The Finite Volume Method for Convection-Diffusion Problems

Chapter 5

The Finite Volume Method for Convection-Diffusion Problems

Prepared by: Prof. Dr. I. Sezai Eastern Mediterranean University Mechanical Engineering Department

Introduction

The steady convection-diffusion equation is

div(u) = div(grad) + S

Integration over the control volume gives :

n (u)dA = n (grad)dA + SdV

A

A

CV

This equation represents the flux balance in a control volume.

The main problem in the discretisation of the convective terms is the calculation of at CV faces and its convective flux across these boundaries.

Diffusion process affects the distribution of in all directions.

Convection spreads influence only in the flow direction. This sets a limit on the grid size for stable convection-diffusion calculations with central difference method.

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Steady one-dimensional convection and diffusion

In the absence of sources, the steady convection and diffusion of a property in a given one-dimensional flow field u is governed by

d (u) = d ( d )

(5.3)

dx

dx dx

The flow must also satisfy continuity, so

d (u) = 0 dx

Integrating Eqn. (5.3) over the CV

( uA )e

-

( uA ) w

=

A

x

e

-

A

x

w

Integrating continuity Eqn.

(5.5)

(uA)e - (uA)w = 0

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(5.6)

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Let F = uA convective mass flux at cell faces D = A/x diffusion conductance

At cell faces: Fw = (uA)w

Fe = (uA)e

Dw

=

w Aw xWP

De

=

e Ae xPE

Using central difference approach for the diffusion terms, Eqn (5.5)

becomes

Fee - Fww = De (E - P ) - Dw (P - W ) Continuity equation becomes

Fe - Fw = 0 We assume that velocity field is known Fe, Fw known.

We need to calculate at faces e and w.

(5.9) (5.10)

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The Central Differencing Scheme

Works well for diffusion terms.

Let us use this method to compute the convective terms by linear interpolation.

For a uniform grid, cell face values are:

e = (P + E ) / 2 w = (W + P ) / 2

Substituting into eqn (5.9)

Fe 2

(P

+ E )

-

Fw 2

(W

+ P )

=

De (E

-P )

-

Dw (P

- W

)

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Rearranging,

Dw

-

Fw 2

+

De

+

Fe 2

P

=

Dw

+

Fw 2

W

+

De

-

Fe 2

E

Dw

+

Fw 2

+

De

-

Fe 2

+ (Fe

-

Fw

)

P

=

Dw

+

Fw 2

W

+

De

-

Fe 2

E

which is of the form aPP = aWW + aEE

where

(5.14)

aW

Dw

+

Fw 2

aE

De

-

Fe 2

aP aW + aE + (Fe - Fw )

This equation has the same general form as the diffusion eqn. (4.11).

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Example 5.1

A property is transported by convection and diffusion through the one dimensional domain shown below. Using central difference scheme, find the distribution of for (L =1, = 1, = 0.1)

(i) Case 1: u = 0.1 m/s (use 5 CV's)

(ii) Case 2: u = 2.5 m/s (use 5 CV's)

Compare the results with the analytical solution.

-o = exp(ux / ) -1 L -o exp(uL / ) -1

(iii) Case 3: u = 2.5 m/s (20 CV's)

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The governing equation is:

A

d dx

( u )

=

d dx

d dx

B

12

3 w4 e5

67

=1

W

P

E

=0

x/2

xWP= x xPE=x

x/2

x

aPP = aWW + aEE + Su

where aP = aW + aE + (Fe - Fw ) - SP

Fw = (uA)w

Fe = (uA)e

Dw

=

w Aw xWP

De

=

e Ae xPE

For interior nodes: xWP = xPE = x For node 2: xWP = x / 2 For node 6: xPE = x / 2

Node 2

3, 4,5 6

aW 0 Dw + Fw / 2 Dw + Fw / 2

aE De - Fe / 2 De - Fe / 2

0

SP -(Dw + Fw / 2)

0 -(De - Fe / 2)

Su (Dw + Fw / 2)A

0 (De - Fe / 2)B

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The resulting system of equations are

-

aP2

aW3

aE 2 - a p3 aW4

aE 3 -a p4

aE 4 aWi

-a pi

aE i aWn-2

-a pn-2 aWn-1

aE n-2 -a pn-1

2

3

4

i

n-2 n-1

=

-Su2

-Su3

-Su4

-Sui

-Sun

-

2

-Sun-1

Solve the system of equations using Tri-diagonal matrix algorithm (TDMA) for 2, 3, 4, ... n-1 , where (n = 7)

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The solution for case 1 is:

1 1

2

0.9421

34

=

0.8006 0.6276

5

6

0.4163

0.1573

7 0

Exact solution is:

(x) = 2.7183 - exp(x) 1.7183

Comparison of the numerical result with the analytical solution.

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