CFD and CHT
Theoretical and Applied Computational Fluid Dynamics and Heat Transfer
Course Overview
The CFD-CHT course is designed around 8 lectures which descend from a structured modelling paradigm. The agenda of each lecture is presented in the following pages. By design, each lecture transitions from one into the next in roughly the same fashion that a systematic analysis is performed. The top-level lectures are titled:
Lecture 1 Introduction to CFD and CHT
Lecture 2 Formulation of the Basic Equations of Fluid Flow and Heat Transfer
Lecture 3 Decoupling Systems and Deriving Boundary Conditions
Lecture 4 Representation through Discretization
Lecture 5 Computational Solutions to the Discrete Equations
Lecture 6 Validation of Computational Models
Lecture 7 Reconstitution and Repackaging Computational Models
Lecture 8 Special Topics in CFD and CHT
Each of these lectures is then expanded into a series of detailed subjects. For example, Lecture 4 on discretization covers: representation, motivation for discretization, connectivity, spatial fields, mathematical formulations, strategic discretization, refinement, heat flux laws, conductance models, enthalpy flux models, vectorization, sparse and flattened matrix representation.
The CFD-CTA course will introduce the topics using practical thermal problems. For example, in lecture 4 a 2D coldplate is discretized using both the sparse and flattened matrix methods. While simple, this example is fruitful in revealing all of the important concepts. For example, both the sparse and flattened representations are setup in the classical explicit and implicit differencing methods in Lecture 5. Both are solved and stability regions are tested using matrix analysis software. With a slight change in one conductance value, stalled iteration is revealed which motivates the remaining discussion on semi-direct solution methods. Similarly, an analysis of numerical consistency is performed in Lecture 6 and finally model replication and reconstitution are demonstrated in Lecture 7.
The course will feature an Interlab sessions which comprises rapid model construction and demonstration of basic concepts. In the past this could not have been possible, but using LCD projectors with today’s rapid analysis software, it is possible to construct models of considerable complexity, demonstrating and reinforcing the content immediately and with substantial graphical appeal. The student will have a personal copy of all software used in the course in order to return to the modelling session and retest the models constructed and evaluated in the class. These codes, tentatively, Hyperion-TFS and TAS will be supplied with a limited student license. Interlab as a concept has been proven effective in the teaching of the CHT short course. It is an effective and preferred instructional medium.
The homework will consist of two facets. The first is to provide instructional fundamentals in creating CFD-CHT models using typical commercial-grade software. This is to be achieved with probing-type activities. For example, a homework assignment may be to construct a computational model of a convection cell and derive the Nusselt number for the geometry, commenting on consistency and computational oddities. The student will be graded in the same fashion that industry grades the engineer, e.g. model utility, robustness, each of modification, utility, ability to explain phenomenon.
The second facet involves the incremental construction of a general purpose advection-diffusion operator and relaxation solution methods to begin to see what is involved in creating a computational analysis code. This facet, while appearing disjoint from the first facet, is actually critically coupled. Considering the ability to successfully use commercial-grade codes is critically dependent on the ability to visualize the solver technology operating behind the scenes, it is imperative that the student acquire the skills though programming on their own. Lecture alone will not convey the setup and execution of a solver or discretization scheme. It must be learned by practice. The instructor will make a compiler recommendation to assist the user. This would allow some common DLL routines that the instructor has constructed to be applied in the students analysis codes, allowing a bypass of the rigorous graphical engine construction that would surmount the programming effort.
CFD and CHT
Theoretical and Applied Computational Fluid Dynamics and Heat Transfer
Course Syllabus
Lecture 1 Introduction to CFD and CHT Introduction: Basic features of computational fluid and thermal analysis
Structuring the analysis approach
Model classification - positives and negatives
Experimental
Analytical
Computational
Hybrid Variants
Emergence of CFD as a general purpose tool
Structuring the analysis approach
Difficulties and failings
Example application of paradigm
Discussion
CFD course syllabus for remaining lectures
Lecture 2 Formulation of the Basic Equations of Heat Transfer An introduction to the control volume and control surface
the control volume attributes
surveying this control volume for the basic conservation process
simplifying the geometry of the control volume
relation to other CFD methods
General conservation equation and conserved quantities
time rate of change of a property within a CV
mass, momentum and energy conservation
Decoupling thermal and fluid analysis
simplifying the number of equations or decoupled CFD
when decoupling is reasonable
when decoupling is not reasonable
Energy conservation equation
Flux terms
the area vector
example of integrating flux quantities
Volume source terms
Final simplified equation
Example application of energy equation
basic ODE
target solution state
concept of 1CV and 1 solution state
revealing important thermal features through discretization
Lecture 3 Decoupling Systems and Deriving Boundary Conditions Extent of full systems and the concept of a boundary condition
Rendering boundary conditions by parting the full system into regions
Types of boundary conditions
Application study - Dirichlet boundary condition
Application study - Neumann boundary condition
Controllability of temperature in heat flux boundaries
Applicability of the Robin boundary condition
Converting Robin boundary conditions
Lecture 4 Discretization of the Governing Equations and Geometry
