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1 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 φ ρφ φ = Γ + Integration over the control volume gives : ( ) ( ) dA grad dA S dV ρφ φ Γ + n u n ( ) ( ) A A CV dA grad dA S dV φ ρφ φ = Γ + n u n This equation represents the flux balance in a control volume. The main problem in the discretisation of the convective terms is the calculation of φ at CV faces and its convective flux across these boundaries. I. Sezai - Eastern Mediterranean University ME555 : Computational Fluid Dynamics 2 Diffusion process affects the distribution of φ 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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• 1

Chapter 5

The Finite Volume Method for Convection-Diffusion Problems

Prepared by: Prof. Dr. I. SezaiEastern Mediterranean UniversityMechanical Engineering Department

( ) ( )div u div grad Sφρφ φ= Γ +

Integration over the control volume gives :( ) ( )dA grad dA S dVρφ φΓ +∫ ∫ ∫n u n( ) ( )

A A CV

dA grad dA S dVφρφ φ⋅ = ⋅ Γ +∫ ∫ ∫n u n

This equation represents the flux balance in a control volume.

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

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 2

Diffusion process affects the distribution of φ 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.

• 2

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

( ) ( )d d dudx dx dx

φρ φ = Γ (5.3)dx dx dx

The flow must also satisfy continuity, so

( ) 0d udx

ρ =

Integrating Eqn. (5.3) over the CV

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 3

( ) ( )e we w

uA uA A Ax xφ φρ φ ρ φ ∂ ∂⎛ ⎞ ⎛ ⎞− = Γ − Γ⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

Integrating continuity Eqn.

( ) ( ) 0e wuA uAρ ρ− =

(5.5)

(5.6)

Let F = ρuA convective mass flux D = ΓA/δx diffusion conductance

at cell faces

At cell faces:( ) ( )w w e eF uA F uA

A Aρ ρ= =Γ Γ w w e ew e

WP PE

Γ Γ= =

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

( ) ( )e e w w e E P w P WF F D Dφ φ φ φ φ φ− = − − −

C i i i b(5.9)

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 4

0e wF F− =

Continuity equation becomes

We assume that velocity field is known → Fe, Fw known. We need to calculate φ at faces e and w.

(5.10)

• 3

The Central Differencing Scheme

Works well for diffusion terms.

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

For a uniform grid, cell face values are:

( ) / 2( ) / 2

e P E

w W P

φ φ φφ φ φ

= += +

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 5

Substituting into eqn (5.9)

( ) ( ) ( ) ( )2 2

e wP E W P e E P w P W

F F D Dφ φ φ φ φ φ φ φ+ − + = − − −

Rearranging,

2 2 2 2w e w e

w e P w W e EF F F FD D D D

F F F F

φ φ φ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− + + = + + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞( )

2 2 2 2w e w e

w e e w P w W e EF F F FD D F F D Dφ φ φ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + − + − = + + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

which is of the formP P W W E Ea a aφ φ φ= +

where

(5.14)

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 6

( )2 2

W E P

w ew e W E e w

a a aF FD D a a F F+ − + + −

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

• 4

Example 5.1

A property φ is transported by convection and diffusion through the one dimensional domain shown below. Using central differencescheme, find the distribution of φ for (L =1, ρ = 1, Γ = 0.1)scheme, find the distribution of φ for (L 1, ρ 1, Γ 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.exp( / ) 1o uxφ φ ρ− Γ −=

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 7

exp( / ) 1L o uLφ φ ρ− Γ −(iii) Case 3: u = 2.5 m/s (20 CV’s)

