numerical methods in heat mass momentum transfer (lecture notes)jayathimurthy
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ME 608Numerical Methods for Heat, Mass
and Momentum Transfer
Jayathi Y. MurthyProfessor, School of Mechanical Engineering
Purdue [email protected]
Spring 2006
Lecture 1: Introduction to ME 608Conservation Equations
Outline of Lecture
Course organization
Introduction to CFD
Conservation equations, general scalar transport equation
Conservation form
Motivation
Huge variety of industrial flows:
•Rotating machinery
•Compressible/incompressible aerodynamics
•Manifolds, piping
•Extrusion, mixing
•Reacting flows, combustion ….
Impossible to solve Navier-Stokes equations analytically for these applications!
History
Earliest “CFD” work by L.F. Richardson (1910)» Used human computers» Iterative solutions of Laplace’s eqn
using finite-difference methods, flow over cylinder etc.
» Error estimates, extrapolation to zero error
“So far I have paid piece rates for the operation (Laplacian) of about n/18 pence per coordinate point, n being the number of digits … one of the quickest boys averaged 2000 operations (Laplacian) per week for numbers of 3 digits, those done wrong being discounted …”
Richardson, 1910
Also researched mathematical models for causes of war :
Generalized Foreign Politics (1939)
Arms and Insecurity(1949)
Statistics of Deadly Quarrels (1950)
Lewis F. Richardson (1881-1953)
History
Relaxation methods (1920’s-50’s)Landmark paper by Courant, Friedrichs and Lewy for hyperbolic equations (1928)Von Neumann stability criteria for parabolic problems (1950)Harlow and Fromm (1963) computed unsteady vortex street using a digital computer. They published a Scientific American article (1965) which ignited interest in modern CFD and the idea of computer experiments Boundary-layer codes developed in the 1960-1970’s (GENMIX by Patankar and Spalding in 1972 for eg.)Solution techniques for incompressible flows published through the 1970’s (SIMPLE family of algorithms by Patankar and Spalding for eg.)Jameson computed Euler flow over complete aircraft (1981)Unstructured mesh methods developed in 1990’s
John
von Neumann (1903-1957)
Richard Courant (1888-1972)
Conservation Equations
Nearly all physical processes of interest to us are governed by conservation equations» Mass, momentum energy conservation
Written in terms of specific quantities (per unit mass basis)» Momentum per unit mass (velocity)» Energy per unit mass e
Consider a specific quantity φ» Could be momentum per unit mass, energy per unit mass..
Write conservation statement for φ for control volume of size
∆x x ∆y x ∆z
Conservation Equations (cont’d)
Accumulation of φ in control volume over time step ∆t =
Net influx of φ into control volume
- Net efflux of φ out of control volume
+ Net generation of φ inside control volume
Conservation Equations (cont’d)
Accumulation:
Generation:
Influx and Efflux:
Diffusion and Convection Fluxes
Diffusion Flux
Convection Flux
Net flux
Velocity Vector
Diffusion coefficient Γ
Combining…
Taking limit as ∆x, ∆y, ∆z -> 0
General Scalar Transport Equation
Or, in vector form:
Conservation Form
Consider steady state. The conservation form of the scalar transport equation is:
Non-Conservation Form
Finite volume methods always start with the conservation form
General Scalar Transport Equation
Storage Convection Diffusion Generation
Recall: φ is a specific quantity (energy per unit mass say)
V : velocity vector
Γ: Diffusion coefficient
ρ: density
S: Source term (Generation per unit volume W/m3)
Continuity Equation
0)( =⋅∇+∂∂ Vρρ
t
Here,
φ= 1
Γ= 0
S = 0
Energy Equation
h = sensible enthalpy per unit mass, J/kg
k = thermal conductivity
Sh = energy generation W/m3
Note: h in convection and storage terms
T in diffusion terms
How to cast in the form of the general scalar transport equation?