Motivation for discretization
Connectivity of control volumes and spatial fields
Example: physical-spatial discretization and mathematical discretization
Strategic discretization and refinement
Cartesian based discretization to be applied
Fitness evaluation - Cartesian grids to represent cylindrical systems
Flux laws and control volume connectivity
thermal resistance
connecting corners
enthalpy flux laws
equivalence of upwind differencing of advection and one-way conductors
Vectorization
pattern conservation equation
vectorizing this equation
types of vectorizations
sparse conductance matrix
flattened conductance matrix
General flux equation
surveying heat flows in specific directions
flow induced by other vectorized quantities and application to solution methods
Application Study: coldplate discretization example
problem definition
CV numbering and spatial discretization
sparse matrix method
flattened matrix method
Restructuring the sparse equation - coldplate discretization example
singularity of the conduCFDnce matrix
how to make non-singular by recognizing the boundary condition
mathematical insight to the Neumann condition - singularity of G
Flattened matrix method
types of compression
memory pointer method
application and survey of differential heat flow into a CV
Lecture 5 Computational Solutions to the Discrete Equations Solving the conservation equations in unison
Direct solutions to the vectorized conservation equation
the exponential matrix function
simplified steady state solutions
Example: direct solution of vectorized coldplate example
survey of Eigenvalues for steady state
direct single inversion of conductance matrix
Failure of direct methods
sparseness and memory limitations
search for new methods
Introduction to time discretization
Steady state solutions
evaluation for singular conduCFDnce matrix
singularity and the Neumann condition
residual forms
Generalized Newton’s method
effect of relaxation factor
stopping criteria
Transient solution methods - explicit differencing
explicit time differencing and the recursive vectorized conservation equation
demonstration of transient solution instability as a function of time step
introduction to solution stability
Stability analysis
modified conservation equation
stability assessment from Eigenvalues
error propagation equation
survey of Eigenvalues for coldplate example
r-factor assessment
controlling solution stability
Transient solution methods - implicit differencing
Euler forward differencing
vectorized conservation equation
solution of the implicit temperature vector
hybrid direct inversion
Example: coldplate simulation
transient solution at large ?t
stability and Eigenvalue assessment
time step assessment
time accuracy and discussion
Development of general time differencing schemes
Crank-Nicholson differencing for second order accuracy
emphasis on inverting systems Ax = f
turning to numerical solution methods
Basic relaxation methods
application to transient solutions
solution to coldplate example
retardation of solution rate due to conductance matrix
search for semi-direct inversion methods
Survey and application of semi-direct inversion methods
ADI methods
ADI Splitting
Work units compared to SOR
Banding effect caused by Splitting
Inplementation of orthogonal heat flow components
ADI Brian Method
Elimination of banding effect as a steady state solver
Conjugate gradient method
Method development
Pseudo code and discussion of pre-conditioners
Application to coldplate problem and other examples
Lecture 6 Validation of Computational Models and Solutions Validation phase
a philosophy
types of errors
responsibility of the analyst
Numerical consistency analysis
modified equation showing effect of finite CV size
effect of reducing CV size
mesh discretization factor
setup of the method
Example application of numerical consistency analysis
determining solution order from results
Bench marking: exact solutions to benchmark transient solutions
Showcase of various exact solutions with comparisons to numerical schemes
1D slab, sine distribution
1D slab semi infinite in length, error function solution
Burgers equation, steady and transient solutions and wave equations
phase change examples
Energy survey methods for validation
Qualitative energy surveys
approach
emphasis on continual model assessment, checking
applying heat flux vectors for rapid order-level model checks
Quantitative energy surveys
defining subdomains
defining area vectors and flux quantities
example integration of heat flux over a subdomain
equivalent form using divergence theorem
Lecture 7 Reconstitution and Repackaging Computational Models Model evolution and reconstitution
Recognizing the model as an object
Unplugging model from test boundary conditions
Model replication and reissue of boundary conditions
Example replication of the coldplate model with a coupled flow loop
Lecture 8 Special Topics in CFD and CHT Modelling the advection terms from fluid flow
differencing methods
artificial numerical diffusion
numerical dispersion
3rd order upwind method
solution examples
Modelling heat transfer couplings to flow loops
closed loop system and implications
boundary conditions
example application study
Neumann condition satisfied naturally
Erroneous coupling of CFD and CFD solutions
review of the enthalpy flux equation
effects of divergent velocity field (nonconverged CFD solution)
example application
Flux Integration
Review and derivation of 1st order method giving G = k A/?X
higher order differencing methods
2nd Order differencing
example derivation
origin of corner and negative conductors
relation to FEA methods
Radiation heat transfer
simple grey body exchange
energy augment equation
adaptation of SOR
application study - radiant fin pair
rapid radiation connection methods
Phase change heat transfer
interface tracking and enthalpy-porosity methods
modification of vectorized conservation equation
benchmark studies
comparisons to exact solutions
application studies
Hyperbolic heat conduction
discussion of ellipticity
concept of finite wave speed in heat conduction
modification of the Fourier constitutive models
hyperbolicity
conservation laws
final hyperbolic heat conduction equation
wave speed and degeneration
Aeroheating
Heat transfer in rarified flows
Dynamic slosh analysis using CFD VOF methods