The governing equation is: ( )d d dudx dx dx

φρ φ ⎛ ⎞= Γ⎜ ⎟⎝ ⎠A B

1 2 3 4 5 6 7φ = 1 W P E

ewφ = 0

P P W W E E ua a a Sφ φ φ= + +where ( )P W E e w Pa a a F F S= + + − −

( ) ( )w w e ew w e e

F uA F uAA AD D

ρ ρ= =Γ Γ

For interior nodes: For node 2: / 2For node 6: / 2

WP PE

WP

PE

x x xx xx x

δ δ δδ δδ δ

= ==

=

δxδx/2 δxWP= δx δx/2δxPE=δx

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 8

2 0 / 2 ( / 2) ( / 2)3, 4,5 / 2 / 2 0 0

6 / 2 0 ( / 2) ( / 2)

W E P u

e e w w w w A

w w e e

w w e e e e B

Node a a S SD F D F D F

D F D FD F D F D F

φ

φ

− − + ++ −+ − − −

w w e ew eWP PE

D Dx xδ δ

= =

• 5

The resulting system of equations are

2 2 2 2P E

W p E

a a Sua a a Su

φφ

−⎡ ⎤ −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥3 3 3

4 4 4

2 2 2

3 3

4 4

2 2

i i i

n n n

W p E

W p E

i iW p E

n nW p E

Sua a a Su

Sua a a

Sua a a

φφ

φ

φ− − −

− −

⎢ ⎥ −⎢ ⎥ ⎢ ⎥⎢ ⎥− ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥−−⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥− ⎢ ⎥ ⎢ ⎥⎢ ⎥

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 9

1 11 1

n n n

n nn nW p

Sua a φ− −

− −

⎢ ⎥ ⎢ ⎥⎢ ⎥ −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦−⎢ ⎥⎣ ⎦

Solve the system of equations using Tri-diagonal matrix algorithm (TDMA) for φ2, φ 3, φ 4, … φ n-1 , where (n = 7)

The solution for case 1 is:

1

2

10.9421

φφ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥2

3

4

5

6

7

0.94210.80060.62760.41630.1573

0

φφφφφφ

⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 10

7φ ⎣ ⎦⎣ ⎦

Exact solution is:

2.7183 exp( )( )1.7183

xxφ −=Comparison of the numerical result with the analytical solution.

• 6

The solution for case 2: (u = 2.5 m/s, 5 CV’s)

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 11

Comparison of the numerical result with the analytical solution.

The solution appears to oscillate about the exact solution.

The solution for case 3: (u = 2.5 m/s, 20 CV’s)

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 12

Comparison of the numerical result with the analytical solution.

Grid refinement has reduced the F/D ratio from 5 to 1.25. Central difference scheme yields accurate results when F/D ratio is low.

• 7

Properties of Discretisation Schemes

The numerical results will only be physically realistic when the discretisation scheme has certain fundamental properties. The most important ones are:

• Conservativeness

• Boundednes

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 13

• Transportiveness

1. ConservativenessTo ensure conservation of φ for the whole solution domain the flux of φ leaving a CV across a certain face must be equal to the flux of φ entering the adjacent CV through the same faceadjacent CV through the same face.To achieve this the flux through a common face must be represented in a consistent manner (by one and the same expression) in adjacent CV’s.

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 14

• 8

Example of consistent specification of diffusive fluxes

2 2 1( )wx

φ φδ

Γ − 2 3 2( )ex

φ φδ

Γ −

Flux entering CV 2 Flux leaving CV 2An overall flux balance may be obtained by summing the net flux through each CV

3 22 1 2 11 2 2

( )( ) ( )e A e wqx x x

φ φφ φ φ φδ δ δ

−− −⎡ ⎤⎡ ⎤Γ − + Γ −Γ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 15

4 3 3 2 4 33 3 4

( ) ( ) ( )e w B w B Aq q qx x x

φ φ φ φ φ φδ δ δ− − −⎡ ⎤ ⎡ ⎤+ Γ −Γ + −Γ = −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Γe1 = Γw2, Γe2 = Γw3 and Γe3 = Γw4 → Fluxes across CV faces are expressed in consistent manner,

→ fluxes cancel out in pairs when summed over the entire domain.

