The Control-Volume Finite-Di erence Approximation to...

The Control-Volume Finite-DifferenceApproximation to the Diffusion Equation

ME 448/548 Notes

Gerald Recktenwald

Portland State University

Department of Mechanical & Materials Engineering

[email protected]

21 January 2014

ME 448/548: 2D Diffusion Equation

Motivation

Numerical solution of the 2D Poisson equation is the next step in developing our

knowledge of CFD technique.

• Introduce the Finite Volume Method

. Naturally deal with material discontinuity

. Can naturally enforce conservation of mass and energy

. Core idea in many (not all) commercial CFD codes

• Extend analysis to two spatial dimensions

. More interesting practical applications

. More complex data structures

. More complex procedures to solve the Ax = b problem

ME 448/548: 2D Diffusion Equation 1

Overview

Goals for this unit

• Introduce the Poisson equation

. A model of steady heat conduction with a source term

. Form of the pressure equation for incompressible flow

. Precursor to the generalized advection diffusion equation

• Use the finite volume method to obtain discrete equations

• Allow for non-uniform mesh, diffusion coefficient, and source term.

• Introduce a set of Matlab codes for 2D Control Volume Finite Difference (CVFD)

• Demonstrate solutions for three model problems.

• Measure truncation error for a model problem with a simple solution

• Apply to fully-developed flow in rectangular ducts


Model Problem

The two dimensional diffusion equation in Cartesian coordinates is

∂

∂x

(Γ∂φ

∂x

)+∂

∂y

(Γ∂φ

∂y

)+ S = 0 (1)

where φ is the scalar field, Γ is the diffusion coefficient, and S is the source term.

Example: Heat conduction in a rectangular domain

Γ = k, thermal conductivity or Γ =k

ρcp= α

S = volumetric heat source


2D Cartesian Finite Volume Mesh

x

y

1 2 nx

i = 0 2j = 0

2

3

3 nx

ny

xu(1) = 0

x(i)

y(j)

xu(i)

1

yv(j)

yv(ny+1)

xu(nx+1)

1

nx+1 nx+2

nx ny

yv(1) = 0

Interior node

Boundary node

Ambiguous corner node


Finite Volume Mesh: Compass Point Notation

Nodes are labelled with their position

relative to the typical point “P”.

N

S

EW

x

y

Capital letters designate node locations.

N, S, E, W are the names of north,

south, east and west neighbors.

φN , φS, φE, φW are the values of φ at

the north, south, east and west neighbor

nodes.

Use of compass point names simplifies

the algebra. For example the value of φ

at point N is φN or φi,j+1, but φN is

simpler to write.

Compass point notation is an historic

convention that does not extend to

unstructured meshes.


Finite Volume Mesh

Detailed notation for a typical 2D control

volume

δxw δx

e

S

PW E

N

∆x

∆y

xexw

yn

ys

δys

δyn

Lowercase letters designate the interface

between the nodes.

xe and xw are the x coordinates of the

east and west faces of the control

volume around P

yn and ys are the y coordinates of the

north and south faces of the control

volume around P

Similarly, φn, φs, φe, φw are the values

of φ at the north, south, east and west

control volume interfaces.


Finite Volume Mesh


volume

δxw δx

e

S

PW E

N

∆x

∆y

xexw

yn

ys

δys

δyn

Regardless of whether the control

volumes sizes are uniform, node P is

always located in the geometric center

of each control volume.

Thus,

xP − xw = xe − xP =∆x

2

yP − ys = yn − yP =∆y

2

for uniform or non-uniform meshes


Finite Volume Mesh


volume

δxw δx

e

S

PW E

N

∆x

∆y

xexw

yn

ys

δys

δyn

In contrast, the distances between the

nodes is not assumed to be uniform

δxe = xE − xP 6= δxw = xP − xWδyn = yN − yP 6= δys = yP − yS

δxe = xE − xP 6= δxw = xP − xWδyn = yN − yP 6= δys = yP − yS

Note the use of upper and lower case

subscripts: Upper case refers to nodes;

lower case refers to interfaces.


Convert the Differential Equation to a Discrete Equation

Integrate over the control volume

∫ yn

ys

∫ xe

xw

∂

∂x

(Γ∂φ

∂x

)dx dy

=

∫ yn

ys

[(Γ∂φ

∂x

)e

−(

Γ∂φ

∂x

)w

]dy

≈[(

Γ∂φ

∂x

)e

−(

Γ∂φ

∂x

)w

]∆y

≈[ΓeφE − φPδxe

− ΓwφP − φWδxw

]∆y

δxw δx

e

S

PW E

N

∆x

∆y

xexw

yn

ys

δys

δyn

The final step is obtained by using central difference approximations for the derivatives at

the interfaces.



