 5 0.3503 x 10-2 0.4112 x 10-2 0.9185 x 10-2 0.7427 x 10-2 20.5

10 x 10 0.7713 x 10-3 0.7678 x 10-3 0.2059 x 10-2 0.1917 x 10-2 64.0

20 x 20 0.1964 x 10-3 0.1794 x 10-3 0.5078 x 10-3 0.4808 x 10-3 582.8

71

3.6. Taylor’s Decaying Vortices

The fourth problem solved is the time-dependent Taylor’s decaying vortices problem

[Taylor 1923]. The problem was chosen because it has an exact analytical solution, allowing for

accurate error analysis. Kim and Moin [Kim 1985] utilized the exact solution to test the

boundary conditions for the fractional step method. Henriksen and Holmen [Henriksen 2002]

used it to test their algebraic splitting scheme for the incompressible Navier-Stokes equations.

Quarteroni et al. [Quarteroni 2000] also tested their factorization methods using the exact

solution.

An exact solution of the two-dimensional, time-dependent Navier-Stokes equations, with

1ρ = , is given by the stream function [Taylor 1923]

2 22 2( , , ) exp[ ( ) ]cos( ) cos( )

( ) x y x yx y

x y t v k k t k x k yk kωψ = − ++

(3.15)

which leads to the following u(x,y,t) and v(x,y,t) velocities:

2 22 2( , , ) exp[ ( ) ]cos( )sin( )

( )y

x y x yx y

ku x y t v k k t k x k y

y k kωψ −∂

= − = − +∂ +

(3.16)

2 22 2v( , , ) exp[ ( ) ]sin( )cos( )

( )x

x y x yx y

kx y t v k k t k x k yx k k

ωψ∂= = − +

∂ + (3.17)

where ω is the initial maximum vorticity, and kx and ky are wave numbers. The field represents

a decaying system of eddies in a rectangular array rotating alternately in opposite directions. The

u and v velocities and pressure at time t = 0 are shown in Figures 3.6.1(a) through 3.6.1(c) over

0 1x≤ ≤ , 0 1y≤ ≤ . Figure 3.6.1(d) shows the velocity vector plot of the flow field. Parameter

values for the flow shown in Figure 3.6.1 are kx = ky = 2π, ω = 2 π2, and ν = 1.

72

00.2

0.40.6

0.81

x

0

0.2

0.4

0.6

0.8

1

y

-1

0

1u

00.2

0.40.6

0.81

x

(a)

(b)

Figure 3.6.1: Velocity fields for the Taylor’s decaying vortices problem at t = 0. (a) u velocity. (b) v velocity.

00.2

0.40.6

0.81

x

0

0.2

0.4

0.6

0.8

1

y

-1

0

1v

00.2

0.40.6

0.81

x

73

00.2

0.40.6

0.81

x

0

0.2

0.4

0.6

0.8

1

y

-1.5-1

-0.50

0.5p

00.2

0.40.6

0.81

x

(c)

(d)

Figure 3.6.1: (c)Pressure field for the Taylor’s decaying vortices problem at t = 0. (d) Corresponding velocity vector plot. Coefficients of three neighboring discrete variables at

two different locations (A and B) are shown in Table 3.6.1.

A

B

74

The problem was solved over 0 1x≤ ≤ , 0 1y≤ ≤ with Dirichlet boundary conditions for

all variables on all surfaces [Taylor 1923] [Kim 1985]. Numerical results for ( )xtu y and ( )xtp y

for 0.4375 0.5x≤ ≤ at four different times are compared with the exact solutions in Figures

3.6.2(a) and 3.6.2(b). These results were obtained on a 16 × 16 uniform mesh, and with tΔ =

0.005. The RMS error at t = 0.01 for the 16 × 16 grid case is 1.1×10-3 for xtu and 7.4 × 10-3 for

xtp . An even coarser, 8 x 8 grid, calculation leads to an RMS error of only 6.8 × 10-3 for xtu

and 2.7x10-2 for xtp at t = 0.01, again showing the near second order accuracy of the method. In

fact, in some cases, the results show a better than second order accuracy.

The numerical scheme developed here, as was pointed out in chapter two, has “inherent

upwinding.” That is, based on flow directions at neighboring cells, the coefficients in the discrete

algebraic equations multiplying the velocities at these neighboring cells are automatically

adjusted. This characteristic of the scheme is demonstrated by evaluating the coefficients that

multiply the velocities in the neighboring cells in determining ytu values at two different

locations in the Taylor’s decaying vortices problem. Two neighboring cells at two different

locations are shown schematically in Figure 3.6.1(d) and identified by the letters A and B. The

flow at A is to the left, and at B it is to the right. The u velocity , ,yt

i j ku at each of these locations is

evaluated using equation (2.90) in terms of the neighboring velocities 1, ,yt

i j ku − and 1, ,yt

i j ku + (in

addition to other discrete variables). Equation (2.90) is re-written as

75

(a)

(b)

Figure 3.6.2: Numerical and exact solutions of the Taylor’s decaying vortices problem at different times. (a) u velocity. (b) Pressure.

y

0.0 0.5 1.0

_ xt

u

-1

0

1t=0

t=0.005

t=0.01

t=0.02

t=0.05

0.4375 < x < 0.5

y

0.0 0.5 1.0

_ xt

p

0

1

t=0

t=0.005

t=0.01

t=0.02

t=0.05

0.4375 < x < 0.5

76

, 1 1, 2 1, 3 , , , 1 4 1, 1, , 1( ) ( )yt yt yt xy xy xy xy

i j i j i j i j i j k i j i j ku c u c u c u u c u u− + − + + −= + + + + + (3.18)

and the coefficients at t = 0.01 (after 20 time steps) are shown in Table 3.6.1. These coefficients

correspond to the simulation over a 10 × 10 grid for ν = 0.001. The magnitude of the

coefficients of 1, ,yt

i j ku − and 1, ,yt

i j ku + , c1 and c2, for the two cases clearly show that the numerical

scheme is automatically “weighting” the coefficients as a function of the flow direction. In

addition, the magnitudes of the coefficients of the local space-averaged u velocities (averaged

over x and y, xyu ) are also adjusted as a result of the flow direction.

Table 3.6.1: Coefficients of discrete variables in equation (3.18) showing inherent upwinding.

Coefficients Location A Location B

1c -0.024 -95.9

2c -95.9 -0.024

3c 1.06 96.9

4c 96.9 1.06

77

Chapter4

Modified Nodal Integral Method for the Three-Dimensional, Time-Dependent, Incompressible Navier-Stokes Equations

Two-dimensional modified nodal integral method for the N-S equations has been

developed in chapter two with two new features: Poisson-type pressure equation was used

instead of the continuity equation; convection terms in the N-S equations are kept on the left

hand side and thus contribute to the homogeneous solution of the transverse-integrated ordinary

differential equations. Here, we extend the two-dimensional modified nodal integral method to

three dimensions. Extension is straightforward but not trivial.

4.1. Reformulation and Discretization of the N-S Equations

The time-dependent, incompressible Navier-Stokes equations in three dimensions are:

v 0u wX Y Z

∂ ∂ ∂∂ ∂ ∂

+ + = (4.1)

2 2 2

2 2 2

1v ( , , , ) 0Xu u u u u u u pu w v g X Y Z TT X Y Z X Y Z X

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ρ ∂

⎡ ⎤+ + + − + + + − =⎢ ⎥

⎣ ⎦ (4.2)

2 2 2

2 2 2

v v v v v v 1v ( , , , ) 0Yw pu w v g X Y Z T

T X Y Z X Y Z ρ Y∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎡ ⎤+ + + − + + + − =⎢ ⎥

⎣ ⎦ (4.3)

2 2 2

2 2 2

1v ( , , , ) 0Zw w w w w w w pu w v g X Y Z TT X Y Z X Y Z ρ Z

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎡ ⎤+ + + − + + + − =⎢ ⎥

⎣ ⎦ (4.4)

78

where, ( , , , )g X Y Z T represents volumetric body forces such as gravity, and capital letters X, Y,

Z and T are used to denote the global coordinates. Similar to the 2D case, a Poisson equation for

pressure is derived to replace the continuity equation. Manipulating Equations (4.2-4.4), the

Poisson equation for pressure is given by [Harlow 1965] [Tannehill 1997]

2 2 22 2 2

2 2 2

2 2 2

2 2 2

v

v v2 2 2

v

X Y Z

p p p u wX Y Z X Y Z

g g gu w w uY X Z Y X Y X Y ZD D D D D D Du w v v vT X Y Z X Y Z

∂ ∂ ∂ ∂ ∂ ∂ρ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ρ ρ ρ ρ ρ ρ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ρ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + = − + +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

− − − + + +

⎡ ⎤− + + + − − −⎢ ⎥

⎣ ⎦

(4.5)

where, the dilatation term D is defined as

vu wDX Y Z

∂ ∂ ∂∂ ∂ ∂

≡ + + . (4.6)

Since equation (4.5) was derived only from the momentum equations (4.2-4.4), the continuity

equation can be incorporated in equation (4.5) by simply setting D equal to zero. However, as

pointed out for the 2D case and in literature by several authors [Ghia 1977] [Tannehill 1997],

setting D in equation (4.5) identically to zero may lead to an unstable numerical scheme. Hence,

while solving the Poisson equation for pressure, retention of the temporal derivative of the local

dilatation is considered essential for the convergence of a numerical scheme.

In the nodal method for the three-dimensional N-S equations, the space-time domain

(X, Y, Z, T) is first discretized into rectangular space-time cells (i, j, k, n) of size (2 2i ja b×

2 2 )k nc τ× × with cell-centered local coordinates ( , , ,i i j j k ka x a b y b c z c− ≤ ≤ − ≤ ≤ − ≤ ≤

n ntτ τ− ≤ ≤ ). The N-S equations are re-written in terms of local coordinates:

79

2 2 2

2 2 2v

1 ( , , , ) ( ) (v v ) ( )

p p p

x p p p

u u u u u u uu w vt x y z x y z

p u u ug x y z t u u w wx x y z

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ρ ∂ ∂ ∂ ∂

⎡ ⎤+ + + − + +⎢ ⎥

⎣ ⎦

= − + − − − − − −

(4.7)

2 2 2

2 2 2

v v v v v v vv

1 v v v( , , , ) ( ) (v v ) ( )

p p p

y p p p

u w vt x y z x y z

p g x y z t u u w wy x y z

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ρ ∂ ∂ ∂ ∂

⎡ ⎤+ + + − + +⎢ ⎥

⎣ ⎦

= − + − − − − − −

(4.8)

2 2 2

2 2 2v

1 ( , , , ) ( ) (v v ) ( )

p p p

z p p p

w w w w w w wu w vt x y z x y z

p w w wg x y z t u u w wz x y z

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ρ ∂ ∂ ∂ ∂

⎡ ⎤+ + + − + +⎢ ⎥

⎣ ⎦

= − + − − − − − −

(4.9)

22 22 2 2

2 2 2

2 2 2

2 2 2

v

v v2 2 2

v

X Y Z

p p p u wx y z x y z

g g gu w w uy x z y x y x y z

D D D D D D Du w v v vt X Y Z X Y Z

∂ ∂ ∂ ∂ ∂ ∂ρ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ρ ρ ρ ρ ρ ρ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ρ∂ ∂ ∂ ∂ ∂ ∂ ∂

⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞+ + = − + +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

− − − + + +

⎡ ⎤− + + + − − −⎢ ⎥

⎣ ⎦

(4.10)

where pu , v p and pw are respectively the cell-averaged u, v and w velocities at the previous time

step [Wang 2003b]. Equations (4.7-4.9) are different from the standard momentum equations

(equations (4.2-4.4)) in that convection terms based on cell-averaged velocities at the previous

time step have been added to both sides of the equations and the original convection terms are

moved to the right hand side. The reason behind writing the momentum equations in this form is

to reduce the computational burden when solving final set of discrete algebraic equations [Wang

2003b].

80

4.2. Transverse Integration Procedure

Time, in this modified nodal scheme, is treated in the same fashion as spatial coordinates.

Transverse integration for the 3D case involves integrating the N-S equations locally over three

of the four independent variables. By applying the local transverse integration procedure, such as

, , , , ,1( ) ( , , , ) , , v, ,

8

jn i

n j i

b axyt

i j k n i j kb ai j n

z x y z t dxdydt u w pa b

τ

τφ φ φ

τ − − −≡ =∫ ∫ ∫ (4.11)

to equations (4.7-4.10), fifteen transverse-integrated ordinary differential equations are obtained,

2

12

( ) ( )yzt

yztd p x S xdx

= (4.12)

2

12

( ) (y)zxt

zxtd p y Sdy

= (4.13)

2

12

( ) ( )xyt

xytd p z S zdz

= (4.14)

2

22

( ) ( ) ( )yzt yzt

yztp

du x d u xu v S xdx dx

− = (4.15)

2

32

v ( ) v ( ) ( )yzt yzt

yztp

d x d xu v S xdx dx

− = (4.16)

2

42

( ) ( ) ( )yzt yzt

yztp

dw x d w xu v S xdx dx

− = (4.17)

2

22

( ) ( )v (y)zxt zxt

zxtp

du y d u yv Sdy dy

− = (4.18)

2

32

v ( ) v ( )v (y)zxt zxt

zxtp

d y d yv Sdy dy

− = (4.19)

2

42

( ) ( )v (y)zxt zxt

zxtp

dw y d w yv Sdy dy

− = (4.20)

81

2

22

( ) ( ) ( )xyt xyt

xytp

du z d u zw v S zdz dz

− = (4.21)

2

32

v ( ) v ( ) ( )xyt xyt

xytp

d z d zw v S zdz dz

− = (4.22)

2

42

( ) ( ) ( )xyt xyt

xytp

dw z d w zw v S zdz dz

− = (4.23)

2( ) ( )

xyzxyzdu t S t

dt= (4.24)

3v ( ) ( )

xyzxyzd t S t

dt= (4.25)

4( ) ( )

xyzxyzdw t S t

dt= (4.26)

where the subscripts (i, j, k, n) on independent variables have been omitted, and terms not

explicit are lumped into the right hand as pseudo-source terms. For example,

22 22 2

2 2

1

v

1 v v( ) 2 2 28

n i k

n i k

a czxt

a ck i n

X Y Z

p p u wx z x y z

u w w uS y dzdxdtc a y x z y x y

g g g Dx y z t

τ

τ

∂ ∂ ∂ ∂ ∂ρ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ρ ρ ρτ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ρ ρ ρ ρ∂ ∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− − − + +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟⎜ ⎟≡ − − −∫ ∫ ∫ ⎜ ⎟⎜ ⎟⎜ ⎟+ + + −⎜ ⎟⎜ ⎟⎝ ⎠

(4.27)

A list of definitions for all the pseudo-source terms can be found in appendix C. Notice that the

transverse-integrated equations (4.12-4.26) are similar in form to equations (2.37-2.44) obtained

for the two-dimensional MNIM developed in chapter two.

