Numerical Heat Transfer

Xi'an Jiaotong University via XuetangX

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The Numerical Heat Transfer MOOC, spearheaded by the Xi'an Jiaotong University teaching team led by Professor Tao Wenquan, Academician of the Chinese Academy of Engineering, is meticulously designed for graduate students in engineering thermophysics, energy and power engineering, and related disciplines. This course synthesizes decades of the team’s cutting-edge research and engineering expertise, systematically establishing a comprehensive knowledge framework spanning fundamental theories to advanced applications, with the overarching goal of cultivating students’ core competencies in addressing complex heat transfer engineering challenges.

Distinguished by its systematic rigor, cutting-edge relevance, and practical orientation, the course provides an in-depth exposition of the theoretical foundations and key methodologies in numerical heat transfer. The team delivers rigorous instruction on the mathematical principles underpinning the finite volume method, SIMPLE algorithm, and grid generation techniques, while elucidating discretization strategies and error control mechanisms for conduction, convection, and radiation equations. With a strong emphasis on disciplinary frontiers, the curriculum encompasses multiscale coupled heat-fluid flow methodologies, intelligent optimization algorithms, and other innovative approaches, including detailed analyses of the team’s proprietary algorithm platforms. On the practical front, the course features industry-grade case studies to demonstrate operational logic and industrial application paradigms of mainstream simulation tools, thereby strengthening students’ proficiency in modeling and solving complex engineering problems.

The pedagogical framework is anchored by Numerical Heat Transfer, the nationally recognized authoritative textbook authored by Academician Tao Wenquan, ensuring both theoretical rigor and forward-looking perspectives. The content deeply integrates the team’s methodological innovations from major scientific research initiatives, offering a holistic view of the translational pathway from algorithm development to industrial deployment. Participants gain exclusive access to the team’s algorithm toolkit, industrial benchmark cases, and error diagnostics modules, substantially enhancing the precision of scientific simulations and the efficacy of engineering practice.

Upon completion, graduate students will develop a systematic mindset for numerical modeling, master self-directed technical approaches to complex heat transfer problems, and establish a robust foundation for academic research or industrial R&D.

Syllabus

Chapter 1 Introduction

0.1 Course Introduction
1.1 Mathematical formulation of heat transfer and fluid flow (HT & FF) problems
1.2 Basic concepts of NHT, its importance and application examples
1.3 Mathematical and physical classification of HT & FF problems and its effects on numerical solution
1.4 Summary&Homework of Chapter 1

Chapter 2 Discretization of Computational Domain and Governing Equations

2.1 Grid generation (domain discretization）
2.2 Control volume and heat balance methods for equation discretization

Chapter 3 Numerical Methods for Solving Diffusion Equation and their Applications

3.1 General equation of 1-D Heat Conduction Equation
3.2 Fully Implicit Scheme of Multi dimensional Heat Conduction Equation
3.3 Treatments of Source Term and B.C.
3.4 TDMA & ADI Methods for Solving ABEs
3.5 Fully Developed HT in Circular Tubes
3.6 Fully Developed HT in Rectangle Ducts

Chapter 4 Discretized Schemes of Diffusion and Convection Equation

4.1 Two ways of discretization of convection term
4.2 CD and UD of the convection term
4.3 Hybrid and power law schemes
4.4 Characteristics of five three-point schemes
4.5 Discussion on false diffusion
4.6 Methods for overcoming or alleviating effects of false diffusion
4.7 Discretization of multidimensional problem and B.C. treatment

Chapter 5 Solution Methods for Algebraic Equations

5.1 Introduction to Solution Methods of ABEqs
5.2 Construction of Iteration Methods of Linear Algebraic Equations
5.3 Convergence Conditions and Acceleration Methods for Solving Linear ABEqs
5.4 Block correction method –promoting conservation satisfaction
5.5 Multigrid Techniques Promoting Simultaneous Attenuation of Different Wave length Components

Chapter 6 Primitive Variable Methods for Elliptic Flow and Heat Transfer

6.1 Source terms in momentum equations and two key issues in numerically solving momentum equation
6.2 Staggered grid system and discretization of momentum equation
6.3 Pressure correction methods for N S equation
6.4 Approximations in SIMPLE algorithm
6.5 Discussion on SIMPLE algorithm and criteria for convergence
6.6 Developments of SIMPLE algorithm
6.7 Boundary condition treatments for open system
6.8 Fluid flow & heat transfer in a closed system

Chapter 7 Mathematical and Physical Characteristics of Discretized Equations

7.1 Consistence, Convergence and Stability of Discretized Equations
7.2 Von Neumann Method for Analysing Stability of Initial Problems
7.3 Conservation of Discretized Equations
7.4 Transportive Property of Discretized Equations
7.5 Sign-preservation Principle for Analyzing Convective Stability
7.6 Mathematical and Physical Characteristics of Discretized Equations

Chapter 8 Numerical Simulation for Turbulent Flow and Heat Transfer

8.1 Introduction to turbulence
8.2 Time-averaged governing equation for incompressible convective heat transfer
8.3 Zero Equation Model and One Equation Model
8.4 Two-Equation Model
8.5 Wall Function Method
8.6 Low Reynolds Number k-epsilon Model
8.7 Brief Introduction to Recent Developments

Chapter 9 Introduction to BFC and Collocated Grid System

9.1 Treatments of Irregular Domain in FDM,FVM
9.2 Introduction to Body-Fitted Coordinates
9.3 Boundary Normalzation for Generating Body-Fitted Coordinates (simple algebraic method)
9.4 Introduction to Collocated Grid System

Chapter 10 General Code for 2D Elliptical Fluid Flow and Heat Transfer Problems

10.1 Format Improvement of General Governing Equation
10.2 Numerical Methods Adopted in Teaching Code
10.3 Code Structure and Module Functions
10.4 Grid System
10.5 Techniques Adopted in the Code
10.6 Methods of Application and Explanation of MAIN Program

Chapter 11 Application Examples of the General Code for 2D Elliptical FF & HT Problems

11.1 2D steady heat conduction without source term in Cartesian coordinate
11.2 Steady heat conduction in a hollow cylinder
11.3 Example 3 Fully-developed heat transfer in a
11.4 Fully developed heat transfer in annular space with straight fin at inner wall
11.5 Fluid flow and heat transfer in a 2-D sudden expansion
11.6 Complicated fully developed fluid flow and heat transfer in a square duct
11.7 Impinging flow on a rotating disc
11.8 Turbulent flow and heat transfer in ductwith a stepwise inlet velocity distribution