Flux Consistency ensures conservation of φ over the entire domain for the central difference formulation of the diffusive flux.

Inconsistent flux interpolation formulae give rise to unsuitable schemes that do not satisfy overall conservation.

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 16

For nodes 1, 2 and 3 → quadratic function 1 is used. For nodes 2, 3 and 4 → quadratic function 2 is used.If gradient of 1 ≠ gradient of 2 at cell face → flux leaving CV 2 will not be equal to flux entering CV 3→ overall conservation is not satisfied.

• 9

2) Boundedness

The sufficient condition for a convergent iterative method is

1 at all nodes< 1 at one node at least

nbaa

≤⎧⎨′ ⎩

∑P P pa a S′ = − (5.22)< 1 at one node at leastPa ⎩

If eqn. (5.22) is satisfied, resulting matrix coefficients are diagonally dominant.

For diagonal dominance, (aP – Sp) should be large and Sp < 0.

Diagonal dominance is a desirable feature for satisfying the boundednesscriterion.

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 17

This states that in the absence of sources the internal nodal values of φshould be bounded by its boundary values.

In a steady conduction problem without sources for which the boundary temperatures are 200 and 500 oC, all interior values of T should be between these temperatures.

Another essential requirement for boundedness is that all coefficients of the discretised equations should have the same sign.

If the discretisation scheme does not satisfy the boundedness criteria the solution may not converge at all Or if it converges it will containthe solution may not converge at all. Or if it converges it will contain wiggles. (See case 2 of Example 5.1). In case 2 most of the aE values were negative (Table 5.3).

Node

2

Table 5.3

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 18

2 2e e e e e

E ePE

F A u Aa Dx

ρδΓ

= − = −

23456

• 10

3) TransportivenessThe transportiveness property of a fluid flow can be illustrated by considering a constant source of φ at a point P

cell Peclet number/

F uPeD x

ρδ

= =Γ

Distribution of φ in the vicinity of a source at different Peclet numbers.

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 19

Lines represent contours of constant φ.

For no convection and pure diffusion Pe = 0

For no diffusion and pure convection Pe → ∞, φE = φP E is influenced only by P.

Assesment of the Central Differencing Scheme for Convection Diffusion Problems

ConservativenessThe central differencing scheme uses consistent expressions to evaluate convective and diffusive fluxes at the CV faces.

The scheme is conservative.

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 20

• 11

Boundedness

(i) The internal coefficients of discretised scalar transport equation (5.14) are

W E Pa a a

( )2 2w e

w e W E e wF FD D a a F F+ − + + −

(Fe – Fw) = 0 from continuity → aP = aW + aEThus, convergence criteria (5.22) is satisfied by the central difference scheme

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 21

scheme.

In the example of section 5.3:For case 2: Pe = 5 → oscillatory For case 1 and 3: Pe < 2

(ii) aE = De – Fe/2

For 0 eFa D> → → <

= <

If Pe > 2 → CD scheme violates boundedness

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 22

→ gives physically unrealistic solutions.

• 12

TransportivenessThe CD scheme does not recognise the direction of the

flow or the strength of convection relative to diffusion. Thus it does not posses the transportiveness property atThus, it does not posses the transportiveness property at high Pe.Accuracy

The CD scheme is stable and accurate only if Pe = F/D < 2.

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 23

The CD scheme satisfies this criteria for low Re numbers or for small grid spacings.

Thus, CD scheme is not a suitable discretisation practice for general purpose flow calculations.

5.6 The upwind differencing schemeThe scheme takes into account the flow direction, φ at cell face = φ at upstream

node formulation is used

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 24

When the flow is in the positive direction, uw>0, ue>0 (Fw>0, Fe>0), the upwind scheme sets φw = φW and φe = φP (5.25)

• 13

The discretised equation (5.9) becomes

Which can be rearranged as( ) ( )e P w w e E P w P WF F D Dφ φ φ φ φ φ− = − − − (5.26)

( ) ( )P W ED D F D F Dφ φ φ+ + = + +to give

When the flow is in the negative direction, uw

• 14

Example 5.2 Solve the problem considered in example 5.1 using the upwind differencing scheme for (i) u = 0.1 m/s,(ii) u = 2.5 m/s ( )

with the coarse five-point grid.