Energy Equation (cont’d)
Equation of State
Substitute to Find
Here,
φ= h
Γ= k/Cp
S = Sh
Momentum Equation
X-Momentum Equation
jiij
j i
uux x
τ µ⎛ ⎞∂∂
= +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠
Here,
φ= u
Γ= µ
S = Su - px∂∂
S is good “dumping ground” for everything that doesn’t fit into the other terms
Species Transport Equation
Yi = kg of specie i /kg of mixture
Γi = diffusion coefficient of i in mixture i
Ri = reaction source
Closure
In this lecture we» Developed the procedure for developing the governing
equation for the transport of a scalar φ» Recognized the commonality of transport of
– Mass, momentum, energy, species» Casting all these different equations into this single form is
very useful» Can devise a single method to solve this class of governing
equation
Lecture 2: The General Scalar Transport Equation
Overview of Numerical Methods
Last time…
Wrote conservation statement for a control volume
Derived a general scalar transport equation
Discovered that all transport processes commonalities» Storage» Diffusion» Convection» Generation
This time…
Examine important classes of partial differential equations and understand their behavior
See how this knowledge applies to the general scalar transport equation
Start a general overview of the main elements of all numerical methods
General Scalar Transport Equation
Storage Convection Diffusion Generation
Recall: φ is a specific quantity (energy per unit mass say)
V : velocity vector
Γ: Diffusion coefficient
ρ: density
S: Source term (Generation per unit volume W/m3)
Classification of PDEs
Consider the second-order partial differential equation for φ (x,y):
Coefficients a,b,c,d,e,f are linear -- not functions of φ, but can be functions of (x,y)
Discriminant
D < 0 Elliptic PDE
D=0 Parabolic PDE
D>0 Hyperbolic PDE
Elliptic PDEs
Consider 1-D heat conduction in a plane wall with constant thermal conductivity
Boundary conditions
Solution:
To
TL
Elliptic PDE’s
To
TL
• T(x) is influenced by both boundaries
• In the absence of source terms, T(x) is bounded by the values on both boundaries
•Can we devise numerical schemes which preserve these properties?
Parabolic PDEs
Consider 1D unsteady conduction in a slab with constant properties:
Boundary and initial conditions Solution:
T0TiT0
Parabolic PDEs (cont’d)
T0TiT0
• The solution at T(x,t) is influenced by the boundaries, just as with elliptic PDEs
•We need only initial condtions T(x,0). We do not need future conditions
•Initial conditions only affect future conditions, not past conditions
• Initial conditions affect all spatial points in the future
• A steady state is reached as t->∞. In this limit we recover the elliptic PDE.
•In the absence of source terms, the temperature is bounded by initial and boundary conditions
•Marching solutions are possible
Hyperbolic PDEs
Consider the convection of a step change in temperature:
Initial and boundary conditions
Solution:
Hyperbolic PDEs (cont’d)
Hyperbolic PDEs (cont’d)
• Upstream conditions can potentially affect the solution at a point x; downstream conditions do not
• Inlet conditions propagate at a finite speed U
•Inlet condition is not felt at location x until a time x/U
Relation to Scalar Transport Equation
• Contains all three canonical PDE terms
• If Re is low and situation is steady, we get an elliptic equation
• If diffusion coefficient is zero , we get a hyperbolic equation
• If Re is low and situation is unsteady, we get a parabolic equation
• For mixed regimes, we get mixed behavior
Components of CFD Solution
Geometry creation
Domain discretization (mesh generation)
Discretization of governing equations
Solution of discrete equations; accounting for non-linearities and inter-equation coupling
Visualization and post-processing
Solution Process
Analytical solution gives us φ(x,y,z,t). Numerical solution gives us φ only at discrete grid points.The process of converting the governing partial differential equation into discrete algebraic equations is call discretization.
Discretization involves » Discretization of space using mesh generation» Discretization of governing equations to yield sets
of algebraic equations
Mesh Types
Regular and body-fitted meshes
Stair-stepped representation of complex geometry
Mesh types (cont’d)
Block-structured meshes
Unstructured meshes
Mesh Types
Non-conformal mesh
Hybrid mesh
Cell shapes
Mesh Terminology
• Node-based finite volume scheme: φ stored at vertex
• Cell-based finite volume scheme: φ stored at cell centroid
Overview of Finite Difference Method
Step 1: Discretize domain using
a mesh.
Unknowns are located at nodes
Step 2: Expand φ in Taylor series about point 2
Subtracting equations yields
Consider diffusion equation:
Finite Difference Method (cont’d)
Step 3: Adding equations yields
Drop truncated terms:
Step 4: Evaluate source term at point 2:
Second order truncation error
Finite Difference Method (cont’d)
Step 5: Assemble discrete equation
Comments» We can write one such equation for each grid point» Boundary conditions give us boundary values» Second-order accurate» Need to find a way to solve couple algebraic equation set
Overview of Finite Volume Method
Consider the diffusion equation:
Step 1: Integrate over control volume
Finite Volume Scheme (cont’d)
Step 2: Make linear profile assumption between cell centroids for φ. Assume S varies linearly over CV
Step 3: Collect terms and cast into algebraic equation:
Comments
Process starts with conservation statement over cell. We find φ such that it satisfies conservation. Thus, regardless of how coarse the mesh is, the finite volume scheme always gives perfect conservation
This does not guarantee accuracy, however.