82

4.3. Local Solutions for the Transverse-Integrated ODEs

This step is identical to that for the 2D case, except now fifteen ODEs are solved

analytically within each cell instead of eight solved for the 2D case. Particular solutions are

obtained after expanding and truncating the modified pseudo-source terms at the zeroth order.

The local solutions of the ODEs for transverse-integrated pressure are quadratic, and for

example, the solution for ( )zxtp y is given by

1 2( ) 1 22

zxtzxt Sp y y C y C= + + . (4.28)

The local solution for ( )zxtu y is of the following form,

v

2( ) 3 4

p yzxt zxtu y S e C y Cν= + + . (4.29)

Solutions for the other transverse-integrated velocities ( ( )yztu x , ( )xytu z , v ( )yzt x , v ( ),zxt y v ( ),xyt z

( )yztw x , ( ),zxtw y ( )xytw z ) are of similar forms.

The solutions for ( )xyzu t , v ( )xyz t and ( )xyzw t are linear in time. For example,

2 5( )xyz xyzu t S t C= + (4.30)

The constants Ci (i = 1,2, …) are eliminated in favor of the discrete unknowns by imposing

boundary conditions on cell surfaces normal to the independent variable. A set of discrete

equations is obtained by imposing continuity of each variable (and its derivative for the second

order ODEs) at the four-dimensional (x, y, z, t) cell interfaces. This process leads to a set of

fifteen coupled, algebraic equations per cell for , v , , ,yzt yzt yzt yztijk ijk ijk ijku w p , v , , ,zxt zxt zxt zxt

ijk ijk ijk ijku w p

, v , , ,xyt xyt xyt xytijk ijk ijk ijku w p , vxyz xyz

ijk ijku and xyzijkw , in terms of the fifteen pseudo-source terms, S’s.

These algebraic equations are,

83

1, , , , 1 , , 1 1 , , 1 1, , , 1

1 1

( ) 1 1 02 2 2

xyt xyt xyt xyt xytk ki j k i j k i j k k i j k k i j k

k k k k

c c p p p c S c Sc c c c

+− + + +

+ +

+− − + + = (4.31)

1, , 1, , 1, , 1 , , 1 1 1, ,

1 1

( ) 1 1 02 2 2

yzt yzt yzt yzt yzti ii j k i j k i j k i i j k i i j k

i i i i

a a p p p a S a Sa a a a

+− + + +

+ +

+− − + + = (4.32)

1, , , 1, , 1, 1 , , 1 1 , 1,

1 1

( ) 1 1 02 2 2

j j zxt zxt zxt zxt zxti j k i j k i j k j i j k j i j k

j j j j

b bp p p b S b S

b b b b+

− + + ++ +

+− − + + = (4.33)

11 2 , , 12 2 , , 1 13 , , 1 13 14 , , 14 , , 1( ) 0xyt xyt xyt xyt xyti j k i j k i j k i j k i j kA S A S A u A A u A u+ − ++ + − + + = (4.34)

11 3 , , 12 3 , , 1 13 , , 1 13 14 , , 14 , , 1v ( ) v v 0xyt xyt xyt xyt xyti j k i j k i j k i j k i j kA S A S A A A A+ − ++ + − + + = (4.35)

11 4 , , 12 4 , , 1 13 , , 1 13 14 , , 14 , , 1( ) 0xyt xyt xyt xyt xyti j k i j k i j k i j k i j kA S A S A w A A w A w+ − ++ + − + + = (4.36)

21 2 , , 22 2 1, , 23 1, , 23 24 , , 24 1, ,( ) 0yzt yzt yzt yzt yzti j k i j k i j k i j k i j kA S A S A u A A u A u+ − ++ + − + + = (4.37)

21 3 , , 22 3 1, , 23 1, , 23 24 , , 24 1, ,v ( ) v v 0yzt yzt yzt yzt yzti j k i j k i j k i j k i j kA S A S A A A A+ − ++ + − + + = (4.38)

21 4 , , 22 4 1, , 23 1, , 23 24 , , 24 1, ,( ) 0yzt yzt yzt yzt yzti j k i j k i j k i j k i j kA S A S A w A A w A w+ − ++ + − + + = (4.39)

31 2 , , 32 2 , 1, 33 , 1, 33 34 , , 34 , 1,( ) 0zxt zxt zxt zxt zxti j k i j k i j k i j k i j kA S A S A u A A u A u+ − ++ + − + + = (4.40)

31 3 , , 32 3 , 1, 33 , 1, 33 34 , , 34 , 1,v ( ) v v 0zxt zxt zxt zxt zxti j k i j k i j k i j k i j kA S A S A A A A+ − ++ + − + + = (4.41)

31 4 , , 32 4 , 1, 33 , 1, 33 34 , , 34 , 1,( ) 0zxt zxt zxt zxt zxti j k i j k i j k i j k i j kA S A S A w A A w A w+ − ++ + − + + = (4.42)

84

, , , , , 2 , ,2 0xyz xyz xyzi j k i j k p i j ku u Sτ− − = (4.43)

, , , , 3 , ,v v 2 0xyz xyz xyzi j i j k p i j kSτ− − = (4.44)

, , , , 4 , ,2 0xyz xyz xyzi j i j k p i j kw w Sτ− − = , (4.45)

where the subscript p denotes previous time step values and all other variables are at the current

time step, and the coefficients A are defined as:

, ,

, ,

Re

11 Re, ,

2 1(1 )

i j k

i j k

wk

wp i j k

c eAwv e

≡ +−

(4.46)

, , 1

112 Re

, , 1

2 1( 1 )i j k

kw

p i j k

cAwv e +

+

+

≡ −− +

(4.47)

, ,

, ,

Re, ,

13 Re(1 )

i j k

i j k

wp i j kw

e wA

v e≡

− (4.48)

, , 1

, , 114 Re(1 )i j k

p i j kw

wA

v e +

+≡−

(4.49)

, ,

, ,

Re

21 Re, ,

2 1(1 )

i j k

i j k

ui

up i j k

a eAuv e

≡ +−

(4.50)

1, ,

122 Re

1, ,

2 1( 1 )i j k

iu

p i j k

aAuv e +

+

+

≡ −− +

(4.51)

, ,

, ,

Re, ,

23 Re(1 )

i j k

i j k

up i j ku

e uA

v e≡

− (4.52)

1, ,

1, ,24 Re(1 )i j k

p i j ku

uA

v e +

+≡−

(4.53)

85

, ,

, ,

Re v

31 Rev, ,

2 1v(1 )

i j k

i j k

j

p i j k

b eA

v e≡ +

− (4.54)

, 1,

132 Re v

, 1,

2 1v( 1 )i j k

j

p i j k

bA

v e +

+

+

≡ −− +

(4.55)

, ,

, ,

Rev, ,

33 Re v

v(1 )

i j k

i j k

p i j keA

v e≡

− (4.56)

, 1,

, 1,34 Re v

v(1 )i j k

p i j kAv e +

+≡−

. (4.57)

The cell Reynolds numbers based on the previous time step velocities are defined as

, ,, ,

2Re i p i j k

i j k

a uu

v≡ (4.58)

, ,, ,

2 vRe v j p i j k

i j k

bv

≡ (4.59)

, ,, ,

2Re k p i j k

i j k

c ww

v≡ . (4.60)

Due to the symmetry in the original transverse-integrated equations (4.12-4.26), the discretized

equations (4.31-4.45) and corresponding coefficients A in equations (4.46-4.57) are also

symmetric in x, y and z directions.

4.4. Constraint Equations

Following the procedures for NIM, the pseudo-source terms are eliminated next using the

constraint equations. For the 3D case, fifteen constraint equations are needed. Four of the

86

constraint equations are obtained by applying the operator, 1

16

jn k i

n k j i

bc a

c b ai j k n

dxdydzdta b c

τ

ττ − − − −∫ ∫ ∫ ∫ ,

on Equations. (4.7-4.10).

1 1 1 1 0yzt zxt xytS S S f+ + + = (4.61)

2 2 2 2 2 0xyz yzt zxt xytS S S S f+ + + + = (4.62)

3 3 3 3 3 0xyz yzt zxt xytS S S S f+ + + + = (4.63)

4 4 4 4 4 0xyz yzt zxt xytS S S S f+ + + + = (4.64)

where

22 2

, , 1, , , , , 1, , , , , 11

, , , 1, , , 1, ,

, , , , 1

v v2 2 2

v v2

2 2

v v2

2

yzt yzt zxt zxt xyt xyti j k i j k i j k i j k i j k i j k

i j k

zxt zxt yzt yzti j k i j k i j k i j k

j i

xyt xyti j k i j k

u u w wf

a b c

u ub a

c

ρ ρ ρ

ρ

ρ

− − −

− −

−

⎛ ⎞⎛ ⎞ ⎛ ⎞− − −≡ + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞− −+ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

−+ , , , 1,

, , 1, , , , , , 1

, , 1, , , , , 1, , , , 1

2

22 2

2 2

zxt zxti j k i j k

k j

yzt yzt xyt xyti j k i j k i j k i j k

i k

yzt yzt zxt zxt xytxi j k xi j k yi j k yi j k zi j k zi j

i j

w wb

w w u ua c

g g g g g ga b

ρ

ρ ρ ρ

−

− −

− − −

⎛ ⎞⎛ ⎞ −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞− −+ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞− − −

+ + +⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

,

, ,

2

2

xytk

k

xyzi j k

c

Dρ

τ

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

+

(4.65)

in which,

, ,

, , 1, , , , , 1, , , , , 1

1 v8

v v2 2 2

jk i

k j i

bc axyzi j k

c b ai j k

yzt yzt zxt zxt xyt xyti j k i j k i j k i j k i j k i j k

i j k

u wD dxdydza b c x y z

u u w wa b c

∂ ∂ ∂∂ ∂ ∂− − −

− − −

⎛ ⎞= + +∫ ∫ ∫ ⎜ ⎟

⎝ ⎠

− − −= + +

, (4.66)

87

2 0 , , 1, , 0 , , , 1,

0 , , , , 1 , , 1, ,

1 1( ) ( ) (v v ) ( )2 2

1 1 1( ) ( ) ( )2 2

yzt yzt zxt zxtp i j k i j k p i j k i j k

i j

xyt xyt yt yt xyztp i j k i j k i j k i j k x

k i

f u u u u u ua b

w w u u p p gc aρ

− −

− −

≡ − − + − −

+ − − + − + (4.67)

3 0 , , 1, , 0 , , , 1,

0 , , , , 1 , , , 1,

1 1( ) (v v ) (v v ) (v v )2 2

1 1 1( ) (v v ) ( )2 2

yzt yzt zxt zxtp i j k i j k p i j k i j k

i j

xyt xyt zxt zxt xyztp i j k i j k i j k i j k y

k j

f u ua b

w w p p gc bρ

− −

− −

≡ − − + − −

+ − − + − + (4.68)

4 0 , , 1, , 0 , , , 1,

0 , , , , 1 , , , , 1

1 1( ) ( ) (v v ) ( )2 2

1 1 1( ) ( ) ( )2 2

yzt yzt zxt zxtp i j k i j k p i j k i j k

i j

xyt xyt xyt xyt xyztp i j k i j k i j k i j k z

k k

f u u w w w wa b

w w w w p p gc cρ

− −

− −

≡ − − + − −

+ − − + − +. (4.69)

To simplify and simultaneously retain the stability of the numerical scheme, similar to the 2D

situation, terms in the square bracket on the RHS of equation (4.10) are replaced with (D/2τ)

evaluated at the current time step, where τ is half time step.

The other eleven constraint equations are obtained by imposing the condition that the

cell-averaged variables be unique, independent of the order of integration [Azmy 1983], i.e.

1 1 1 1( ) ( ) ( ) ( )2 2 2 2

ji k n

i j k n

ba cyzt zxt xyt xyz

i j k na b c

u x dx u y dy u z dz u t dta b c

τ

ττ

− − − −

= = =∫ ∫ ∫ ∫ (4.70)

1 1 1 1v ( ) v ( ) v ( ) v ( )2 2 2 2

ji k n

i j k n

ba cyzt zxt xyt xyz

i j k na b c

x dx y dy z dz t dta b c

τ

ττ

− − − −

= = =∫ ∫ ∫ ∫ (4.71)

1 1 1 1( ) ( ) ( ) ( )2 2 2 2

ji k n

i j k n

ba cyzt zxt xyt xyz

i j k na b c

w x dx w y dy w z dz w t dta b c

τ

ττ

− − − −

= = =∫ ∫ ∫ ∫ (4.72)

1 1 1( ) ( ) ( )2 2 2

ji k

i j k

ba cyzt zxt xyt

i j ka b c

p x dx p y dy p z dza b c

− − −

= =∫ ∫ ∫ (4.73)

88

After plugging in the local solutions such as equations (4.28-4.30), the above uniqueness

constraint equations are of the following form:

2 , , 41 2 , , , , , 42 , , 1 42 , ,(1 ) 0xyz xyt xyz xyt xyti j k i j k i j k p i j k i j kS A S u A u A uτ −− + − + + − = (4.74)

2 , , 51 2 , , , , , 52 1, , 52 , ,(1 ) 0xyz yzt xyz yzt yzti j k i j k i j k p i j k i j kS A S u A u A uτ −− + − + + − = (4.75)

2 , , 61 2 , , , , , 62 , 1, 62 , ,(1 ) 0xyz zxt xyz zxt zxti j k i j k i j k p i j k i j kS A S u A u A uτ −− + − + + − = (4.76)

3 , , 41 3 , , , , , 42 , , 1 42 , ,v v (1 )v 0xyz xyt xyz xyt xyti j k i j k i j k p i j k i j kS A S A Aτ −− + − + + − = (4.77)

3 , , 51 3 , , , , , 52 1, , 52 , ,v v (1 )v 0xyz yzt xyz yzt yzti j k i j k i j k p i j k i j kS A S A Aτ −− + − + + − = (4.78)

3 , , 61 3 , , , , , 62 , 1, 62 , ,v v (1 )v 0xyz zxt xyz zxt zxti j k i j k i j k p i j k i j kS A S A Aτ −− + − + + − = (4.79)

4 , , 41 4 , , , , , 42 , , 1 42 , ,(1 ) 0xyz xyt xyz xyt xyti j k i j k i j k p i j k i j kS A S w A w A wτ −− + − + + − = (4.80)

4 , , 51 4 , , , , , 52 1, , 52 , ,(1 ) 0xyz yzt xyz yzt yzti j k i j k i j k p i j k i j kS A S w A w A wτ −− + − + + − = (4.81)

4 , , 61 4 , , , , , 62 , 1, 62 , ,(1 ) 0xyz zxt xyz zxt zxti j k i j k i j k p i j k i j kS A S w A w A wτ −− + − + + − = (4.82)

2 2

, , , , 1 , , 1, , 1 , , 1 , ,1 1 1 1 02 2 2 2 3 3

xyt xyt yzt yzt xyt yztk ii j k i j k i j k i j k i j k i j k

c ap p p p S S− −+ − − − + = (4.83)

22

, , 1, , , , , 1, 1 , , 1 , ,1 1 1 1 02 2 2 2 3 3

jyzt yzt zxt zxt yzt zxtii j k i j k i j k i j k i j k i j k

bap p p p S S− −+ − − − + = , (4.84)

where,

89

Re , ,, ,

Re , ,

(1 )

( 1 )41 2

, ,

wi j kk p i j k

wi j k

c w e

e

p i j k

vA

w

+

− +− +

≡ (4.85)

, ,

, ,

Re

42 Re, ,

1Re1

i j k

i j k

w

wi j k

eAwe

≡ −− +

(4.86)

Re , ,, ,

Re , ,

(1 )

( 1 )51 2

, ,

ui j ki p i j k

ui j k

a u e

e

p i j k

vA

u

+

− +− +

≡ (4.87)

, ,

, ,

Re

52 Re, ,

1Re1

i j k

i j k

u

ui j k

eAue

≡ −− +

(4.88)

Re v , ,, ,

Re v , ,

v (1 )

( 1 )61 2

, ,v

i j kj p i j k

i j k

b e

e

p i j k

vA

+

− +− +

≡ (4.89)

, ,

, ,

Re v

62 Re v, ,

1Re v1

i j k

i j ki j k

eAe

≡ −− +

. (4.90)

Again, symmetry in x, y and z directions is seen in the constraint equations (4.74-4.84) and the

corresponding coefficients in equations (4.85-4.90).