Integrate the source term ∫ xe

xw

∫ yn

ys

S dy dx ≈ SP∆x∆y (2)

Note: The Control Volume Finite Difference (CVFD) method treats the source term and

diffusion coefficients as piecewise constants. This is a rather crude approximation, say,

compared to allowing the source term and diffusion coefficient to vary linearly within the

control volume.

However, piecewise constant profiles of S and Γ allow the method to be conservative,

i.e., conserving mass or energy, automatically. The conservative nature of the CVFD

method is one of its primary strengths.



Putting pieces back into the model equation gives the discrete system of equations

−aSφS − aWφW + aPφP − aEφE − aNφN = b (3)

where

aE =Γe

∆x δxe, aW =

Γw

∆x δxw, aN =

Γn

∆y δyn, aS =

Γs

∆y δys

aP = aE + aW + aN + aS (4)

b = SP (5)

This is not a tridiagonal system of equations.


Non-uniform Γ

Continuity of fluxes at the interface requires

ΓP∂φ

∂x

∣∣∣∣xe−

= ΓE∂φ

∂x

∣∣∣∣xe+

= Γe∂φ

∂x

∣∣∣∣xe

Use central difference approximations

ΓeφE − φPδxe

= ΓPφe − φPδxe−

(6)

ΓeφE − φPδxe

= ΓEφE − φeδxe+

(7)δx

e

material 1 material 2

δxe+

δxe

P E


Non-uniform Γ

Equations 6 and 7 can be rearranged as

φe − φP =δxe−

ΓP

Γe

δxe(φE − φP ) (8)

φE − φe =δxe+

ΓE

Γe

δxe(φE − φP ) (9)

Add Equation 8 and Equation 9

φE − φP =Γe

δxe(φE − φP )

[δxe−

ΓP+δxe+

ΓE

].

δxe

material 1 material 2

δxe+

δxe

P E

Cancel the factor of (φE − φP ) and solve for Γe/δxe to get

Γe

δxe=

[δxe−

ΓP+δxe+

ΓE

]−1

=ΓE ΓP

δxe−ΓE + δxe+ΓP.


Non-uniform Γ

Thus, the diffusion coefficient at the interface that results in flux continuity is

Γe =ΓE ΓP

βΓE + (1− β)ΓP(10)

where

β ≡δxe−

δxe=xe − xPxE − xP

(11)

An analogous derivation gives formulas for Γw, Γn, and Γs.


Solving the System of Equations

Regardless of uniform or variable Γ, the discrete equation has a five-point stencil, and the

discrete equation for any interior node can be written.

−aSφS − aWφW + aPφP − aEφE − aNφN = b (12)

To set up the matrix for this system of equations, we need to re-number the unknowns.

Important: i and j subscripts for the mesh are not the same as the row and

column indices in the system Ax = b.



Order the nodes:

n = i+ (j − 1)nx

n is the node number (row number)

in Ax = b. i and j are the mesh

indices corresponding to xi and yj.

With natural ordering the neighbors

in the compass point notation have

these indices

np = i + (j-1)*nx

ne = np + 1

nw = np - 1

nn = np + nx

ns = np - nx

x

y

1 2 nx

i = 0 2j = 0

2

3

3 nx

ny

xu(1) = 0

x(i)

y(j)

xu(i)

1

yv(j)

yv(ny+1)

xu(nx+1)

1

nx+1 nx+2

nx ny

yv(1) = 0

Interior node

Boundary node

Ambiguous corner node



This leads to a vector of unknowns

φ1,1

φ2,1...

φnx,1φ1,2

φ2,2...

φi,j...

φnx,ny

⇐⇒

φ1

φ2...

φnxφnx+1

φnx+2...

φn...

φN

(13)



The coefficient matrix has 5 non-zero diagonals

A = a S apa W a E a N


Algorithm for obtaining the numerical solution

1. Define physical parameters: Lx, Ly, Γ(x, y) and boundary conditions

2. Define the mesh: nx and ny if uniform

3. Compute the coefficient matrix

4. Solve the system of equations Aφ = b, where φ is the vector of unknowns.

5. Post-process to visualize the solution

The Poisson equation is steady. Each step is performed only once.


Structured Mesh

The CVFD Matlab codes use structured meshes. in size.

• The cells in the domain are topologically equivalent to a rectangular array

• The cells need not be uniform in size.

• Each cell not on a boundary touches four other cells

• Each row has the same number of cells

• Each column has the same number of cells

Uniform Mesh Block-Uniform Mesh

Lx1, nx1 Lx2, nx2

Ly1, ny1

Ly2, ny2

Ly3, ny3

x

y

∆x

∆y


Boundary Conditions (part 1)

Boundarytype

BoundaryCondition Post-processing in fvpost

1 Specified T Compute q′′ from discrete approximation to Fourier’s law.

q′′

= kTb − Tixb − xi

where Ti and Tb are interior and boundary temperatures, respectively.