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 27

The governing equation is: ( )d d dudx dx dx

φρ φ ⎛ ⎞= Γ⎜ ⎟⎝ ⎠A B

1 2 3 4 5 6 7φ = 1 W P E

ewφ = 0

P P W W E E ua a a Sφ φ φ= + +where ( )P W E e w Pa a a F F S= + + − −

( ) ( )w w e ew w e e

F uA F uAA AD D

ρ ρ= =Γ Γ

For interior nodes: For node 2: / 2For node 6: / 2

WP PE

WP

PE

x x xx xx x

δ δ δδ δδ δ

= ==

=

δxδx/2 δxWP= δx δx/2δxPE=δx

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 28

2 0 max(0, ) ( max( ,0)) ( max( ,0))3, 4,5 max( ,0) max(0, ) 0 0

6 max( ,0) 0 ( max(0, )) ( max(0, ))

W E P u

e e w w w w A

w w e e

w w e e e e B

Node a a S SD F D F D F

D F D FD F D F D F

φ

φ

+ − − + ++ + −+ − + − + −

w w e ew eWP PE

D Dx xδ δ

= =

• 15

u = 0.l m/s:

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 29

u = 2.5 m/s

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 30

Upwind scheme produced a much more realistic solution compared with central difference scheme.

However, the solution is not very close to the exact value.

• 16

5.6.1 Assessment of the upwind differencing scheme

Conservativeness the upwind differencing scheme utilises consistent expressions to calculate fluxes through cell faces: therefore it can be easily shown that the formulation is conservativeBoundednessthe coefficients of the discretised equation are always positive and satisfy the requirements for boundedness

Fe – Fw = 0 → aP = aW + aE Stable iterative solutionAll coefficients are positive No wiggles in

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 31

Coefficient matrix is diagonally dominantTransportiveness

The scheme accounts for the direction of the flow so transportiveness is build into the formulation.

solution

Accuracythe scheme is based on the backward differencing formula so the accuracy is only first order on the basis of the Taylor series truncation error (see Appendix A):A major drawback of the scheme:it produces erronous results when the flow is not aligned with the grid lines.p g gφ is smeared error has a diffusion-like appearance false diffusion

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 32

• 17

Consider pure convection without diffusion and no source term.the true solution is:all nodes above diagonal should be 100all nodes below diagonal should be 0

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 33

Upwind method is not suitable for accurate flow calcualtions

5.7 The hybrid differencing schemeCentral differencing scheme: accurate to second order → Not transportiveUpwind differencing scheme: accurate to first order → is transportive

Hybrid difference scheme uses:central difference scheme for Pe < 2upwind difference scheme in which diffusion has been set to zero for Pe ≥ 2

For a west face

The hybrid differencing formula for the net flux through the west face is as follows:

( )/

w ww

w w WP

uFPeD x

ρδ

= =Γ (5.35)

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 34

1 2 1 21 1 2 22 2

2

2

w w W P ww w

w w W w

w w P w

q F for PePe Pe

q F for Pe

q F for Pe

φ φ

φ

φ

⎡ ⎤⎛ ⎞ ⎛ ⎞= + + − − <

• 18

The general form of the discretised equation is

The central coefficient is given byP P W W E Ea a aφ φ φ= + (5.37)

After some re-arrangement it is easy to establish that the neighbour coefficients for the hybrid differencing scheme for steady one -dimensional convection – diffusion can be written as follows:

( )P W E e wa a a F F= + + −

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 35

max , ,0 max , ,02 2

W E

w ew w e e

a a

F FF D F D⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞+ − −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