Up to this stage, there are thirty equations (fifteen transverse-averaged equations and

fifteen constraint equations) for thirty unknowns (fifteen transverse-averaged variables and

fifteen pseudo-source terms). The fifteen pseudo-source terms in the set of discrete algebraic

equations (4.31-4.45) are eliminated next using these constraint equations (4.61-4.64) and (4.74-

4.84), leading to a final set of fifteen equations and fifteen unknowns per cell. This is done in

two steps.

90

First, the expressions for fifteen pseudo-source terms are obtained. Specifically,

equations (4.61)(conservation of pressure equation), (4.83) and (4.84)(uniqueness of pressure)

are used to solve for 1 , ,yzti j kS , 1 , ,

zxti j kS and 1 , ,

xyti j kS , equations (4.43-4.45) (local solutions of

, , ( )xyti j ku t , , ,v ( )xyt

i j k t , , , ( )xyti j kw t ) are used to solve for 2 , ,

xyzi j kS , 3 , ,

xyzi j kS and 4 , ,

xyzi j kS , equations (4.74-

4.76) (uniqueness of u velocity) are used to solve for 2 , ,xyti j kS , 2 , ,

yzti j kS , 2 , ,

zxti j kS , equations (4.77-

4.79) (uniqueness of v velocity) are used to solve for 3 , ,xyti j kS , 3 , ,

yzti j kS , 3 , ,

zxti j kS , equations (4.80-

4.82) (uniqueness of w velocity) are used to solve for 4 , ,xyti j kS , 4 , ,

yzti j kS , 4 , ,

zxti j kS .

Second, the solutions for these pseudo-source terms are substituted into the other fifteen

equations to solve for the transverse-integrated velocities and pressure.

The final set of fifteen equations is

17 , , 11 , , 1 12 , , 1 13 , , 1, , 14 , , 1 1, , 1

15 , , , 1, 16 , , 1 , 1, 1 18 1 , , 19 1 , , 1

( ) ( )

( ) ( )

xyt xyt xyt yzt yzt yzt yzti j k i j k i j k i j k i j k i j k i j k

zxt zxt zxt zxti j k i j k i j k i j k i j k i j k

F p F p F p F p p F p p

F p p F p p F f F f− + − + − +

− + − + +

= + + + + +

+ + + + + + (4.91)

27 , , 21 1, , 22 1, , 23 , , , 1, 24 1, , 1, 1,

25 , , , , 1 26 1, , 1, , 1 28 1 , , 29 1 1, ,

( ) ( )

( ) ( )

yzt yzt yzt zxt zxt zxt zxti j k i j k i j k i j k i j k i j k i j k

xyt xyt xyt xyti j k i j k i j k i j k i j k i j k

F p F p F p F p p F p p

F p p F p p F f F f− + − + + −

− + + − +

= + + + + +

+ + + + + + (4.92)

37 , , 31 , 1, 32 , 1, 33 , , , , 1 34 , 1, , 1, 1

35 , , 1, , 36 , 1, 1, 1, 38 1 , , 39 1 , 1,

( ) ( )

( ) ( )

zxt zxt zxt xyt xyt xyt xyti j k i j k i j k i j k i j k i j k i j k

yzt yzt yzt yzti j k i j k i j k i j k i j k i j k

F p F p F p F p p F p p

F p p F p p F f F f− + − + + −

− + − + +

= + + + + +

+ + + + + + (4.93)

47 , , 41 , , 1 42 , , 1 43 , , , , , 44 , , 1 , , 1,( ) ( )xyt xyt xyt xyz xyz xyz xyzi j k i j k i j k i j k i j k p i j k i j k pF u F u F u F u u F u u− + + += + + + + + (4.94)

47 , , 41 , , 1 42 , , 1 43 , , , , , 44 , , 1 , , 1,v v v (v v ) (v v )xyt xyt xyt xyz xyz xyz xyzi j k i j k i j k i j k i j k p i j k i j k pF F F F F− + + += + + + + + (4.95)

47 , , 41 , , 1 42 , , 1 43 , , , , , 44 , , 1 , , 1,( ) ( )xyt xyt xyt xyz xyz xyz xyzi j k i j k i j k i j k i j k p i j k i j k pF w F w F w F w w F w w− + + += + + + + + (4.96)

91

57 , , 51 1, , 52 1, , 53 , , , , , 54 1, , 1, , ,( ) ( )yzt yzt yzt xyz xyz xyz xyzi j k i j k i j k i j k i j k p i j k i j k pF u F u F u F u u F u u− + + += + + + + + (4.97)

57 , , 51 1, , 52 1, , 53 , , , , , 54 1, , 1, , ,v v v (v v ) (v v )yzt yzt yzt xyz xyz xyz xyzi j k i j k i j k i j k i j k p i j k i j k pF F F F F− + + += + + + + + (4.98)

57 , , 51 1, , 52 1, , 53 , , , , , 54 1, , 1, , ,( ) ( )yzt yzt yzt xyz xyz xyz xyzi j k i j k i j k i j k i j k p i j k i j k pF w F w F w F w w F w w− + + += + + + + + (4.99)

67 , , 61 , 1, 62 , 1, 63 , , , , , 64 , 1, , 1, ,( ) ( )zxt zxt zxt xyz xyz xyz xyzi j k i j k i j k i j k i j k p i j k i j k pF u F u F u F u u F u u− + + += + + + + + (4.100)

67 , , 61 , 1, 62 , 1, 63 , , , , , 64 , 1, , 1, ,v v v (v v ) (v v )zxt zxt zxt xyz xyz xyz xyzi j k i j k i j k i j k i j k p i j k i j k pF F F F F− + + += + + + + + (4.101)

67 , , 61 , 1, 62 , 1, 63 , , , , , 64 , 1, , 1, ,( ) ( )zxt zxt zxt xyz xyz xyz xyzi j k i j k i j k i j k i j k p i j k i j k pF w F w F w F w w F w w− + + += + + + + + (4.102)

77 , , 71 , , 72 , , 1 73 , , 74 1, ,

75 , , 76 , 1, 78 , , , 2 , ,

xyz xyt xyt yzt yzti j k i j k i j k i j k i j k

zxt zxt xyzi j k i j k i j k p i j k

F u F u F u F u F u

F u F u F u f− −

−

= + + +

+ + + + (4.103)

77 , , 71 , , 72 , , 1 73 , , 74 1, ,

75 , , 76 , 1, 78 , , , 3 , ,

v v v v v

v v v

xyz xyt xyt yzt yzti j k i j k i j k i j k i j k

zxt zxt xyzi j k i j k i j k p i j k

F F F F F

F F F f− −

−

= + + +

+ + + + (4.104)

77 , , 71 , , 72 , , 1 73 , , 74 1, ,

75 , , 76 , 1, 78 , , , 4 , ,

xyz xyt xyt yzt yzti j k i j k i j k i j k i j k

zxt zxt xyzi j k i j k i j k p i j k

F w F w F w F w F w

F w F w F w f− −

−

= + + +

+ + + + (4.105)

where the subscript p denotes variables evaluated at previous time step. The coefficients F are

functions of , , , , , ,, , , , ,Re ,Re v ,Rei j k i j k i j k i j ka b c v u wτ . [Coefficients A are also functions of

, , , , , ,, , , , ,Re ,Re v ,Rei j k i j k i j k i j ka b c v u wτ .] For example,

2 2 2 2 2

11 2 2 2 2 2

2 ( 2 )2 ( ( ))

j k i j k

k j k i j k

b c a b cF

c b c a b c− −

≡+ +

(4.106)

92

11 4241 13

41

A AF AA

≡ − (4.107)

7741 51 61

1 1 1 1 12

FA A A τ

⎛ ⎞≡ − + + +⎜ ⎟

⎝ ⎠. (4.108)

A list of definitions for all the coefficients F can be found in appendix D.

4.5 Boundary Conditions

Boundary conditions for the 3D case are similar to those developed for the 2D case. No

slip boundary conditions are imposed on solid surfaces. In addition, Dirichlet condition can also

be specified, for example, on inlet surfaces. Boundary conditions for pressure on no-slip surfaces

for the 3D case are derived using the x, y and z momentum equations [Ghia 1977] [Gresho 1987]

[Ferziger 1996] [Tannehill 1997]. For example, on the no-slip surface at x = xmax, u = v = w = 0,

ut

∂∂

= 2 2

2 2 0u uy z

∂ ∂∂ ∂

= = , and thus, the u-momentum equation, averaged locally over y, z and t,

becomes (see Figure 4.1)

2

2

1 ( ) ( ) 0yzt yzt

yztx

dp x d u xv bdx dxρ

− + = . (4.109)

Following the process in the 2D case, the following expression with second order accuracy for

2

2

yztd udx

can be derived,

93

h4 h3 h2 h1

(i-3, j, k) (i-2, j, k) (i-1, j, k) (i, j, k)

yztu = 3, ,yzt

i j ku − 2, ,yzt

i j ku − 1, ,yzt

i j ku − , ,yzt

i j ku

x = (x0 – h1 – h2 – h3) ( x0 – h1 – h2) ( x0 – h1) x0

Figure 4.1: Boundary condition for pressure at the surface x = xmax = x0.

94

2 2 220 1 1 2 2

,2 2 2 21 1 2

21 2 11, 2,2 2

1 2 2 1 2

( ) 2( 3 3 ) ( )

2( ) 2 ( ) ( )

yztyztyzt

i jwall

yzt yzti j i j

d u x x h h h hd u udx dx h h h

h h hu u O hh h h h h− −

= + += = − +

+

+− +

+

(4.110)

A second order accurate scheme for the first derivative of pressure at the wall has the

following form

0

, , 1, , , , 2 1, , 1 2 2, , 1 2

1 2 1 2( )

- ( )( )

( )

yzt yzt yzt yzt yztyzti j k i j k i j k i j k i j k

wall x x

p p p h p h h p hdp O hdx h h h h

− − −

=

− + += + +

+ (4.111)

These expressions for 2

2

yztd udx

and yztdp

dx are substituted in equation (4.109) to obtain the discrete

form of the pressure boundary condition for , ,yzti j kp on the wall at x = xmax. Pressure boundary

conditions for the other walls are similarly derived.

The numerical scheme for the 3D case has all the same characteristics as the 2D case (See

section 1.6 and section 2.4.). Code implementation and iterative solvers for the 3D case are also a

straightforward extension of the 2D case (See section 2.9.).

95

Chapter 5

Application of Modified Nodal Integral Method for Time-Dependent Navier-Stokes equations – Three Dimensional Case

The three dimensional modified nodal integral method has been applied to the following

problems: three-dimensional developed flow and developing flow in a rectangular channel,

three-dimensional lid-driven cavity problems in a cube and in a prism with aspect ratio of two.

5.1. Three-Dimensional Fully Developed Flow in a Rectangular Channel

Unlike the two-dimensional fully developed flow between parallel plates, three-

dimensional fully developed flow in a rectangular channel does not have an exact solution. The

boundary conditions for this problem are shown in Figure 5.1.1. As in the two-dimensional case,

the fully developed feature is captured by setting the derivative with respect to x equal to zero

( 0=∂∂x

) at the inlet and exit planes. Wall boundary conditions described in chapter two are used

for top, bottom, front and back surfaces. Constant pressure is enforced at the inlet and exit

planes.

Numerical results for the following parameter values are shown in Figure 5.1.2 and 5.1.3.

Computational domain size is [0,1] [0,1] [0,1]× × and ( 0) 4.0, ( 1) 0, 0.005p x p x μ= = = = = ,

1ρ = . The mesh size used to solve this problem is 8 8 8× × . Three-dimensional effect is shown

clearly in the velocity profile: the u velocity distribution is similar to parabolic in both y and z

directions (see Figure 5.1.2 and 5.1.3).

96

2

2

0v 0

1 0

xyt

xyt

xyt xytxyt

z

u

dp d uv bdz dzρ

=

=

− + + =

2

2

0v 0

1 0

zxt

zxt

zxt zxtzxt

y

u

dp d uv bdy dyρ

=

=

− + + =

, ,

0

v 0

0

yzt

yzt

yti j k

dudx

ddx

p

=

=

=, , 0

0

v 0

yzt

yzt

yzti j k

dudx

ddx

p p

=

=

=

x

yz

2

2

0v 0

1 0

zxt

zxt

zxt zxtzxt

y

u

dp d uv bdy dyρ

=

=

− + + =

2

2

0v 0

1 0

xyt

xyt

xyt xytxyt

z

u

dp d uv bdz dzρ

=

=

− + + =

front surface

back surface

top surface

bottom surface

Figure 5.1.1: Boundary conditions for 3D fully developed flow in a rectangular channel.