2 Specified q′′ Compute Tb from discrete approximation to Fourier’s law.

Tb = Ti + q′′ xb − xi

kwhere Ti and Tb are interior and boundary temperatures, respectively.


Boundary Conditions (part 2)

Boundarytype

BoundaryCondition Post-processing in fvpost

3 Convection From specified h and T∞, compute boundary temperature and heat fluxthrough the cell face on the boundary. Continuity of heat flux requires

−kTb − Tixb − xi

= h(Tb − Tamb)

which can be solved for Tb to give

Tb =hTamb + (k/δxe)Ti

h+ (k/δxe)

where δxe = xb − xi

4 Symmetry q′′ = 0. Set boundary Tb equal to adjacent interior Ti.


Matlab codes for obtaining the numerical solution

A set of general purpose codes has been written to facilitate experimentation with the

CVFD method.

Algorithm Tasks Core Routines

Define the mesh fvUniformMesh or

fvUniBlockMesh

Define boundary conditions

Compute finite-volume coefficients for interior cells fvcoef

Adjust coefficients for boundary conditions fvbc

Solve system of equations

Assemble coefficient matrix fvAmatrix

Solve

Compute boundary values and/or fluxes fvpost

Plot results


Model Problem 1

Choose a source term that may be physically unrealistic, but one that gives an exact

solution that is easy to evaluate

S =

[(π

Lx

)2

+

(2π

Ly

)2]

sin

(πx

Lx

)sin

(2πy

Ly

)

The exact solution is

φ = sin

(πx

Lx

)sin

(2πy

Ly

)Main code to solve this problem is in demoModel1.m


Model Problem 1

The exact solution is

φ = sin

(πx

Lx

)sin

(2πy

Ly

)

0

0.5

1

00.2

0.40.6

0.81

0

0.5

1

1.5

2

yx


Solutions to Model Problem 1 Show Correct Truncation Error

The local truncation error at each node is

ei ∼ O(∆x2).

Since the exact solution is known we can

compute

‖e‖2

N=

√∑e2i

N∼√Ne2

N=

e√N.

where N = nxny is the total number of

interior nodes in the domain, and e is the

average truncation error per node.10

−310

−210

−110

010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

∆ x

Me

asu

red

err

or

Measured

Theoretical error ~ (∆ x)3

Since ei ∼ O(∆x2), N ∼ n2x, and ∆x = Lx/(nx + 1), we can estimate

‖e‖2

N∼

e√N

=O(∆x2)

nx=O(L2x/(nx + 1)2

)nx

∼ O(

1

nx

)3

= O(∆x3).


Model Problem 2

Uniform source term: S = 1.

Analytical solution is an infinite

series

Code in demoModel2.m

0

0.5

1

00.2

0.40.6

0.81

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

yx


Model Problem 3

Heat conduction in a rectangle consisting of

two material regions.

• Inner rectangle with high conductivity

• Outer rectangle with low conductivity

• Inner region has uniform heat source

0.25Lx

0.25Ly

0.5Ly

0.5Lx

Lx

Ly

Γ1 = 1

S1 = 0

Γ2 = αΓ

1

S2 = 1000

The discontinuity and difference in material properties can be used to stress the solution

algorithm.

α =Γ2

Γ1

The analytical solution does not exist. Code in demoModel3.m


Model Problem 3

0.25Lx

0.25Ly

0.5Ly

0.5Lx

Lx

Ly

Γ1 = 1

S1 = 0

Γ2 = αΓ

1

S2 = 1000

Solution with α = 100

0

0.5

1

00.2

0.40.6

0.81

0

5

10

15

20

25

30

35

yx


Model Problem 4: Fully Developed Flow in a Rectangular Duct

For simple fully-developed flow the governing equation for the axial velocity w is

µ

[∂2w

∂x2+∂2w

∂y2

]−dp

dz= 0

This corresponds to the generic model equation with

φ = w, Γ = µ (= constant), S = −dp

dz.



The symmetry in the problem allows alternative ways of defining the numerical model

Full Duct

x

y

Lx

Ly

Quarter Duct

x

y

Lx

Ly

For the full duct simulation depicted on the left hand side, the boundary conditions are no

slip conditions on all four walls.

w(x, 0) = w(x, Ly) = w(0, y) = w(Lx, y) = 0. (full duct)



The symmetry in the problem allows alternative ways of defining the numerical model

Full Duct

x

y

Lx

Ly

Quarter Duct

x

y

Lx

Ly

For the quarter duct simulation depicted on the right hand side, the boundary conditions

are no slip conditions on the solid walls (x = Lx and y = Ly)

w(Ly, y) = w(x, Lx) = 0 (quarter duct)

and symmetry conditions on the other two planes

∂u

∂x

∣∣∣∣x=0

=∂u

∂y

∣∣∣∣y=0

= 0.


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