Example 5.2 Solve the problem considered in case 2 of example 5.1 using the hybrid scheme for u=2.5 m/s.Compare a 5 node solution with a 25 node solution

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 36

• 19

Comparison with the analytical solutionThe numerical results are compared with the analytical solution in table 5.9

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 37

5.7.1 Assessment of the hybrid differencing scheme

Is fully conservative Is unconditionally bounded (since the coefficients are always positive)Satisfies the transportiveness propertyProduces physical realistic solutionsHighly stable compared with higher order schemeIs only first order accurate

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 38

• 20

Hybrid differencing scheme for multi-dimensional convection-diffusionThe discretised equation that covers all cases is given by

with central coefficient

P P W W E E S S N N B B T Ta a a a a a aφ φ φ φ φ φ φ= + + + + +

P W E S N B Ta a a a a a a F= + + + + + + Δ

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 39

The coefficient of this equation for the hybrid differencing schemeare as follows:

One-dimensional flow two-dimensional flow three-dimensional flow

max[F (D +F /2) 0] max[F (D +F /2) 0] max[F (D +F /2) 0]aW max[Fw,(Dw+Fw/2),0] max[Fw,(Dw+Fw/2),0] max[Fw,(Dw+Fw/2),0]

aE max[-Fe,(De-Fe/2),0] max[-Fe,(De-Fe/2),0] max[-Fe,(De-Fe/2),0]

aS - max[Fs,(Ds+Fs/2),0] max[Fs,(Ds+Fs/2),0]

aN - max[-Fn,(Dn-Fn/2),0] max[-Fn,(Dn-Fn/2),0]

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 40

aB - - max[Fb,(Db+Fb/2),0]

aT - - max[-Ft,(Dt-Ft/2),0]

ΔF Fe-Fw Fe-Fw+Fn-Fs Fe-Fw+Fn-Fs+Ft-Fb

• 21

In the above expressions the value of F and D are calculated with the following formulae

F bFace w e s n b t

F (ρu)wAw (ρu)eAe (ρu)sAs (ρu)nAn (ρu)bAb (ρu)tAt

D ΓwAw/δxWP ΓeAe/δxPE ΓsAs/δySP ΓnAn/δyPN ΓbAb/δzPN ΓtAt/δzPT

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 41

The Power Law SchemeIs a more accurate approximation to the 1-D exact solutionProduces better results than the hybrid schemeyfor Pe > 10 diffusion is set to zerofor 0 < Pe < 10 the flux is evaluated by a polynomial expression

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 42

• 22

For example, the net flux per unit area at the west control volume face is evaluated using

( )( )5

for 0 10where 1 0 1

w w W w P Wq F PePe Pe

φ β φ φβ⎡ ⎤= − − < (5.44b)

( )a a a F F= + + −

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 43

Central coefficient:and

Wa

( )P W E e wa a a F F= + +

Ea

( ) [ ]5max 0, 1 0.1 max ,0e e eD Pe F⎡ ⎤− + −⎢ ⎥⎣ ⎦( ) [ ]5

max 0, 1 0.1 max ,0w w wD Pe F⎡ ⎤− +⎢ ⎥⎣ ⎦

5.9 Higher order differencing schemes for convection-diffusion problems

Hybrid and Upwind schemes areStableObey the transportiveness requirementBut have first order accuracyAre prone to numerical diffusion errors

Such errors can be minimized by employing higher orderdiscretisations.

CentralDifference scheme is second order accurate but is not stable.