97

0

0.5

1

z0 0.25 0.5 0.75

x0

0.25

0.5

0.75

y

2.7192.5592.3992.2402.0801.9201.7601.6011.4411.2811.1210.9620.8020.6420.482

Figure 5.1.2: Velocity profile for 3D fully developed flow in a rectangular channel at planes y = 0.1, 0.5 and 0.9.

98

0

0.25

0.5

0.75

z0 0.25 0.5 0.75

x0

0.51

y

2.7192.5592.3992.2402.0801.9201.7601.6011.4411.2811.1210.9620.8020.6420.482

Figure 5.1.3: Velocity profile for 3D fully developed flow in a rectangular channel at planes z = 0.1, 0.5 and 0.9.

99

5.2. Three-Dimensional Developing Flow in a Rectangular Channel

Similar to the two-dimensional developing flow between parallel plates, in this problem,

velocity at the inlet plane is uniform, while pressure is specified at the exit. The dimension of the

channel in the main flow direction is chosen to be twice the entrance length so that the flow is

fully developed at the exit [Schlicting 1968]. The flow field develops from uniform along y and z

direction to a parabolic distribution at the exit, where the fully developed boundary conditions

are enforced. Boundary conditions for this problem are shown in Figure 5.2.1.

Numerical results for the three-dimensional developing flow in a rectangular channel

with the following parameter values are shown in Figure 5.2.2-5.2.4. Computational domain size

chosen is [0,1] [0,1] [0,1]× × , 0.01, ( 1) 0, 0.1, 1inu p x μ ρ= = = = = . The mesh size is 8 8 8× × . The

u velocity develops from uniform in the inlet to a distribution similar to parabolic at the exit in

both y and z directions. The center plane velocity pattern is very similar to the two-dimensional

results (see Figure 5.2.2 and Figure 3.2.2). The velocities at planes close to the boundaries (at

planes y = 0.1, y = 0.9, z = 0.1 and z = 0.9) are distorted in the Figure 5.2.2 and 5.2.3, because of

the view angle. The velocity at y = 0.9 plane is plotted at a front view in Figure 5.2.4, from

which the symmetry to the centerline is seen.

100

2

2

0v 0

1 0

xyt

xyt

xyt xytxyt

z

u

dp d uv bdz dzρ

=

=

− + + =

2

2

0v 0

1 0

zxt

zxt

zxt zxtzxt

y

u

dp d uv bdy dyρ

=

=

− + + =

, ,

0

v 0

0

yzt

yzt

yti j k

dudx

ddx

p

=

=

=

x

yz

2

2

0v 0

1 0

zxt

zxt

zxt zxtzxt

y

u

dp d uv bdy dyρ

=

=

− + + =

2

2

0v 0

1 0

xyt

xyt

xyt xytxyt

z

u

dp d uv bdz dzρ

=

=

− + + =

front surface

back surface

top surface

bottom surface

2

2

v 01 0

yztin

yzt

yzt yztyzt

x

u u

dp d uv bdz dzρ

=

=

− + + =

Figure 5.2.1: Boundary conditions for 3D developing flow in a rectangular channel.

101

0

0.25

0.5

0.75

z0 0.2 0.4 0.6 0.8

x0

0.5

1y

0.01800.01700.01590.01490.01380.01270.01170.01060.00950.00850.00740.00640.00530.00420.0032

Figure 5.2.2: Velocity profile for 3D developing flow in a rectangular channel at planes z = 0.1, 0.5 and 0.9.

102

00.51

z 0 0.2 0.4 0.6 0.8

x0

0.25

0.5

0.75

y

0.01800.01700.01590.01490.01380.01270.01170.01060.00950.00850.00740.00640.00530.00420.0032

Figure 5.2.3: Velocity profile for 3D developing flow in a rectangular channel at planes y = 0.1, 0.5 and 0.9.

103

0

0.25

0.5

0.75

z

0 0.2 0.4 0.6 0.8x

00.5

1

y

0.01800.01700.01590.01490.01380.01270.01170.01060.00950.00850.00740.00640.00530.00420.0032

Figure 5.2.4: Velocity profile for 3D developing flow in a rectangular channel at plane y = 0.9 (different view angle to show the vector direction).

104

5.3. Lid Driven Cavity Flow in a Cube

Three-dimensional, lid driven cavity problem has been used extensively to test numerical

algorithms [Ku 1987] [Babu 1994] [Cortes 1994] [Baloch 2002], in which the lid at the top (z =

1 for the cubic) moves with a constant velocity in the x direction (see Figure5.3.1). Centerline

velocity profiles for this problem are compared with reference solutions in Figure 5.3.2-5.3.5.

Figures 5.3.2 and 5.3.3 illustrate the u and w velocities in the cube along the center line parallel

to the z-axis and x-axis respectively for Reynolds number of 100. Figures 5.3.4 and 5.3.5 are

corresponding velocity profiles for Reynolds number of 1000. These results are obtained using a

20 x 20 x 20 non-uniform mesh and are compared with those from [Ku 1987] and [Babu 1994].

It is clear from these results that even for coarse meshes, the MNIM for the time-dependent N-S

equations leads to fairly accurate results. For example, for the Re = 1000 case, Babu et al. [Babu

1994] used an 81 x 81 x 81 mesh, while a 20 x 20 x 20 mesh is used in MNIM. The MNIM code

takes about 60 minutes for Re = 100 and 126 minutes for Re = 1000 on a 1.5 GHz PC running

LINUX operating system. The CPU time for this three-dimensional problem is low despite the

fact that very simple Gauss-Seidel sweeps are used repeatedly at each time step till convergence.

For larger problems, significant savings in CPU time can be achieved by incorporating more

efficient solvers.

105

11

1

Lid motion

Cube

x y

z

Figure 5.3.1: Configuration of the lid driven cavity problem in a cube.

106

0.0 0.2 0.4 0.6 0.8 1.0-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0 Re=100

u-ve

loci

ty

z

MNIM Ku et al Babu et al

Figure 5.3.2: U-velocity along the vertical centerline for the 3D lid driven cavity cube problem for Re = 100.

107

0.0 0.2 0.4 0.6 0.8 1.0-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6 Re=100

w-v

eloc

ity

x

MNIM Ku et al Babu et al

Figure 5.3.3. W-velocity along the horizontal centerline for the 3D lid driven cavity cube problem for Re = 100.

108

0.0 0.2 0.4 0.6 0.8 1.0-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0 Re=1000

u-ve

loci

ty

z

MNIM Ku et al Babu et al

Figure 5.3.4: U-velocity along the vertical centerline for the 3D lid driven cavity cube problem for Re = 1000.

109

0.0 0.2 0.4 0.6 0.8 1.0-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6 Re=1000

w-v

eloc

ity

x

MNIM Ku et al Babu et al

Figure 5.3.5: W-velocity along the horizontal centerline for the 3D lid driven cavity cube problem for Re = 1000.

110

5.4. Lid Driven Cavity Flow in a Prism

In the three-dimensioal lid driven cavity flow in a prism, The dimension in the z direction

is increased to two (see Figure 5.4.1). Numerical results for this problem are shown in Figures

5.4.1-5.4.6. For the Reynolds number = 100 case, a 20 x 20 x 40 nonuniform mesh with a

geometric factor of 1.1 was used for MNIM. The velocity field in the center plane is very similar

to the flow field in two-dimensional lid driven cavity flow in a prism (see Figure 5.4.2 and

Figure 3.4.1 (a)).

A comparison of the centerline velocity with reference solution from [Cortes 1994] and

results obtained using commercial software Fluent on a very fine mesh shows that the MNIM

results agree very well with the reference solution and Fluent results(see Figure 5.4.3). The

reference results was obtained using a 35 x 35 x 70 mesh [Cortes 1994]. For MINIM, a time step

of 0.03 is used and steady state is reached after 300 time steps, while the reference solution

[Cortes 1994] was obtained using time step of 0.00025 and 60000 time steps.

The center plane velocity vector profile for flow in the prism with Reynolds number of

1000 is shown in Figure 5.4.4 (vector length proportional to velocity magnitude) and Figure

5.4.5 (unifor vecotr length). The centerline velocity profile obtained using MNIM on a 30 x 30 x

60 mesh agrees very well with those obtained using Fluent with a denser 60 x 60 x 120 mesh

(see Figure 5.4.6). However, both results differ somewhat in the middle portion of the cavity

from those reported in reference [Cortes 1994] obtained using a 35 x 35 x 70 mesh (see Figure

5.4.7). Though it is difficult to conclude with certainty which of the two solutions in the middle

part of the cavity is the correct one, agreement between results obtained using MNIM and Fluent

suggests that the mesh used in [Cortes 1994] may still be too coarse for the numerical scheme.

111

11

2

Lid motion

Prism

x y

z

Figure 5.4.1: Configuration of the lid driven cavity problem in a prism

112

01

z 0 0.5 1

x0

0.5

1

1.5

y

0.8840.8250.7660.7070.6480.5890.5300.4710.4120.3540.2950.2360.1770.1180.059

Figure 5.4.2: Center plane velocity vectors for three-dimensional lid-driven cavity problem in a prism with aspect ratio of 2 for Re = 100.

Vector length is proportional to the velocity magnitude.

113

0.0

0.5

1.0

1.5

2.0

-0.5 0.0 0.5 1.0

MNIM Cortes et al Fluent

Re=100

u velocity

Cav

ity H

eigh

t

Figure 5.4.3: Comparison of centerline velocity profiles for a prismatic cavity with an aspect ratio of 2 for Reynolds number of 100

114

Frame 00

01

z 0 0.5 1

x0

0.5

1

1.5

y

0.8050.7320.6590.5860.5120.4390.3660.2930.2200.1460.073

Figure 5.4.4: Center plane velocity vectors for three-dimensional lid-driven cavity problem in a prism with aspect ratio of 2 for Re = 1000.

Vector length is proportional to the velocity magnitude.

115

Frame 00

01

z 0 0.5 1

x0

0.5

1

1.5

y

0.80520.73200.65880.58560.51240.43920.36600.29280.21960.14640.0732

Figure 5.4.5: Center plane velocity vectors for three-dimensional lid-driven cavity problem in a prism with aspect ratio of 2 for Re = 1000. Vector length is uniform.

116

0.0

0.5

1.0

1.5

2.0

-0.5 0.0 0.5 1.0

Re=1000

u velocity

Cav

ity H

eigh

t MNIM Fluent Cortes et al

Figure 5.4.6: Comparison of centerline velocity profiles for a prismatic cavity with an aspect ratio of 2 for Reynolds number of 1000

117

Chapter 6

Parallel Implementation of the MNIM for the Navier-Stokes Equations

Numerical simulation of nonlinear, complex systems, such as weather, electronic circuits

and nuclear reactors has motivated development of faster computers. Today, the data intensive

commercial applications, such as video conferencing and virtual reality, have become the driving

force behind the development of advanced computers. The speed of a single processor has been

increasing, but history suggests that the speed of computers can never meet the need of

applications. Parallel computers provide a solution to achieve much faster speed under current

processor speed level. Today, parallel computers with hundreds and even thousands of

processors are not uncommon.

Navier-Stokes equations are nonlinear equations. Numerical schemes for solving N-S

equations are computation-intensive. Thus, it is desirable to develop parallelized version of

existing numerical schemes. The goal in this chapter is to develop a parallel version of the

MNIM code for Navier-Stokes equations developed in the previous chapters.

6.1. Shared Memory v.s. Distributed Memory

There are two parallel programming models [Foster 1995]: shared memory and

distributed memory. In shared memory programming model, data are shared by different

processors through shared memories. It is easier to program, but locality and scalability are

major issues for shared-memory programming. Shared memory model is often implemented

using OpenMP. OpenMP is a collection of compiler directives, library routines, and environment

variables that can be used to specify shared memory parallelism.

118

In distributed memory model, each processor has its own memory and encapsulated local

data. Message Passing Interface (MPI) is the most popular parallel programming model in

distributed memory systems. In MPI, each processor is identified by a unique identification

number (ID). Processors interact with each other by sending and receiving messages through

library function calls. MPI is harder to program than shared-memory OpenMP, but easier to scale

up to a large number of processors. Because of its advantage in scalability, MPI is chosen to

develop a parallel version of the MNIM code for the N-S euqaitons.

6.2. Domain Decomposition

Domain decomposition is often employed to develop parallel version of serial codes

[Carey 1989] [Smith 1996] [Foster 1995]. In this approach, the computational domain is

decomposed into sub-domains with number equal to the number of processors. Each domain is

assigned to a processor. The set of discrete equations is solved in each domain for certain

number of iterations. Each domain, then, exchanges the boundary information with its

neighboring nodes. This process is repeated until the difference between the newly received

boundary information and the boundary information received at the end of previous set of

iterations is within certain tolerance. The flow chart of the parallelized computer code is shown

in Figure 6.1. A pseudo-code for domain decomposition is given below:

Set initial and boundary condition for the whole domain For each time step For each processor while(the boundary values of each sub-domain not converged ) { Relax the discrete equations in each sub-domain

for certain number of iterations Each sub-domain exchanges boundary variables with its neighbors by calling sendrecv

}

119

Start

Read data Calculate geometry parameters

Set initial conditions and boundary conditions

Generate child processes and broadcasting parameters

Each child process iterates on u, v, p for certain number of iterations

No

Next time step? Yes

End

Each child process calculates coefficients of the discrete algebraic equations

Exchange boundary values u, v, p

Synchronization of processes

Converged at the boundaries?

No

Yes

Figure 6.1: Flow chart of parallelization process with domain decomposition

120

6.3. The Ghost Nodes

In the domain decomposition approach, the boundary nodes of the neighboring sub-

domains are overlapped, so that each sub-domain can exchange information with its neighbors.

These overlapped nodes are called ghost nodes. Hence, the boundary of one sub-domain is

actually the interior of another sub-domain. In Figure 6.2, the computational domain is

decomposed into 4 sub-domains 0, 1, 2 and 3. Each sub-domain is assigned to a processor.

Region a is the overlapping nodes of domain 0 and 1. Right boundary of sub-domain 0 is the first

set of vertical interior surfaces in sub-domain 1. Similarly, the left boundary of sub-domain 1 is

the last set of vertical interior surfaces in sub-domain 0. After each set of sub-domain specific

iterations, each sub-domain receives a new set of boundary values from neighboring sub-

domains. For example, sub-domain 0 receives ytη values along S0 from sub-domain 1, and sub-

domain 1 receives ytη values along S1 from sub-domain 0.

In traditional numerical methods, only the node values need to be exchanged between the

neighboring domains. In MNIM, since the surface averaged variables are the discrete unknowns,

unlike the traditional numerical method, more than one variable need to be exchanged between

two neighboring domains. For example, ytη values along S0 and xtη values along S2 and S3

from sub-domain 0 need to be exchanged with ytη values along S1 and xtη values along S2

and S3 need from sub-domain 1.