The process of discretization yields a flux balance involving face values of the diffusion flux, for example:
Profile assumptions for φ and S need not be the same.
eex
φ∂⎛ ⎞−Γ ⎜ ⎟∂⎝ ⎠
Comments (cont’d)
As with finite difference method, we need to solve a set of coupled algebraic equations
Though finite difference and finite volume schemes use different procedures to obtain discrete equations, we can use the same solution techniques to solve the discrete equations
Closure
In this lecture we
Considered different canonical PDEs and examined their behavior
Understood how these model equations relate to our general scalar transport equations
Started an overview of the important elements of any numerical method
In the next lecture we will complete this overview and start looking more closely at the finite volume method for diffusion problems.
Lecture 3: Overview of Numerical Methods
Last time…
Examined important classes of partial differential equations and understood their behavior
Saw how this knowledge would apply to the general scalar transport equation
Started an overview of numerical methods including mesh terminology and finite difference methods
This time…
We will continue the overview and examine
Finite difference, finite volume and finite element methods
Accuracy, consistency, stability and convergence of a numerical scheme
Overview of Finite Element Method
Consider diffusion equation
Let be an approximation to φSince is an approximation, it does not satisfy the diffusion equation, and leaves a residual R:
Galerkin finite element method minimizes R with respect to a weight function:
φφ
2
2
d S Rdxφ
Γ + =
Finite Element Method (cont’d)
A family of weight functions Wi, I = 1,…N, (N: number of grid points) is used. This generates N discrete equations for the N unknowns:
Weight function is local – i.e. zero everywhere except close to i
i+1wi
Element iElement i-1
ii-1
Finite Element Method (cont’d)
In addition a local shape function Ni is used to discretize R. Under a Galerkin formulation, the weight and shape functions are chosen to be the same.
Ni
Shape function is non-zero only in the vicinity of node i => “local basis”
Ni-1
i+1i-1 iElement iElement i-1
Finite Element Method (cont’d)
The discretization process again leads to a set of algebraic equations of the form:
Comments» Note how the use of a local basis restricts the relationship
between a point i and its neighbors to only nearest neighbors» Again, we have an algebraic equation set to solve – can use
the same solvers as for finite volume and finite difference methods
, , 1 1 , 1 , 1i i i i i i i i i i ia a a bφ φ φ+ + − −= + +
Comparison of methodsAll three yield discrete algebraic equation sets which must be solved
Local basis – only near-neighbor dependence
Finite volume method is conservative; the others are not
Order or accuracy of scheme depends on » Taylor series truncation in finite difference schemes» Profile assumptions in finite volume schemes» Order of shape functions in finite element schemes
Solution of Linear Equations
Linear equation set has two important characteristics» Matrix is sparse, may be banded» Coefficients are provisional for non-linear problems
Two different approaches» Direct methods» Iterative methods
Approach defines “path to solution”» Final answer only determined by discretization
Direct MethodsAll discretization schemes lead to
Here φ is solution vector [φ1 , φ2 ,…, φN]T.Can invert:
Inversion is O(N3 ) operation. Other more efficient methods exist.» Take advantage of band structure if it exists» Take advantage of sparsity
Direct Methods (cont’d)
Large storage and operation count » For N grid points, must store NxN matrix » Only store non-zero entries and fill pattern
For non-linear problems, A is provisional and is usually updated as a part of an outer loop» Not worth solving system too “exactly”
As a result, direct methods not usually preferred in CFD today
Iterative Methods
Guess and correct philosophyGauss-Seidel scheme is typical:» Visit each grid point
Update using
» Sweep repeatedly through grid points until convergence criterion is met
» In each sweep, points already visited have new values; points not yet visited have old values
Iterative Methods (cont’d)
Jacobi scheme is similar to Gauss-Seidel scheme but does not use latest available values» All values are updated simultaneously at end of sweep.
Iterative are not guaranteed to converge to a solution unless Scarborough criterion is satisfied
Scarborough Criterion
Scarborough criterion states that convergence of an iterative scheme is guaranteed if:
This means that coefficient matrix must be diagonally dominant
Gauss-Seidel Scheme
No need to store coefficient matrix
Operation count per sweep scales as O(N)
However, convergence, even when guaranteed, is slow for large meshes
Will examine alternatives later in course
Accuracy
While looking at finite difference methods, we wrote:
Halving grid size reduces error by factor of four for second-order schemeCannot say what absolute error is – truncation error only gives rate of decrease
Second-order truncation error
Accuracy
Order of discretization scheme is n if truncation error is O(∆xn )When more than one term is involved, the order of the discretization scheme is that of the lowest order term.