Formulations that do not take into account the flow direction are

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unstableFor more accuracy: use higher order schemes, which preserve upwinding for stability

• 23

5.9.1 Quadratic upwind differencing scheme: the QUICK scheme

The quadratic upstream interpolation for convective kinetic (QUICK) scheme of Leonard(1979) uses a three-point upstream-weighted upstream quadratic interpolation for cell face values. The face value of φ is obtained from a quadratic function through two bracketing nodes ( h id f h f ) d d h id ( i 1 )(on each side of the face) and a node on the upstream side (Fig. 5.17)

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 45

Two upstream nodes and one downstream node is used to calculate the face value of φ

It can be shown that for a uniform grid the value of φ at the cell face between two bracketing nodes i and i-1, and upstream node i-2 is given by the following formula:

h h b k i d f h f d h(5.45)1 2

6 3 18 8 8face i i i

φ φ φ φ− −= + −

When uw > 0, the bracketing nodes for the west face ’w’ are W and P, the upstream node is WW (Figure 5.17), and

When ue > 0, the bracketing nodes for the east face ’e’ are P and E, the upstream node is W ,so

6 3 18 8 8w W P WW

φ φ φ φ= + − (5.46)

6 3 1φ φ φ φ

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The diffusion terms may be evaluated using the gradient of the appropriate parabola.

It is interesting to note that on a uniform grid this practice gives the same expressions as central differencing for diffusion.

(5.47)8 8 8e P E Wφ φ φ φ= + −

• 24

If Fw>0 and Fe>0 and if we use equations (5.46-5.47) for the convective terms and central differencing for the diffusion terms, the discretised form of the one-dimensional convection-diffusion transport equation(5.9) may be written as ( ) ( )e e w w e E P w P WF F D Dφ φ φ φ φ φ− = − − − (5.9)

( ) ( )6 3 1 6 3 1F F D Dφ φ φ φ φ φ φ φ φ φ⎡ ⎤⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟

which can be rearranged to give

This is now written in the standard form for discretised equations

( ) ( )8 8 8 8 8 8e P E W w W P WW e E P w P W

F F D Dφ φ φ φ φ φ φ φ φ φ⎡ ⎤⎛ ⎞ ⎛ ⎞+ − − + − = − − −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

3 6 6 1 3 18 8 8 8 8 8w w e e P w w e W e e E w WW

D F D F D F F D F Fφ φ φ φ⎡ ⎤ ⎡ ⎤ ⎡ ⎤− + + = + + + − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦(5.48)

a a a aφ φ φ φ+ + (5 49)

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 47

whereP P W W E E WW WWa a a aφ φ φ φ= + + (5.49)

For Fw < 0 and Fe < 0 the flux across the west and east boundaries is given by the expressions

6 3 18 8 8w P W E

φ φ φ φ= + −(5 50)

Substitution of these two formulae for the convective terms in the discretised convection-diffusion equation (5.9) together with central differencing for the diffusion terms leads, after re-arrangement as above, to the following coefficients:

6 3 18 8 8e E P EE

φ φ φ φ= + −(5.50)

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• 25

General expressions, valid for positive and negative flow directions, can be obtained by combining the two sets of coefficients above.The QUICK scheme for one-dimensional convection-diffusion problems can be summarised as follows:

With central coefficient

And neighbour coefficients

P P W W E E WW WW EE EEa a a a aφ φ φ φ φ= + + + (5.51)

( )P W E WW EE e wa a a a a F F= + + + + −

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where αw=1 for Fw > 0 and αe=1 for Fe > 0 αw=0 for Fw < 0 and αe=0 for Fe < 0

Example 5.4 Using the QUICK scheme solve the problem considered in example 5.1 for u=0.2 m/s on a five-point grid. Compare the quick solution with the exact and central differencing solution.

Boundary Points :

A B

δxδx/2 δxWP= δx δx/2

1 2 3 4 5 6 7φ = 1 W P E

ewφ = 0

δxPE=δx

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y

Consider node 2. φw = φA

To calculate φe : φw is needed. But there is no φw use linear interpolation to create a mirror node at δx/2 to the west of boundary A.

• 26

It can be easily shown that the linearly extrapolated value at the minor node is given by

(5.52)0 2 A Pφ φ φ= −

Node 2Mirror Node Domain boundary

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The extrapolation to the ‘mirror’ node has given us the required W node for the formula (5.47) that calculates φe at the east face of control volume 2:

(5.53)( )6 3 1 7 3 228 8 8 8 8 8e P E A P P E A

φ φ φ φ φ φ φ φ= + − − = + −

At the boundary nodes the gradients in diffusive flux terms can be evaluated using central difference scheme similar to calculation of diffusion terms in interior nodes.