121

0 1

2 3d

cbS1

aS0

S2

S3

y

x

Figure 6.2: Domain decomposition and the ghost nodes.

122

6.4. Load Balancing and Synchronization

In order to keep load on each processor balanced, each sub-domain is designed to have

approximately the same number of computational nodes. Let N be the number of processors

available. The assignment of processors for the case with a total number of 1 to 6 processors is

shown in Figure 6.3.

But because the details of the architecture of each processor may be different, and the

geometry and the boundary conditions for each sub-domain may also be different, each processor

may reach the point for exchanging boundary conditions at different time. When each processor

has finished its iterations on its sub-domain, it calls the MPI function sendrecv to send and

receive information from its neighbors.

The synchronization of different processors is achieved through the MPI function

sendrecv. When this function is called, each processor will wait for its neighbors. Because the

sub-domains are connected, only when all the processors have finished the relaxation on their

respective sub-domains and reach the function sendrecv, will the processors move on from the

sendrecv function. In this way, the different processors are synchronized at the location where

sendrecv is called.

123

(a) (b) (c)

(d) (e) (f)

Figure 6.3: Domain decomposition for different number of processors a) 1 processor b) 2 processors c) 3 processors d) 4 processors e) 5 processors f) 6 processors

124

6.5. Numerical Results

The domain decomposition and MPI based scheme is implemented for the MNIM for the

Navier-Stokes equations. The lid-driven cavity problem with exact solution is solved numerically

using the parallel version of the code. The mesh size over the whole domain is 33x33. A fixed

number of 15 iterations are used for the relaxation on each sub-domain. The FORTRAN code is

run on the SUN Ultra Enterprise 3000 workstations with 6 processors at host

raphson.cse.uiuc.edu.

Speed-up is usually the metric used to evaluate the performance of a parallel algorithm.

The definition of speed-up S is

pTTS 1=

where 1T is the CPU execution time of the sequential code, and pT is the execution time of

parallel code on p processors.

Figure 6.4 shows close to ideal speed-up for the case of 2, 3 and 4 processors. When 5 or

more processors are used, because the problem is too small, the time spent on exchanging

boundary information becomes more than the time gained in iterations for solving the physical

problem. The more number of processors are used, the more time spent on this communication.

Thus, the speed-up drops as the number of processors increases above four. For all the cases, the

RMS errors are within 3107 −× .

125

0 1 2 3 4 5 6 70

1

2

3

4

Spee

d-up

Number of Processors

Figure 6.4: Speed-up of parallelized MNIM for the lid-driven cavity problem with exact solution

(on a Sun Ultra Enterprise 3000 workstation, mesh size 33x33)

126

6.5. Conclusion

The modified nodal integral method developed in the previous chapters is parallelizable,

though the process of parallelization for MNIM is somewhat different from the parallelization

for traditional numerical methods. Good speedup for up to four processors is achieved on a

simple lid-driven cavity problem. Good speedup for more processors is expected for more

complex flows and larger computational domains.

127

Chapter 7

Summary and Conclusion

A modified nodal integral method is developed first for the two-dimensional, time-

dependent, incompressible Navier-Stokes equations, and then extended to three-dimensional.

The two dimensional modified nodal integral method has been applied to the following

problems: fully-developed flow and developing flow between parallel plates, lid driven cavity

problem with exact solutions, classic lid driven cavity problem, lid driven cavity problem with

aspect ratio of two and a two-dimensional, time-dependent Taylor’s decaying vortices problem.

The three dimensional modified nodal integral method has been applied to the following

problems: three-dimensional developed flow and developing flow in a rectangular channel,

three-dimensional lid-driven cavity problems in a cube and in a prism with aspect ratio of two.

Results obtained using the modified scheme are compared with those reported in

literature. Good agreement is found between results obtained using MNIM and reference

solutions. Moreover, grid size used in the MNIM is much coarser than those used earlier to solve

the same problems.

The developed modified nodal integral method is parallelizable.

A modified nodal integral method for Navier-Stokes equations coupled with energy and

specie concentration equations are developed in collaboration with Allen Toreja in appendix E.

128

Appendix A

Definition of Coefficients A for Two-Dimensional MNIM

The definition of coefficients A that appear in the discrete set of algebraic equations in

the MNIM for the two-dimensional N-S equations are given below. They appear in chapter two

in equations (2.60-2.63) and (2.81-2.84) respectively.

,

,

Re v

21 Re v,

2 1v(1 )

i j

i j

j

p i j

b eA

v e≡ +

− (A.1)

, 1

122 Re v

, 1

2 1v( 1 )i j

j

p i j

bA

v e +

+

+

≡ −− +

(A.2)

,

,

Rev,

23 Re v

v(1 )

i j

i j

p i jeA

v e≡

− (A.3)

, 1

, 124 Re v

v(1 )i j

p i jAv e +

+≡−

(A.4)

,

,

Re

51 Re,

2 1(1 )

i j

i j

ui

up i j

a eAuv e

≡ +−

(A.5)

1,

152 Re

1,

2 1( 1 )i j

iu

p i j

aAuv e +

+

+

≡ −− +

(A.6)

,

,

Re,

53 Re(1 )

i j

i j

up i ju

e uA

v e≡

− (A.7)

1,

1,54 Re(1 )i j

p i ju

uA

v e +

+≡−

(A.8)

129

Re v ,,

Re v ,

v (1 )

( 1 )91 2

,v

i jj p i j

i j

b e

e

p i j

vA

+

− +− +

≡ (A.9)

,

,

Re v

92 Re v,

1Re v1

i j

i ji j

eAe

≡ −− +

(A.10)

Re ,,

Re ,

(1 )

( 1 )1 2

,

ui ji p i j

ui j

a u e

ea

p i j

vA

u

+

− +− +

≡ (A.11)

,

,

Re

2 Re,

1Re1

i j

i j

u

a ui j

eAue

≡ −− +

(A.12)

130

Appendix B

Definition of Coefficients F for Two-Dimensional MNIM

The definition of coefficients F that appear in the discrete set of algebraic equations in

the MNIM for the two-dimensional N-S equations are given below. They appear in chapter two

in equations (2.86-2.93) respectively.

11 2 2

312 2( )

j

j i j

bF

b a b≡ − +

+ (B.1)

112 2 2

1 1

312 2( )

j

j i j

bF

b a b+

+ +

≡ − ++

(B.2)

13 2 2

32( )

j

i j

bF

a b≡ −

+ (B.3)

114 2 2

1

32( )

j

i j

bF

a b+

+

≡ −+

(B.4)

2

15 2 2i j

i j

a bF

a b≡ −

+ (B.5)

21

16 2 21

i j

i j

a bF

a b+

+

≡ −+

(B.6)

17 11 12 13 142 2F F F F F≡ + + + (B.7)

131

21 2 2

312 2( )

i

i i j

aFa a b

≡ − ++

(B.8)

122 2 2

1 1

312 2( )

i

i i j

aFa a b

+

+ +

≡ − ++

(B.9)

23 2 2

32( )

i

i j

aFa b

≡ −+

(B.10)

124 2 2

1

32( )

i

i j

aFa b

+

+

≡ −+

(B.11)

2

25 2 2i j

i j

a bF

a b≡ −

+ (B.12)

21

26 2 21

i j

i j

a bF

a b+

+

≡ −+

(B.13)

27 21 22 23 242 2F F F F F≡ + + + (B.14)

21 9231 23

91

A AF AA

≡ − (B.15)

22 92 12232 24

91 1 91 1

j

j j

A AAF AA A

+

+ +

≡ − + (B.16)

2133

912AFA

≡ (B.17)

2234

91 12 j

AFA +

≡ (B.18)

37 31 32 33 342 2F F F F F≡ + + + (B.19)

132

51 251 53

1

a

a

A AF AA

≡ − (B.20)

52 52 2 152 54

1 1 1 1

a i

a j a i

A A AF AA A

+

+ +

≡ − + (B.21)

5153

12 a

AFA

≡ (B.22)

5254

1 12 a i

AFA +

≡ (B.23)

57 51 52 53 542 2F F F F F≡ + + + (B.24)

9271

91

1 AFA

− +≡ (B.25)

9272

91

AFA

≡ − (B.26)

7391 1

1 1 1 12 a

FA A τ

⎛ ⎞≡ + −⎜ ⎟

⎝ ⎠ (B.27)

274

1

1 a

a

AFA

− +≡ (B.28)

275

1

a

a

AFA

≡ − (B.29)

77 71 72 73 74 75F F F F F F≡ + + + +

133

Appendix C

Pseudo-Source Terms for Three-Dimensional MNIM

The definition of the pseudo-source terms that appear in equations (4.12-4.26) in chapter

four in the MNIM for the three-dimensional N-S equations are given below.

22 22 2

2 2

1

v

1 v v( ) 2 2 28

n i k

n i k

a czxt

a ck i n

X Y Z

p p u wx z x y z

u w w uS y dzdxdtc a y x z y x y

g g g Dx y z t

τ

τ

∂ ∂ ∂ ∂ ∂ρ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ρ ρ ρτ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ρ ρ ρ ρ∂ ∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− − − + +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟⎜ ⎟≡ − − −∫ ∫ ∫ ⎜ ⎟⎜ ⎟⎜ ⎟+ + + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.1)

22 22 2

2 2

1

v

1 v v( ) 2 2 28

jn i

n j i

b axyt

b ai j n

X Y Z

p p u wx y x y z

u w w uS z dxdydta b y x z y x y

g g g Dx y z t

τ

τ

∂ ∂ ∂ ∂ ∂ρ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ρ ρ ρτ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ρ ρ ρ ρ∂ ∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− − − + +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟⎜ ⎟≡ − − −∫ ∫ ∫ ⎜ ⎟⎜ ⎟⎜ ⎟+ + + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.2)

22 22 2

2 2

1

v

1 v v( ) 2 2 28

jn k

n j k

b cyzt

b cj k n

X Y Z

p p u wy z x y z

u w w uS x dzdydtb c y x z y x y

g g g Dx y z t

τ

τ

∂ ∂ ∂ ∂ ∂ρ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ρ ρ ρτ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ρ ρ ρ ρ∂ ∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− − − + +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟⎜ ⎟≡ − − −∫ ∫ ∫ ⎜ ⎟⎜ ⎟⎜ ⎟+ + + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.3)

134

2 2

2 2

2

11( )

8( , , , ) ( ) (v v ) ( )

n i k

n i k

p pa czxt

a ck i nx p p p

u u u u u pu w vt x z y z x

S y dzdxdtc a u u ug x y z t u u w w

x y z

τ

τ

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂

τ ∂ ∂ ∂∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤+ + − + +⎜ ⎟⎢ ⎥

⎣ ⎦⎜ ⎟≡ − ∫ ∫ ∫ ⎜ ⎟− + − + − + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.4)

2 2

2 2

2

1v1( )

8( , , , ) ( ) (v v ) ( )

jn i

n j i

p pb axyt

b ai j nx p p p

u u u u u pu vt x y x y x

S z dxdydta b u u ug x y z t u u w w

x y z

τ

τ

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂

τ ∂ ∂ ∂∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤+ + − + +⎜ ⎟⎢ ⎥

⎣ ⎦⎜ ⎟≡ − ∫ ∫ ∫ ⎜ ⎟− + − + − + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.5)

2 2

2 2

2

1v1( )

8( , , , ) ( ) (v v ) ( )

jn k

n j k

p pb cyzt

b cj k nx p p p

u u u u u pw vt y z y z x

S x dzdydtb c u u ug x y z t u u w w

x y z

τ

τ

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂

τ ∂ ∂ ∂∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤+ + − + +⎜ ⎟⎢ ⎥

⎣ ⎦⎜ ⎟≡ − ∫ ∫ ∫ ⎜ ⎟− + − + − + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.6)

2 2

2 2

3

v v v v v 11( )

8 v v v( , , , ) ( ) (v v ) ( )

n i k

n i k

p pa czxt

a ck i ny p p p

pu w vt x z x z y

S y dzdxdtc a

g x y z t u u w wx y z

τ

τ

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂

τ ∂ ∂ ∂∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤+ + − + +⎜ ⎟⎢ ⎥

⎣ ⎦⎜ ⎟≡ − ∫ ∫ ∫ ⎜ ⎟− + − + − + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.7)

2 2

2 2

3

v v v v v 1v1( )

8 v v v( , , , ) ( ) (v v ) ( )

jn i

n j i

p pb axyt

b ai j ny p p p

pu vt x y x y y

S z dxdydta b

g x y z t u u w wx y z

τ

τ

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂

τ ∂ ∂ ∂∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤+ + − + +⎜ ⎟⎢ ⎥

⎣ ⎦⎜ ⎟≡ − ∫ ∫ ∫ ⎜ ⎟− + − + − + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.8)

2 2

2 2

3

v v v v v 1v1( )

8 v v v( , , , ) ( ) (v v ) ( )

jn k

n j k

p pb cyzt

b cj k ny p p p

pw vt y z y z y

S x dzdydtb c

g x y z t u u w wx y z

τ

τ

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂

τ ∂ ∂ ∂∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤+ + − + +⎜ ⎟⎢ ⎥

⎣ ⎦⎜ ⎟≡ − ∫ ∫ ∫ ⎜ ⎟− + − + − + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.9)

2 2

2 2

4

11( )

8( , , , ) ( ) (v v ) ( )

n i k

n i k

p pa czxt

a ck i nz p p p

w w w w w pu w vt x z x z z

S y dzdxdtc a w w wg x y z t u u w w

x y z

τ

τ

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂

τ ∂ ∂ ∂∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤+ + − + +⎜ ⎟⎢ ⎥

⎣ ⎦⎜ ⎟≡ − ∫ ∫ ∫ ⎜ ⎟− + − + − + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.10)

135

2 2

2 2

4

1v1( )

8( , , , ) ( ) (v v ) ( )

jn i

n j i

p pb axyt

b ai j nz p p p

w w w w w pu vt x y x y z

S z dxdydta b w w wg x y z t u u w w

x y z

τ

τ

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂

τ ∂ ∂ ∂∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤+ + − + +⎜ ⎟⎢ ⎥

⎣ ⎦⎜ ⎟≡ − ∫ ∫ ∫ ⎜ ⎟− + − + − + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.11)

2 2

2 2

4

1v1( )

8( , , , ) ( ) (v v ) ( )

jn k

n j k

p pb cyzt

b cj k nz p p p

w w w w w pw vt y z y z z

S x dzdydtb c w w wg x y z t u u w w

x y z

τ

τ

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂

τ ∂ ∂ ∂∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤+ + − + +⎜ ⎟⎢ ⎥

⎣ ⎦⎜ ⎟≡ − ∫ ∫ ∫ ⎜ ⎟− + − + − + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.12)

2 2 2

2 2 2

2

1v1( )

8( , , , ) ( ) (v v ) ( )

jk i

k j i

p p pbc axyz

c b ai j kx p p p

u u u u u u pu w vx y z x y z x

S t dxdydza b c u u ug x y z t u u w w

x y z

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ρ ∂

∂ ∂ ∂∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤+ + − + + +⎜ ⎟⎢ ⎥

⎣ ⎦⎜ ⎟≡ − ∫ ∫ ∫ ⎜ ⎟− + − + − + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.13)

2 2 2

2 2 2

3

v v v v v v 1v1( )

8 v v v( , , , ) ( ) (v v ) ( )

jk i

k j i

p p pbc axyz

c b ai j ky p p p

pu w vx y z x y z y

S t dxdydza b c

g x y z t u u w wx y z

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ρ ∂

∂ ∂ ∂∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤+ + − + + +⎜ ⎟⎢ ⎥

⎣ ⎦⎜ ⎟≡ − ∫ ∫ ∫ ⎜ ⎟− + − + − + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.14)

2 2 2

2 2 2

4

1v1( )

8( , , , ) ( ) (v v ) ( )

jk i

k j i

p p pbc axyz

c b ai j kz p p p

w w w w w w pu w vx y z x y z z

S t dxdydza b c w w wg x y z t u u w w

x y z

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ρ ∂

∂ ∂ ∂∂ ∂ ∂

− − −

⎛ ⎞⎡ ⎤+ + − + + +⎜ ⎟⎢ ⎥

⎣ ⎦⎜ ⎟≡ − ∫ ∫ ∫ ⎜ ⎟− + − + − + −⎜ ⎟⎜ ⎟⎝ ⎠

(C.15)

136

Appendix D

Definition of Coefficients F for Three-Dimensional MNIM

The definition of coefficients F that appear in the discrete set of algebraic equations in

the MNIM for the three-dimensional N-S equations are given below. They appear in chapter four

in equations (4.91-4.105) respectively.

2 2 2 2 2

11 2 2 2 2 2

2 ( 2 )2 ( ( ))

j k i j k

k j k i j k

b c a b cF

c b c a b c− −

≡+ +

(D.1)

2 2 2 2 2 2 21 1

12 2 2 2 2 21

( 3 ) ( 3 )2 ( ( ))

j k k i j k k

k j k i j k

b c c a b c cF

c b c a b c+ +

+

− + + −≡ −

+ + (D.2)

2

13 2 2 2 2 2

32( ( ))

j k

j k i j k

b cF

b c a b c≡ −

+ + (D.3)

21

14 2 2 2 2 2

32( ( ))

j k

j k i j k

b cF

b c a b c+≡

+ + (D.4)

2

15 2 2 2 2 2

32( ( ))

i k

j k i j k

a cFb c a b c

≡ −+ +

(D.5)

21

16 2 2 2 2 2

32( ( ))

i k

j k i j k

a cFb c a b c

+≡ −+ +

(D.6)

2 2 21 1 1

17 2 2 2 2 21

( )( ( 3 ) ( ( 3 )))2 ( ( ))

k k j k k k i j k k k

k k j k i j k

c c b c c c a b c c cF

c c b c a b c+ + +

+

+ + + + +≡ −

+ + (D.7)

2 2

18 2 2 2 2 2( )i j k

j k i j k

a b cF

b c a b c≡ −

+ + (D.8)

2 21

19 2 2 2 2 2( )i j k

j k i j k

a b cF

b c a b c+≡ −

+ + (D.9)

137

2 2 2 2 2

21 2 2 2 2 2

2 ( )2 ( ( ))

j k i j k

i j k i j k

b c a b cF

a b c a b c− + +

≡+ +

(D.10)

2 2 2 2 2 2 2 21

22 2 2 2 2 21

( ) 3 ( )2 ( ( ))

j k i j k i j k

i j k i j k

b c a b c a b cF

a b c a b c+

+

+ + − +≡ −

+ + (D.11)

2

23 2 2 2 2 2

32( ( ))

i k

j k i j k

a cFb c a b c

≡ −+ +

(D.12)

21

24 2 2 2 2 2

32( ( ))

i k

j k i j k

a cFb c a b c

+≡ −+ +

(D.13)

2

25 2 2 2 2 2

32( ( ))

i j

j k i j k

a bF

b c a b c≡ −

+ + (D.14)

21

26 2 2 2 2 2

32( ( ))

i j

j k i j k

a bF

b c a b c+≡ −

+ + (D.15)

2 2 2 2 2 2 21 1

27 2 2 2 2 21

( )( ( ) 3 ( ))2 ( ( ))

i i j k i j k i i j k

i i j k i j k

a a b c a b c a a b cF

a a b c a b c+ +

+

+ + + + +≡ −

+ + (D.16)

2 2

28 2 2 2 2 2( )i j k

j k i j k

a b cF

b c a b c≡ −

+ + (D.17)

2 21

29 2 2 2 2 2( )i j k

j k i j k

a b cF

b c a b c+≡ −

+ + (D.18)

138

2 2 2 2 2

31 2 2 2 2 2

2 (2 )2 ( ( ))

j k i j k

j j k i j k

b c a b cF

b b c a b c+ −

≡+ +

(D.19)

2 2 2 2 2 2 21 1

32 2 2 2 2 21

( 3 ) ( 3 )2 ( ( ))

k j j i j k j

j j k i j k

c b b a b c bF

b b c a b c+ +

+

− + + −≡ −

+ + (D.20)

2

33 2 2 2 2 2

32( ( ))

i j

j k i j k

a bF

b c a b c≡ −

+ + (D.21)

21

34 2 2 2 2 2

32( ( ))

i j

j k i j k

a bF

b c a b c+≡ −

+ + (D.22)

2

35 2 2 2 2 2

32( ( ))

j k

j k i j k

b cF

b c a b c≡ −

+ + (D.23)

21

36 2 2 2 2 2

32( ( ))

j k

j k i j k

b cF

b c a b c+≡ −

+ + (D.24)

2 2 2 21 1 1

37 2 2 2 2 21

( )( ( 3 ) ( 3 ))2 ( ( ))

j j j j j k i j j j k

j j j k i j k

b b b b b c a b b b cF

b b b c a b c+ + +

+

+ + + + +≡ −

+ + (D.25)

2 2

38 2 2 2 2 2( )i j k

j k i j k

a b cF

b c a b c≡ −

+ + (D.26)

2 21

39 2 2 2 2 2( )i j k

j k i j k

a b cF

b c a b c+≡ −

+ + (D.27)

139

11 4241 13

41

A AF AA

≡ − (D.28)

12 42 11242 14

41 1 41 1

k

k k

A AAF AA A

+

+ +

≡ − + (D.29)

1143

412AFA

≡ (D.30)

1244

41 12 k

AFA +

≡ (D.31)

12 42 111 11 4247 13 14

41 41 41 1

k

k

A AA A AF A AA A A

+

+

≡ + + − + (D.32)

21 5251 23

51

A AF AA

≡ − (D.33)

22 52 12252 24

51 1 51 1

i

i i

A AAF AA A

+

+ +

≡ − + (D.34)

2153

512AFA

≡ (D.35)

2254

51 12 i

AFA +

≡ (D.36)

21 52 22 52 12157 23 24

51 51 51 1

i

i

A A A AAF A AA A A

+

+

≡ + + − + (D.37)

140

31 6261 33

61

A AF AA

≡ − (D.38)

32 62 13262 34

61 1 61 1

j

j j

A AAF AA A

+

+ +

≡ − + (D.39)

3163

612AFA

≡ (D.40)

3264

61 12 j

AFA +

≡ (D.41)

32 62 131 31 6267 33 34

61 61 61 1

j

j

A AA A AF A AA A A

+

+

≡ + + − + (D.42)

141

4271

41

1 AFA

− +≡ (D.43)

4272

41

AFA

≡ − (D.44)

5273

51

1 AFA

− +≡ (D.45)

5274

51

AFA

≡ − (D.46)

6275

61

1 AFA

− +≡ (D.47)

6276

61

AFA

≡ − (D.48)

7741 51 61

1 1 1 1 12

FA A A τ

⎛ ⎞≡ − + + +⎜ ⎟

⎝ ⎠ (D.49)

7841 51 61

1 1 1 1 12

FA A A τ

⎛ ⎞≡ + + −⎜ ⎟

⎝ ⎠ (D.50)

142

Appendix E

Modified Nodal Integral Method for Navier-Stokes Equations Coupled with Energy and Specie Concentration Equations+

Buoyancy is the force due to density variation in the presence of a gravitational field. In

many buoyancy-driven flows, the density variation is important only in the body force of the

Navier-Stokes equations [Davis 1983]. Natural convection is such an example. Many proposed

designs of next generation of nuclear reactors rely on natural circulation as safety feature. Hence,

computational fluid dynamics codes for nuclear reactor thermal-hydraulics must be capable of

simulating this important phenomenon.

Convection can be driven not only by temperature gradient (buoyant convection), but

also by concentration (diffusocapillary flow) [Jue 1998]. In many cases, like zero-gravity

environment, concentration can be very important. In pressurized water reactor, boron

distribution in the coolant, though less likely to impact the velocity field directly, is an important

parameter for neutronic analysis. Therefore a numerical method coupling the N-S equations with

energy and concentration equations is desirable.

With Boussinesq approximation, the energy and concentration equations are coupled with

the N-S equations only through the gravity terms. The modified nodal integral method developed

in chapter two can be easily modified to couple the energy and concentration equations.

+ This coupling of the energy and N-S equations was carried out in collaboration with Allen Toreja [Rizwan-uddin 2001b]

143

E.1. The Boussinesq Approximation

For flows where the variation of density has strong influence only on the gravity term,

Boussinesq approximation is introduced [Currie 1993], i.e., the density in all the other terms is

treated as a constant except in the gravity term. Thus, the N-S equations are written as

0v=+

yxu

∂∂

∂∂ (E.1.1)

xp

yu

xu

yu

xuu

tu

∂∂

∂∂

∂∂μ

∂∂ρ

∂∂ρ

∂∂ρ

*

2

2

2

2)(v −+=++ (E.1.2)

2 2 *

02 2

v v v v vv ( ) ( )pρ ρu ρ gt x y x y y

∂ ∂ ∂ ∂ ∂ ∂μ ρ ρ∂ ∂ ∂ ∂ ∂ ∂

+ + = + − − − , (E.1.3)

where *p is the pressure relative to the static pressure.

E.2. Thermal Convection

In thermal convection problem, density variation is caused by temperature variations.

The density in the gravitational term can be approximated as

)](1[)( 00 TTT T −−= βρρ , (E.2.1)

or

)( 000 TTT −−=− βρρρ , (E.2.2)

where Tβ is the thermal expansion coefficient and 0T is a reference temperature such that

00 )( ρρ == TT . Substituting equation (E.2.2) into equation (E.1.3) and dropping the subscript 0

for density yields

144

0v=+

yxu

∂∂

∂∂ (E.2.4)

2 2 *

2 2

1v ( )u u u u u pu vt x y x y x

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ρ ∂

+ + = + − (E.2.5)

2 2 *

02 2

v v v v v 1v ( ) ( )Tpu v g T T

t x y x y y∂ ∂ ∂ ∂ ∂ ∂ β∂ ∂ ∂ ∂ ∂ ρ ∂

+ + = + − + − , (E.2.6)

where temperature ),,( tyxT is governed by the energy equation

qy

Tx

TyT

xTu

tT

++=++ 2

2

2

2v

∂∂α

∂∂α

∂∂

∂∂

∂∂

. (E.2.7)

Here pc

kρ

α = and pc

qqρ

'''= .

For flow induced by thermal convection, the dissipation term ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛=Φ

22 v2yx

u∂∂

∂∂μ

2v⎟⎠

⎞⎜⎝

⎛++

xyu

∂∂

∂∂μ and the

tp∂∂ term in the energy equation are small compared with other terms

and were dropped.

E.3. Non-Dimensional Form

For comparison with previous work [Azmy 1983] [Davis 1983], the non-dimensional

form of equations (E.2.4 - E.2.7) is given below:

0~v~

~~

=+yx

u∂∂

∂∂

(E.3.1)

xp

yu

xu

yu

xuu

tu

~~

)~~

~~

Pr(~~

v~~~~

~~ *

2

2

2

2

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

−+=++ (E.3.2)

145

TRay

pyxyx

ut

~Pr~~

)~v~

~v~Pr(~

v~v~~v~~

~v~ *

2

2

2

2+−+=++

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂ (E.3.3)

qy

Tx

TyT

xTu

tT ~

~

~

~

~~~

v~~~

~~~

2

2

2

2++=++

∂∂

∂∂

∂∂

∂∂

∂∂

. (E.3.4)

The definitions of the non-dimensional variables and parameters are

Lxx =~ ,

Lyy =~ (E.3.5)

0

0~TT

TTTref −−

= (E.3.6)

αuLu =~ ,

αLvv~ = (E.3.7)

2~

Ltt α

= (E.3.8)

)(~

0

2

TTkqLq

ref −= (E.3.9)

αv

=Pr (Prandl number) (E.3.10)

vLTTg

Ra refTα

β 30 )( −

= (Rayleigh number). (E.3.11)

E.4. MNIM for Navier-Stokes Equations Coupled with Energy Equation

Comparison of equations (E.2.5 – E.2.6) with equations (2.2-2.3) in chapter two shows

that the differences between these two sets of equations are:

(a) Pressure relative to static pressure is used instead of total pressure in the former set,

146

(b) Boussinesq approximation is used to evaluate the density in the gravity term instead of a

constant, and yb is replaced by 0( )Tg T Tβ− − .

The first is a trivial difference and only requires that the correct physical meaning of *p be kept

in mind. The second difference leads to the coupling of the energy equation with the Navier-

Stokes equations. The energy equation is coupled with the Navier-Stokes equations through the

velocities in the convection terms, and the Navier-Stokes equations are coupled with the energy

equation through the temperature in the gravity term.

When developing MNIM for the above set of equations, there are two possible

approaches to treat the temperature terms in the momentum equations. One approach is to keep

them on the left-hand side in the transverse-averaged ODE's as unknowns. This will result in a

set of coupled differential equations for u, v, T and p that have to be solved simultaneously. The

second approach is to move the temperature terms to the right hand side and lump them into the

pseudo-source terms. This approach will decouple the Navier-Stokes equations and the energy

equation before the transverse integration step. Each of them can then be solved separately and

then coupled via the pseudo-source terms.

The second approach is simpler to develop and implement. It is chosen for this work. The

MNIM developed in chapter two is used for the Navier-Stokes equations here. The only change

needed is to replace the yb term with 0( )Tg T Tβ− − . Coupling of the Navier-Stokes

equations and the energy equation in the iteration process is schematically shown in Figure E.1.

The development of MNIM for the energy equation is reported in the following section.

147

Figure E.1: Coupling of the Navier-Stokes equations and the energy equation

Navier-Stokes equations

Energy equation

u, v in convection terms T in the gravity term

148

E.5. Development of MNIM for the Energy Equation

The energy equation is rewritten as

2 2

2 2 v TT T T T Tu q

x x y y t∂ ∂ ∂ ∂ ∂α α∂ ∂ ∂ ∂ ∂

− + − = + . (E.5.1)

Steady-state, two-dimensional energy equation has been solved using a nodal approach earlier by

Michael et al [Michael 1993]. Extension of this scheme to arbitrary geometry is carried out by

Toreja [Toreja 2003]. Equation (E.5.1) is solved following the same approach used earlier to

solve the momentum equations.

E.5.1. Transverse Integration Procedure

Transverse integrating the above equation locally in (x, t), (y, t) and (x, y) direction yields

xtxt

xtxt

SdyTdy

dyTd

=− )(v2

2α (E.5.2)

ytyt

ytyt

SdxTdxu

dxTd

=− )(2

2α (E.5.3)

xyxydT S

dt= . (E.5.4)

where, again, average of the product in the convection term has been approximated by product of

the averages in equations (E.5.2) and (E.5.3), and the pseudo-source terms have been

approximated by constants.

149

E.5.2. Local Solutions and Continuity

Equations (E.5.2) and (E.5.3) are ODE's with variable coefficients. Expressions for cell

interior variation of transverse-averaged velocities )(v yxt and )(xu yt are of the form

ηη 4321CeCCC ++ , where yx,=η . Realizing that the source terms are being approximated by

constants, leading to a second order scheme, the cell interior velocities are also approximated by

the cell-averaged velocities (constants). This approximation is known to lead to third order error

in the numerical scheme [Michael 1994]. Hence, ytxyt uxu ≈)( and xtyxt y v)(v ≈ , where ytxu

and xtyv are the cell-averaged u and v velocities. Solving equations (E.5.2 – E.5.4) and imposing

continuity on the cell boundaries generate the discrete algebraic equations for xtjiT , , yt

jiT , and xyjiT ,

with the pseudo-source terms xtjiS , , yt

jiS , and xyjiS , :

0)( ,1,4,71,1,41,,71,1,5,,6 =++−−+ +++−++xtjijiji

xtjiji

xtjiji

xtjiji

xtjiji THHTHTHSHSH (E.5.5)

0)( ,,14,7,1,14,1,7,1,15,,6 =++−−+ +++−++ytjijiji

ytjiji

ytjiji

ytjiji

ytjiji TGGTGTGSGSG (E.5.6)

02 ,1,,, =−+ −xyji

xykji

xyji TTSτ , (E.5.7)

where the coefficients jiG , and jiH , are functions of ia , jb , α , ytxu and xtyv . For simplicity,

subscripts ji, are omitted from now on. For example,

612

1

Peuyt

i Peuyt

eG ae Peuyt

⎛ ⎞≡ −⎜ ⎟− +⎝ ⎠

(E.5.8)

v

6 v

121 v

Pe xt

j Pe xt

eH be Pe xt

⎛ ⎞≡ −⎜ ⎟− +⎝ ⎠

, (E.5.9)

where the nodal Peclet numbers are defined as

150

α

ytxiuaPeuyt 2

≡ (E.5.10)

α

xtyjb

xtPev2

v ≡ . (E.5.11)

Notice that there are only three equations (E.5.5 – E.5.7) for six unknowns ( xtT , ytT , xyT , xtS ,

ytS and xyS ) per cell. Hence, three additional equations are obtained by imposing cell

constraint equations over the cell.

E.5.3. Constraint Equations

The first constraint equation is obtained by averaging equation (E.5.1) over the cell,

which leads to the cell-averaged conservation equation:

xytxyytxt qSSS +=+ . (E.5.12)

The uniqueness of the cell averaged temperature leads to the other two constraint equations.

Requiring that xytxty TT = yields

xyxyk

xtxt

j

xtj

jSTSHT

bHT

bH τ+=++ −− 13

51

622

. (E.5.13)

Similarly, xytytx TT = yields

xyxyk

ytyt

i

yti

iSTSGT

aGT

aG τ+=++ −− 13

51

622

. (E.5.14)

Equations (E.5.5), (E.5.6), (E.5.7) and constraint equations (E.5.12), (E.5.13), (E.5.14)

form a closed set of six discrete algebraic equations with six unknowns xtT , ytT , xyT ,

xtS , ytS and xyS per cell. To reduce the number of discrete equations, the pseudo-source

151

terms xtS , ytS and xyS are eliminated, leaving three algebraic equations for three unknowns

xtT , ytT and xyT :

0113121,111110198

1,171615413211

=++++++

++++++

+−++−

+−+−+−

xytj

xytxykj

xyj

xyk

xy

ytji

ytj

yti

ytxtj

xtxtj

qzqzTzTzTzTz

TzTzTzTzTzTzTz (E.5.15)

0113121,111110198

1,171615413211

=++++++

++++++

+−++−

−++−+−xyti

xytxyki

xyi

xyk

xy

xtji

xti

xtj

xtyti

ytyti

qtqtTtTtTtTt

TtTtTtTtTtTtTt (E.5.16)

07615143211 =++++++ −−−xytxtxt

jyt

iytxyxy

k qeTeTeTeTeTeTe , (E.5.17)

where the coefficients iii etz ,, are functions of ii HG , . For example,

1313

52

33

617142

+++ +

−+

−+=jj

j HGHH

HGHHHHz (E.5.18)

1313

52

33

617142

+++ +

−+

−+=ii

i HGGG

HGGGGGt (E.5.19)

τττ

3333

331

)(21HGHG

HGe−−

++= (E.5.20)

Equation (E.5.15), (E.5.16), (E.5.17) are used to solve for xtkjiT ,, , yt

kjiT ,, and xykjiT ,, , respectively.

Both Drichlet and/or Neumann boundary conditions can be imposed on surfaces. This completes

the development of the MNIM for the energy equation.

E.6. Development of the MNIM for the Specie Concentration Equation

The concentration equation for a solute is similar to the energy equation:

2 2

2 2vi i i i ici Ci

C C C C Cu D qt x y x y

∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂

⎛ ⎞+ + = + +⎜ ⎟

⎝ ⎠, (E.6.1)

152

where Ci is the concentration of the ith specie (i =1,2, …, K), K is the total number of species,

and Dci is the mass diffusion coefficient for the solute. Because of the complete similarity

between the concentration equation and the energy equation, the concentration equation can be

solved using the MNIM developed for the energy equation. Adding another set of discrete

variables for concentration and replacing α by mass diffusion coefficient ciD allow simultaneous

solution of energy and concentration equations. Obviously, if the velocity field does not depend

upon concentration, concentration distribution can be determined after the velocity and

temperature fields have been determined.

In general, like the Boussinesq approximation for temperature, the joint temperature and

concentration effect on density can be approximated in the following form [Jue 1998]

0 0 0( , ) [1 ( ) ( )]T Ci i iT C T T C Cρ ρ β β= − − − − , (E.6.2)

or

0 0 0 0[ ( ) ( )]T Ci i iT T C Cρ ρ ρ β β− = − − − − . (E.6.3)

where Tβ is the thermal expansion coefficient and Cβ is the expansion coefficient for

concentration. Substituting equation (E.6.3) into equation (E.1.10) gives

2 2 *

0 02 2

v v v v vv ( ) ( ) 0T C i ipρ ρu ρ g T T g C C

t x y x y y∂ ∂ ∂ ∂ ∂ ∂μ μ ρ β ρ β∂ ∂ ∂ ∂ ∂ ∂

+ + − − + − − − − = , (E.6.4)

or

2 2 *

0 02 2

v v v v v 1v ( ) ( ) 0T C i ipu v v g T T g C C

t x y x y ρ y∂ ∂ ∂ ∂ ∂ ∂ β β∂ ∂ ∂ ∂ ∂ ∂

+ + − − + − − − − = (E.6.5)

153

where the subscript 0 on 0ρ has been dropped. Numerical solution of equation (E.6.5) using the

numerical scheme developed in chapter two requires the definition of ),,( tyxby term to be

modified to

0 0( , , ) ( ) ( )y T Ci i ib x y t g T T g C Cβ β= − − − − . (E.6.6)

Since the original code is written in a modular form, iterations between the N-S variables (u, v

and p) on one hand and temperature and concentration on the other hand are fairly straight

forward. The code developed for the N-S equation is modified to solve the complete set of

equations for u, v, p, T and Ci. In general, the iteration procedure followed is the same as that

used when solving the N-S and energy equations. Now, the discrete variables for concentration,

xtkjiC ,, , yt

kjiC ,, and xykjiC ,, , are also evaluated every time the discrete variables for temperature,

xtkjiT ,, , yt

kjiT ,, and xykjiT ,, , are evaluated. The procedure is schematically shown in Figure E.2.

Numerical results for the coupled N-S energy and N-S-energy-concentration problems are

presented in the next section.

154

Figure E.2: Coupling of the Navier-Stokes equations, energy and concentration equations.

Navier-Stokes equations

Energy equation Concentration equation

u, v in convection terms T and C in the gravity term

155

E.7. Numerical Results of the MNIM for the Coupled N-S, Energy and Specie Concentration Equations

An exact solution of the coupled, steady-state Navier-Stokes-Energy-Specie

Concentration equations for a modified lid driven cavity problem (0 < x, y < 1) with

Ci0 = T0 = 0, βT = βCi, is given by

)24)(2(8),( 3234 yyxxxyxu −+−=

))(264(8),( 2423 yyxxxyxv −+−−=

})]('[)('')(){(64)](')(')(''')([8),( 22 ygygygxFygxfygxFyxp −++= ν

CiiiT gyxbyxCandgyxbyxT βγβγ /]/),(1[),(/]/),(1[),( 0 +=+=

where γj are weight factors such that their sum is equal to one, and

)]()(')()()([64)]()(''')('')('2)(24[8),( 112 xFygygyGxFygxfygxfxFyxb −−++−= ν

).('')(')(''')()(;)]([5.0)(;)]('[)('')()(

;)()();()();2()(

12

22

1

24234

ygygygygyGxfxFxfxfxfxF

dxxfxFyyygxxxxf

−==−=∫=−=+−=

The flow in the cavity is due to shear caused by the non-uniformly moving lid,

)2(16)1,( 234 xxxyxu +−== ,

as well as due to body forces caused by energy and specie sources/sinks, given by

),(),();,(),( 22 yxCDTvyxqandyxTkTvcyxq iiipT ∇−∇⋅=∇−∇⋅= rrρ

The nodal scheme described above for the time-dependent Navier-Stokes-Energy-Specie

Concentration equations is used to solve the modified lid driven cavity with thermal and specie

sources/sinks. The steady-state problem was solved by marching in time starting from a spatially

uniform initial condition for all variables. Gauss-Seidel iterations are used at each time step.

156

Numerical results and comparison with the exact solution are shown in Figure E.3 for K

= 1 and γ0 = 0.75. The exact solution for u(x,y), v(x,y), P(x,y) and T(x,y) are shown in Figures

E.3a, E.3c, E.3e and E.3g (C(x,y) distribution, except for a multiplication factor, is identical to

that of temperature, T(x,y)). Corresponding L1 errors (node-averaged values plotted at the center

of the node) are shown in Figures E.3b, E.3d, E.3f and E.3h. RMS errors for node-averaged u

velocity, xyjiu , , for the 8 x 8 and 16 x 16 grid sizes are 4.008 x 10-3 and 1.056 x 10-3, respectively,

indicating a second order scheme. Maximum L1 errors for xyjiu , and xy

jiv , for the 16 x 16 case are

respectively, 0.00236 and 0.00139 corresponding to 0.9 % and 0.8% errors. Small RMS errors,

even for a coarse 8 x 8 mesh, show that the modified nodal scheme is accurate and efficient

157

Figure E.3: Exact solution and corresponding L1 error surfaces for the Navier-Stokes-Energy-Concentration equations for the lid driven cavity with energy and specie sources/sinks (mesh size16 x 16). (a) u velocity. (b) L1 error for u velocity. (c) v velocity. (d) L1 error for v velocity.

158

Figure E.3: Exact solution and corresponding L1 error surfaces for the Navier-Stokes-Energy-Concentration equations for the lid driven cavity with energy and specie sources/sinks (mesh size16 x 16). (e) Pressure. (f) L1 error for pressure. (g) Temperature. (h) L1 error for temperature.

159

References [Anderson 1995] J.D. Anderson, Jr., Computational Fluid Dynamics: The Basics with Applications, McGraw Hill, 1995. [Azmy 1982] Y.Y. Azmy, “A Nodal Method for the Numerical Solution of Incompressible Fluid Flow Problems,” M.S. thesis, University of Illinois, 1982. [Azmy 1983] Y.Y. Azmy and J.J. Dorning, “A Nodal Integral Approach to the Numerical Solution of Partial Differential Equations,” in Advances in Reactor Computations, volume II, 893-909, American Nuclear Society, LaGrange Park, IL, 1983. [Azmy 1985] Y.Y. Azmy, “Nodal Method for Problems in Fluid Mechanics and Neutron Transport,” Ph.D. thesis, University of Illinois, 1985. [Babu 1994] V. Babu and S. A. Korpela, “Numerical Solution of the Incompressible, Three-Dimensional Navier-Stokes Equations,” Computers Fluids, 23, 675-691, 1994. [Baloch 2002] A. Baloch, P.W. Grant and M.F. Webster, “Homogeneous and Heterogeneous Distributed Cluster Processing for Two and Three-Dimensional Viscoelastic Flows,” International Journal for Numerical Methods in Fluids, 40, 1347-1363, 2002. [Burns 1975a] T.J. Burns and J.J. Dorning, “A New Computational Method for the Solution of Multidimensional Neutron Problems,” in Proc. of the Joint NEACRP/CSNI Specialists’ Meeting on New Developments in Three-Dimensional Neutron Kinetics and Review of Kinetics Benchmark Calculations, 109-130, Laboratorium fur Reaktorregelung and Anlagensicherung, Garching(Munich), Germany, 1975. [Burns 1975b] T.J. Burns, “The Partial Current Balance Method: A Local Green’s Function Technique for the Numerical Solution of Multidimensional Diffusion Problems,” Ph.D. thesis, University of Illinois, 1975. [Caretto 1972] L.S. Caretto, A.D. Gosman, S.V. Patankar and D. Spalding, “Two Calculation Procedures for Steady, Three-Dimensional Flows with Recirculation,” in Proc. Third Int. Conf. Num. Methods Fluid Mech., Lect. Notes Phys. 19, Springer-Verlag, New York, 60-68, 1972. [Carey 1989] G.F. Carey, Parallel Supercomputing: Methods, Algorithms and Applications, John Wiley & Sons, 1989. [Caughey 1998] D.A. Caughey, M.M. Hafez (Ed.), Frontiers of CFD 1998, World Scientific, 1998. [Choi 1985] D. Choi and C.L. Merkle, “Application of Time-Iterative Schemes to Incompressible Flow,” AIAA J., 23, 1518-1524, 1985.

160

[Chorin 1968] A.J. Chorin, “Numerical Solution of the Navier-Stokes Equations,” Math. Comput., 22, 745-762, 1968. [Cortes 1994] A. B. Cortes and J. D. Miller, “Numerical Experiments with the Lid Driven Cavity Flow Problem,” Computers Fluids, 23, 1005-1027, 1994. [Currie 1993] I.G. Currie, Fundamental Mechanics of Fluids, 2nd Ed. McGraw-Hill Inc., 1993. [Davis 1983] G.D.V. Davis, I.P. Jones, “Natural convection in a square cavity: a comparison exercise,” International Journal for Numerical Methods in Fluids, 3, 227-248, 1983. [Delp 1964] D.L. Delp, D.L. Fischer, J.M. Harriman and M.J. Stewell, “FLARE, A Three-Dimensional Boiling Water Reactor Simulator,” GEAP-4598, General Electric Company, 1964. [Dennis 1979] S.C.R. Dennis, D.B. Ingham, and R.N. Cook, “Finite-Difference Methods for Calculating Steady Incompressible Flows in Three Dimensions,” J. Computational Physics, 33, 325-339, 1979. [Elnawawy 1990] O.A. Elnawawy, A.J. Valocchi and A.M. Ougouag, "The Cell Analytical-Numerical Method for Solution of the Advection-Dispersion Equation: Two-Dimensional Problems," Water Resources Research, 26, 2705-2716, 1990. [Esser 1993a] P.D. Esser, K.S. Smith, “A Semi-analytic Two-Group Nodal Model for SIMULATE-3,” in Transactions of the American Nuclear Society, 68, Part A, 220-222, San Diego, California, 1993. [Esser 1993b] P.D. Esser, R.J. Witt, “An Upwind Nodal Integral Method for Incompressible Fluid Flow,” Nuclear Science and Engineering, 114, 20-35, 1993. [Fasel 1984] H. Fasel and O. Booz, “Numerical Investigation of Supercritical Taylor-Vortex Flow for a Wide Gap”, J. Fluid Mech., 138, 21-52, 1984. [Ferziger 1996] J.H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, Springer, 1996. [Fischer 1981] H.D. Fischer and H. Finnemann, “The Nodal Integration Method—A Diverse Solver for Neutron Diffusion Problems,” Atomkernenergie, Kerntechnik, 39, 229-236, 1981. [Fletcher 1991] C.A.J. Fletcher, Computational Techniques for Fluid Dynamics, Volume II (2nd Ed), Springer-Verlag, 1991.

161

[Foster 1995] I. Foster, Designing and Building Parallel Programs, Addison-Wesley, 1995. [Gatski 1982] T.B. Gatski, C.E. Grosch and M.E. Rose, “A Numerical Study of the Two-Dimensional Navier-Stokes Equations in Vorticity-Velocity Variables,” J. Computational Physics, 48, 1-22, 1982. [Ghia 1977] K.N. Ghia, W.L. Hankey Jr. and J.K. Hodge, “Study of Incompressible Navier-Stokes Equations in Primitive Variables Using Implicit Numerical Technique,” AIAA Paper 77, 648, 1977. [Ghia 1982] U. Ghia, K.N. Ghia and C.T. Shin, “High-Re Solutions for Incompressible Flow Using the Navier-Stokes Equations and a Multigrid Method,” J. Computational Physics, 48, 387-411, 1982. [Gresho 1987] P.M. Gresho and R.L. Sani, “On Pressure Boundary Conditions for the Incompressible Navier-Stokes Equation,” Int. J. Num. Methods Fluids, 7, 1111-1145, 1987. [Guj 1988] G. Guj and F. Stella, “Numerical Solutions of High Re Recirculating Flows in Vorticity-Velocity Form,” Int. J. for Numerical Methods in Fluids, 8, 405-416, 1988. [Gunther 1992] C. Gunther, “Conservative Versions of the Locally Exact Consistent Upwind Scheme of Second Order (LECUSSO-SCHEME),” Inter. J. Num. Methods Eng., 34, 793, 1992. [Harlow 1965] F.H. Harlow and J.E. Welch, “Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluids with Free Surface,” Phys. Fluids, 8, 2182-2189, 1965. [Hartwich 1988] P.M. Hartwich, C.H. Hsu and C.H. Liu, “Vectorizable Implicit Algorithms for the Flux-Difference Split, Three-Dimensional Navier-Stokes Equations,” J. Fluids Eng., 110, 297-305, 1988. [Hennart 1986] J.P. Hennart, “A General Family of Nodal Schemes,” SIAM J. Sci. Stat. Comp., 7, 1, 264-287, 1986. [Hennart 1988] J.P. Hennart, “On the Numerical Analysis of Analytical Nodal Methods,” Num. Methods for Partial Diff. Equations, 4, 233-254, 1988. [Henriksen 2002] M.O. Henriksen and J. Holmen, “Algebraic Splitting for Incompressible Navier-Stokes Equations,” J. Computational Physics, 175, 438-453, 2002.

162

[Horak 1980] W.C. Horak, “Local Green’s Function Techniques for the Solution of Heat Conduction and Incompressible Fluid Flow Problems,” Ph.D. Thesis, University of Illinois, Urbana, 1980. [Horak 1985] W.C. Horak and J.J. Dorning, “A Nodal Coarse-Mesh Method for the Efficient Numerical Solution of Laminar Flow Problems,” J. Comp. Physics, 59, 405-440, 1985. [Joo 1997] H. G. Joo, G. Jiang, and T. J. Downar, “A Hybrid ANM/NEM Interface Current Technique for the Nonlinear Nodal Calculation,” Proc. ANS Conf. Math. Comp., Saratoga, NY, Oct. 1997. [Jue 1998] T. S. Jue, “Numerical Analysis of Thermosolutal Marangoni and Natural Convection Flows,” Numerical Heat Transfer, Part A, 34, 633-652, 1998. [Ku 1987] H.C. Ku, R.S. Hirsh and T.D. Taylor, “A Pseudo-spectral Method for Solution of the Three-Dimensional Incompressible Navier-Stokes Equations,” Journal of Computational Physics, 70, 439-462, 1987. [Issa 1986] R.I. Issa, A.D. Gosmanand and A.P. Watkins, “The Computation of Compressible and Incompressible Recirculating Flows by a Non-iterative Implicit Scheme,” J. Computational Physics, 62, 66-82, 1986. [Iwamoto 1998] T. Iwamoto and M. Yamamoto, “Development of A Multi-group Nodal BWR Core Simulator, NEREUS,” in Proc. Int. Conf. on the Physics of Nuclear Science and Technology, 2, 1106-1113, Hauppauge, New York, 1998. [Jiang 1998] G. Jiang, “A Harmonic Analytic Nodal Method,” in Proc. Int. Conf. on the Physics of Nuclear Science and Technology, 2, 1114-1121, Hauppauge, New York, 1998. [Kim 1985] J. Kim and P. Moin, “Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations,” J. Computational Physics, 59, 308, 1985. [Kwak 1986] D. Kwak, J.L.C. Chang, S.P. Shanks and S.K. Chakravarthy, “A Three-Dimensional Incompressible Navier-Stokes Solver Using Primitive Variables,” AIAA J., 24, 390-396, 1986. [Lawrence 1979] R.D. Lawrence, “A Nodal Green’s Function Method for Multidimensional Diffusion Calculations,” Ph.D. thesis, University of Illinois, 1979. [Lawrence 1980a] R.D. Lawrence and J.J. Dorning, "A Nodal Green's Function Method for Multidimensional Neutron Diffusion Calculations," Nuclear Science and Engineering, 76, 218-231, 1980.

163

[Lawrence 1980b] R.D. Lawrence and J.J. Dorning, "A Discrete Nodal Integral Transport Theory Method for Multidimensional Reactor Physics and Shielding," in Proc. ANS Conf. Advances in Reactor Physics and Shielding, Sun Valley, Idaho, 1980. [Lawrence 1986] R.D. Lawrence, “Progress in Nodal Methods for the Solutions of the Neutron Diffusion and Transport Equations,” Progress in Nuclear Energy, 17, 271-301, 1986. [Michael 1993] E.P.E. Michael, J. J. Dorning, E.M. Gelbard, Rizwan-uddin, “A Nodal Integral Method for the Convection-Diffusion Heat Equation,” Transaction of the American Nuclear Society, 69, 239-241, 1993 [Michael 1994] E.P.E. Michael, J. J. Dorning, Rizwan-uddin, “Third-Order Nodal Integral Method for the Convection-Diffusion Equation,” Transaction of the American Nuclear Society, 71, 195-197, 1994. [Michael 2000a] E.P.E. Michael and J.J. Dorning, “A Primitive-Variable Nodal Method for Steady-State Fluid Flow,” Transaction of the American Nuclear Society, 83, 420-422, 2000. [Michael 2000b] E.P.E. Michael and J.J. Dorning, “A Primitive-Variable Nodal Method for the Time-Dependent Navier-Stokes Equations,” in Proc. ANS Topical Meeting on Math. & Comp., Salt Lake City, 2001. [Michael 2001] E.P.E. Michael, J.J. Dorning and Rizwan-uddin, “Studies on Nodal Integral Methods for the Convection-Diffusion Heat Equation,” Nuclear Science & Engineering, 137, 380-399, 2001. [Morton 1996] K. W. Morton, Numerical Solution of Convection Diffusion Problems, Chapman & Hall, 1996. [O’Rourke 1998] P.J. O’Rourke and M.S. Sahota, “A Variable Explicit/Implicit Numerical Method for Calculating Advection on Unstructured Meshes,” J. Computational Physics, 143, 312-345, 1998. [Pan 1967] F. Pan and A. Acrivos, “Steady Flows in Rectangular Cavities,” J. Fluid Mechanics,” 28, 643-655, 1967. [Patankar 1980] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, 1980. [Quarteroni 2000] A. Quarteroni, F. Saleri and A. Veneziani, “Factorization Methods for the Numerical Approximation of Navier-Stokes Equations,” Comput. Methods Appl. Mech. Engrg., 188, 505-526, 2000.

164

[Rizwan-uddin 1997] Rizwan-uddin, “An Improved Coarse-Mesh Nodal Integral Method for Partial Differential Equations,” Num. Methods for Partial Differential Equations, 13, 113-145, 1997. [Rizwan-uddin 2001a] Rizwan-uddin, “Comparison of the Nodal Integral Method and Non-Standard Finite-Difference Schemes for the Fisher Equation,” SIAM J. Scientific Computing, 22, 1926-1942, 2001. [Rizwan-uddin 2001b] Rizwan-uddin, A. J. Toreja, F. Wang (invited), “Nodal Methods Extended: Modified Nodal Methods; Curved boundaries; Adaptive mesh refinement; Parallelization; etc.,” Transaction of the American nuclear society, 85, 289-291, 2001. [Shatilla 1997] Y. A. Shatilla, “Westinghouse Advanced Nodal Code with Pin-Power Reconstruction for MOX Applications,” Transactions of the American Nuclear Society, 76, 179-181, Orlando, Florida, 1997. [Shin 1989] T.M. Shin, C.H. Tan and B.C. Hwang, “Effects of Grid Staggering on Numerical Schemes,” Inter. J. for Numerical Methods in Fluids, 9, 193-212, 1989. [Schlicting 1968] H. Schlicting, Boundary Layer Theory, McGraw-Hill, New York, 1968. [Smith 1996] B. Smith, P. Bjorstad, and W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, 1996. [Steger 1976] J.L. Steger and P. Kutler, “Implicit Finite-Difference Procedures for the Computation of Vortex Wakes,” AIAA Paper 76-385, San Diego, California, 1976. [Tannehill 1997] J.C. Tannehill, D.A. Anderson and R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer (2nd Ed), Taylor and Francis, 1997. [Taylor 1923] G.I. Taylor, “On the Decay of Vortices in a Viscous Fluid,” Philos. Mag., 46, 671-675, 1923. [Toreja 1999] A.J. Toreja and Rizwan-uddin, “Hybrid Numerical Methods for the Convection-Diffusion Equation in Arbitrary Geometries,” in the Proc. M&C-99 International Conference on Mathematics and Computation, Reactor Physics and Environmental Analysis in Nuclear Applications, 1705-1714, Madrid, Spain, September, 27-30, 1999. [Toreja 2003] A.J. Toreja and Rizwan-uddin, “Hybrid Numerical Methods for Convection-Diffusion Problems in Arbitrary Geometries,” Computers and Fluids, 32, 835 – 872, 2003. [Wang 2000] F. Wang and Rizwan-uddin, “A Nodal Scheme for the Time-Dependent, Incompressible Navier-Stokes Equations,” Trans. Amer. Nucl. Soc., 83, 422-424, 2000.

165

[Wang 2003a] F. Wang and Rizwan-uddin, “Modified Nodal Integral Method for the Three-Dimensional, Time-Dependent Incompressible Navier-Stokes Equations,” Nuclear Mathematical and Computational Sciences: A Century in Review, A Century Anew, Gatlinburg, Tennessee, April 6-11, 2003, on CD-ROM, American Nuclear Society, LaGrange Park, IL, 2003. [Wang 2003b] F. Wang and Rizwan-uddin, “A Nodal Scheme for the Time-Dependent Incompressible Navier-Stokes Equations,” Journal of Computational Physics, 187,168-196, 2003. [Wescott 2001] B. Wescott and Rizwan-uddin, “An Efficient Formulation of the Modified Nodal Integral Method and Application to the Two-Dimensional Burgers’ Equation,” Nuclear Science & Engineering, 139, 293-305, 2001. [Wilson 1983] G.L. Wilson, “Multidimensional Nonlinear Time-Dependent Nodal Integral Methods in Heat Transfer and Fluid Dynamics, ” Ph.D. thesis, University of Illinois, 1987. [Wilson 1988] G.L. Wilson, R.A. Rydin, and Y.Y. Azmy, “Time-Dependent Nodal Integral Method for the Investigation of Bifurcation and Nonlinear Phenomena in Fluid Flow and Natural Convection, ” Nucl. Sci. Eng. 100, 414-425, 1988.