Accuracy is a property of the discretization scheme, not the path to solution
Consistency
A discretization scheme is consistent if the truncation error vanishes as ∆x ->0
Does not always happen: What if truncation error is O(∆x/∆t) ?Consistency is a property of the discretization scheme, not the path to solution
Convergence
Two uses of the term» Convergence to a mesh-independent solution through mesh
refinement» Convergence of an iterative scheme to a final unchanging
answer (or one meeting convergence criterion)We will usually use the latter meaning
Stability
Property of the path to solution Typically used to characterize iterative schemesDepending on the characteristics of the coefficient matrix, errors may either be damped or may grow during iterationAn iterative scheme is unstable if it fails to produce a solution to the discrete equation set
Stability
Also possible to speak of the stability of unsteady schemes» Unstable: when solving a time-dependent problem,
the solution “blows up”Von-Neumann (and other) stability analyses determine whether linear systems stable under various iteration/time-stepping schemesFor non-linear/coupled problems, stability analysis is difficult and not much used» Take guidance from linear analysis in appropriate
parameter range; intuition
Closure
This time we completed an overview of the numerical discretization and solution process » Domain discretization» Discretization of governing equations» Solution of linear algebraic set» Properties of discretization and path to solution
– Accuracy, consistency, convergence, stability
Next time, we will start looking at finite volume discretization of diffusion equation
Lecture 4: The Diffusion Equation – A First Look
Last Time…
We completed an overview of the numerical discretization and solution process » Domain discretization» Discretization of governing equations – finite
difference, finite volume, finite element» Solution of linear algebraic set» Properties of discretization and path to solution
– Accuracy, consistency, convergence, stability
This Time…
We willApply the finite volume scheme to the steady diffusion equation on Cartesian structured meshesExamine the properties of the resulting discretizationDescribe how to discretize boundary conditions
2D Steady Diffusion
• Consider steady diffusion with a source term:
• Here
• Integrate over control volume to yield
2D Steady Diffusion
Apply divergence theorem to yield
Discrete Flux Balance
• Writing integral over control volume:
•Compactly:
Discrete Flux Balance (cont’d)
Area vectors given by:
Fluxes given by
DiscretizationAssume φ varies linearly between cell centroids
Note:» Symmetry of (P, E ) and
(P,W) in flux expression» Opposite signs on (P,E)
and (P,W) terms
Source Linearization
Source term must be linearized as:
Assume SP <0
Final Discrete Equation
P
N
S
EW
Comments
Discrete equation reflects balance of flux*area with generation inside control volume
As in 1-D case, we need fluxes at cell faces
These are written in terms of cell-centroid values using profile assumptions.
Comments (cont’d)
Formulation is conservative: Discrete equation was derived by enforcing conservation. Fluxes balance source term regardless of mesh density
For a structured mesh, each point P is coupled to its four nearest neighbors. Corner points do not enter the formulation.
Properties of Discretization
aP, anb have same sign: This implies that if neighbor φgoes up, φP also goes up
If S=0:
Thus φ is bounded by neighbor values, in keeping with properties of elliptic partial differential equations
Properties of Discretization (cont’d)
What about Scarborough Criterion ?Satisfied in the equality
What about this?
Boundary Conditions
Flux Balance
Different boundary conditions require different representations of Jb
Dirichlet BCs
Dirichlet boundary condition:
φb = φgiven
Put in the requisite flux into the near-boundary cell balance
Dirichlet BC’s (cont’d)
P nbnb
a a>∑
For near-boundary cells:
Satisfies Scarborough Criterion !
Also, φP bounded by interior neighbors and boundary value in the absence of source terms
Neumann BC’s
Neumann boundary conditions : qb given
Replace Jb in cell balance with given flux
Neumann BC’s (cont’d)
P nbnb
a a=∑For Neumann boundaries
So inequality constraint in Scarborough criterion is not satisfied
Also, φP is not bounded by interior neighbors and boundary value even in the absence of source terms – this is is fine because of the added flux at the boundary
Boundary Values and Fluxes
Once we solve for the interior values of φ, we can recover the boundary value of the flux for Dirichletboundary conditions using
Similarly, for Neumann boundary conditions, we can find the boundary value of φ using
Closure
In this lecture we» Described the discretization procedure for the
diffusion equation on Cartesian meshes» Saw that the resulting discretization process
preserves the properties of elliptic equations» Since we get diagonal dominance with Dirichlet bc,
the discretization allows us to use iterative solversNext time, we will look at one more boundary condition (Robbins or mixed bc), source linearization and conjugate heat transfer