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• 27

The discretised equations for nodes 2, 3 and 6 are now written to fit into the standard form to give:

withP P WW WW W W E E ua a a a Sφ φ φ φ= + + + (5.59)

( )F F SThe solution is

( )P W E WW EE e w Pa a a a a F F S= + + + + − −

2

3

4

0.96480.87070.7309

φφφ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ (5 60)

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5

6

0.52260.2123

φφ

⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

(5.60)

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• 28

5.9.2 Assessment of the QUICK schemeThe scheme:

Uses consistent quadratic profiles is conservativeIs based on a quadratic function has 3rd order truncation errorIs based on 2 upstream and 1 downstream nodes hasIs based on 2 upstream and 1 downstream nodes has transportiveness

aP = Σ anb if flow field satisfies continuity desirable for boundedness

aE and aW may not be positive aWW and aEE are negativeIf uw > 0 and ue > 0 :

8F

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Then aE = De – 3 / 8 Fe becomes negative forGives rise to stability problems and unbounded solutions.QUICK scheme is conditionally stable

Involves φWW and φEE which are not immediate-neighbour nodes

83

ee

e

FPeD

= >

5.9.3 Stability problems of the QUICK scheme and remedies

May have negative main coefficients can be unstableAlso other higher order schemes may be oscillatory and unstableunder certain conditions

In this case use:Method of deferred correctionIn this method the cell face value φf is formulated as the sum of the

upwind value and other higher order terms which are evaluated at the previous iteration.

HO u

u of f f

o o of f f

φ φ φ

φ φ φ

= + Δ

Δ = −

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where:value to be computed by 1st order upwind methodvalue computed by high order scheme from previous old valuesvalue computed by 1st order upwind method from previous old values

f f fφ φ φuf fφ φ=0HO

f fφ φ=0u

f fφ φ=

• 29

Let us apply the deferred correction method to QUICK scheme.For uw > 0 QUICK scheme is

6 3 1w W P WWφ φ φ φ= + −

This can be written as

Similarly:ufφ

8 8 8w W P WWφ φ φ φ

[ ]1 3 2 08w W P W WW w

For Fφ φ φ φ φ= + − − >0 is added to source term f uSφΔ

[ ]1 3 2 0F Fφ φ φ φ φ+ >

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(5.62)[ ]

[ ]

[ ]

3 2 081 3 2 081 3 2 08

e P E P W e

w P W P E w

e E P E EE e

For F

For F

For F

φ φ φ φ φ

φ φ φ φ φ

φ φ φ φ φ

= + − − >

= + − − <

= + − − <

The discretisation equation takes the form

The central coefficient isP P W W E E ua a a Sφ φ φ= + + (5.63)

( )P W E e wa a a F F= + + − (5.64)where

( )P W E e w

max[ ,0]( ) max[ ,0]( )max( ,0) max(0, )

max[ ,0]( ) max[ ,0]( )

w e u

w w W w w Pw w e e

e e E e e P

a a SF F

D F D FF F

φ φ φ φφ φ φ φ− − − −

+ + −+ − − − −

( )

(5.65)

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Note that aw and ae are the same as that of the upwind method.

The advantage of this approach is that the coefficients are always positive and now satisfy the requirements for conservativeness, boundedness and transportiveness

( )

• 30

5.9.4 general comments on the QUICK differencing scheme

QUICK schemeHas greater accuracy than central, upwind or power schemesRetains the upwind weighted characteristicsResultant false diffusion is smallResultant false diffusion is small

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Can give (minor) undershoots and overshoots (see Fig. 5.20)

To prevent this problem use:1. Limiters

Limit the scheme to have the face value φf to be between φfcertain values (ULTRA SHARP)

2. Total variation diminishing schemes (TVD)

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• 31

Homework

yH

•Consider a fluid at a uniform temperature Ti entering a channel whose surface is maintained at a different temperature Ts. A Thermal boundary layer along the tube developes, after which the form of the temperature profile does not change. Assume that the flow profile is constant in the channel where the velocities are given by

2

max

21 1 and 0u y vu H

⎛ ⎞= − − =⎜ ⎟⎝ ⎠

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where umax = 1.5umean. The energy equation is

( ) ( ) p p

uT vT k T k Tx y x c x y c y

ρ ρ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂+ = +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

Find the temperature profile in the channel for Re = ρumeanH/μ = 10, Pr = μcp/k = 1. Use Lx/H = 5, where Lx is the length of the solution domain. Use UPWIND method. (Note: k/cp = μ/Pr for fluids.). Also, choosing ρ = 1, find μ from Re relation. Take Tin = 0, Twalls = 100, umean = 1 m/s

Generalisation of Upwind-biased SchemesFor convection terms, an estimate of φ value at the faces of a CV is required. Consider east face, assuming ue > 0

1) Standard Upwind Differencing Scheme (UD)

EEEPWWWew

φeφP

veUPWIND

φe = φP

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φe = φP

The face value of φ is taken to be equal to the value of the upstream node;

(5.66)

• 32

Generalisation of Upwind-biased Schemes

φeφPφW

φe = φP +(φP – φW) / 2

2) Linear Upwind Differencing Scheme (LUD) also called the second order upwind differencing scheme (SOU)

E EEPWWWew

veSOU (LUD)

φ is assumed to vary linearly between W and e. Then φe is found by extrapolating the two upstream node values φW and φP to face e.

( )P W xφ φ δφ φ −

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( )2

1 ( )2

P We P

P P W

xφ φφ φδ

φ φ φ

= +

= + −

The term ½(φP – φW) can be thought as a second order correction to the standard upwind scheme.

(5.67)

Generalisation of Upwind-biased Schemes3) Central Differencing Scheme (CD)

φeφPφE

φe = (φP + φW) / 2

EEEPWWWew

ueCENTRAL

The value of φ is assumed to vary linearly between the two nodes straddling the face, that is;

( )P Eφ φφ + (5 68)

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( )2

or1 ( )2

P Ee

e P E P

φ φφ

φ φ φ φ

=

= + −

(5.68)

(5.69)

• 33

Generalisation of Upwind-biased Schemes4) QUICK Scheme

φeφPφEφW

φe = 6/8φP + 3/8φE – 1/8φW)

E EEPWWWew

ueQUICK

φW

The scheme is based on the assumption that φ varies in terms of a second degree polynomial between two upstream (W and P) and the downstream node E

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 65

downstream node E.6 3 18 8 8e P E W

φ φ φ φ= + −

1 (3 2 )8e P E P W

φ φ φ φ φ= + − −or

(5.70)

(5.71)

Generalisation of Upwind-biased SchemesAll higher order schemes can be expressed in the form:

1 ( )2e P E P

φ φ ψ φ φ= + −

ψ = an appropriate function

(5.72)

ψ pp p

Convective flux at face e is FeφeFor a higher order scheme convective flux consist of two parts:

1) Upwind flux, FeφP2) Additional flux, Feψ(φE – φP)/2

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E P

Additional flux is connected to the gradient of φ at face e, as indicated by (φE – φP)

• 34

Generalisation of Upwind-biased Schemes

LUD scheme may be written as

ψ = 0 for UD scheme

ψ = 1 for CD scheme

LUD scheme may be written as1 ( )2

= for LUD scheme

P We P E P

E P

P W

E P

φ φφ φ φ φφ φ

φ φψφ φ

⎛ ⎞−= + −⎜ ⎟−⎝ ⎠

⎛ ⎞−→ ⎜ ