Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
ME 608
Numerical Methods for Heat, Mass
and Momentum Transfer
Jayathi Y. Murthy
Professor, School of Mechanical Engineering
Purdue University
[email protected]
Spring 2011
1
ME 608
Lecture 1: Introduction to ME 608
Conservation Equations
2
Outline of
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
SIMPLE SOLVER FOR DRIVEN CAVITY FLOW PROBLEM
Vaidehi Ambatipudi
Department of Mechanical Engineering
Purdue University
West Lafayette, Indiana 47906
Email: [email protected]
ABSTRACT
This report presents the solution to the NavierStokes equations. Stand
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 24:
Flux Limiters
1
Last Time
Developed a set of limiter functions
Second order accurate
2
This Time
Examine physical rationale for limiter functions
Application to unstructured meshes
3
Recall HigherOrder Scheme for e
Consider finding face value
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 38:
SIMPLE Algorithm for CoLocated
Scheme
1
Last Time
Completed the development of a momentuminterpolation procedure for the face velocities
Developed a p equation for the colocated formulation
2
This Time
Devise a SIMPLE algorithm that works w
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 39:
CoLocated SIMPLE: Example Problem
Unstructured Storage Schemes
1
Last Time
Looked at how to integrate the SIMPLE algorithm with
a colocated scheme for pressurevelocity storage
2
This Time
Will do a colocated SIMPLE problem
Will consider st
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 40:
Introduction to Multigrid Methods
1
Last Time
Completed colocated SIMPLE problem
2
This Time
We will:
Consider performance of Jacobi and GaussSeidel
schemes for different types of initial guesses
Exploit this fact in devising multigrid schem
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 41:
Multigrid Methods
1
Last Time
Looked at the relationship between mesh size and the
wave number content of error
Deduced that involving coarse meshes would help
reduce lowk errors
2
This Time
We will
Analyze the Jacobi scheme to get a deeper
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 42:
Algebraic Multigrid Method
1
Last Time
Discussed the basics of the multigrid method
Use coarse mesh only to accelerate convergence
Therefore coarse mesh corrections go to zero at
convergence
Can choose coarselevel coefficient matrix for
c
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 43:
Algebraic Multigrid Method
Cycling Strategies
1
Last Time
We
Considered the algebraic multigrid method
Developed coarselevel discrete equations by addtion
of finelevel equations together
Considered an agglomeration strategy based on
coeffici
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 44:
Cycling Strategies
Example Problem
1
Last Time
Completed the discussion of coefficientbased
agglomeration strategy
Discussed fixed and flexible cycling strategies for
switching between coarse and fine levels
2
This Time
We will:
Look at som
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 37:
CoLocated Schemes (continued)
1
Last Time
Started looking at colocated pressurevelocity storage
Developed momentuminterpolation scheme
Figured out discretization accuracy
2
This Time
Complete development of colocated scheme
Start looking
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 36:
CoLocated Schemes
1
Last Time
Completed looking at SIMPLE on staggered
structured Cartesian meshes
Found that staggered meshes are difficult to use on
bodyfitted meshes and unstructured meshes
2
This Time
Start looking at colocated pressur
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 25:
Introduction to Fluid Flow
1
Last Time
Examined physical rationale for limiter functions
Applied to multidimensional situations and
unstructured meshes
This brings us to the end of the convectiondiffusion
part
2
This Time
Start looking at the
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
ME 608
Numerical Methods in Heat, Mass, and Momentum Transfer
Example Problems
Instructor: J. Murthy
1. Consider steady 1D conduction in a 1D domain consisting of 3 equalsized cells as shown in Fig. 1. The right boundary
(x=0) is irradiated with a radiat
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 31:
Fluid Flow: Staggered Mesh
1
Last Time
Started looking at the computation of fluid flow
Sequential schemes
How to introduce pressure into continuity?
Checkerboarding and where to store (u,P)
2
This Time
Reexamine where to store pressure v
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 32:
The SIMPLE Algorithm
1
Last Time
Looked at where to store pressure visvis velocity
Checkerboarding problems
Looked at staggered mesh as a remedy for checkerboarding for structured meshes
2
This Time
Look into problem of introducing pressur
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 33:
The SIMPLE Algorithm (Contd)
1
Last Time
Looked into problem of introducing pressure into
continuity equation for incompressible flows
Introduced SIMPLE algorithm
Derived the pressure correction equation
2
This Time
Look at the SIMPLE algor
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 34:
Boundary Conditions for the Pressure
Correction Equation
Problem Solving
1
Last Time
Looked at solution loop for the SIMPLE algorithm
Examined auxiliary issues
Underrelaxation and convergence
Started looking at boundary conditions
Nature o
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 35:
Problem Solving
Application to BodyFitted Meshes
1
Last Time
Completed our consideration of boundary conditions
for the pressure correction equation
Did an example using the SIMPLE algorithm
2
This Time
Do another example problem using SIMP
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 45:
Sensitivity Analysis
1
Last Time
Completed consideration of multigrid methods
Solved an example problem
2
This Time
We will:
Look at emerging methods to perform sensitivity
analysis using CFD
3
Why Sensitivity and Adjoints?
Tangent Problem:
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Lecture 46:
Final Exam Preparation
1
Last Time
We
Looked at emerging methods to perform sensitivity
analysis using CFD
2
This Time
We will do some exam prep
3
AMG Problem
4
SIMPLE Problem (Staggered)
5
SIMPLE Problem (contd)
6
SIMPLE Problem (CoLocated
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Derivation of Model Equation for Explicit UDS
Following (Tannehill et al. , 1997), we can derive the model equation for the explicit UDS scheme in the following way. The explicit UDS scheme may be written as: 0 P P u 0 0 W (1) t x P
0 0 0 Expanding P in a
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Extra Material
Variants of SIMPLE: SIMPLER &
SIMPLEC
1
SIMPLER Algorithm: Motivation
The SIMPLE algorithm has been used widely but
suffers from a drawback
'
'
Dropping the nb anbunb , nb anbvnb terms produces
toolarge pressure corrections, needing underr
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Proceedings of HT2005 2005 ASME Summer Heat Transfer Conference July 1722, 2005, San Francisco, California, USA
HT200572172
Thermal Transport in Nanotube Composites for LargeArea Macroelectronics
Satish Kumar Department of Mechanical Engineering Purdue
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
ME 608
Numerical Methods in Heat, Mass, and Momentum Transfer
MidTerm Examination
Date: March 5, 2008
6:00 7:30 PM
Instructor: J. Murthy
Open Book, Open Notes
Total: 50 points
Use the nite volume method in all problems.
NAME:
Problem Points
1
30
2
20
TOT
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
ME 608
Numerical Methods in Heat, Mass, and Momentum Transfer
MidTerm Examination Solutions
Date: March 5, 2008
6:00 7:30 PM
Instructor: J. Murthy
Open Book, Open Notes
Total: 50 points
1. We are given the following problem statement. The governing equat
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
ME 608
Numerical Methods in Heat, Mass, and Momentum Transfer
Sample Final Exam Problems
April 25, 2006
Instructor: J. Murthy
1. (a) For the ne mesh, the discrete equations for representative points are given below.
Cell 1 :
Cell 2 :
0
a0 = 10, aW = 0, a0
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
ME 608
Numerical Methods in Heat, Mass, and Momentum Transfer
Sample Final Exam Problems
April 25, 2006
Instructor: J. Murthy
1. Consider 1D steady conduction on a uniform mesh of twelve cells, as shown in Fig.1. The domain length is L = 1.2 m.
Cells 14
Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608

Fall 2010
Partners
Anantharaman, S. and Hu, J.
Devendran, R.S. and Dhar, S.
Ganguly, A. and Weaver, A.
Gavilan, ME.
Guo, L. and Srivastava, I.
Jin, M. and Li, S.
Kharangte, C. and Lee, H.
Kim, SM, and Kim, W.
Konichi, C.
Kumar, S. and Sarangi, S.
Lee, S. and Rau,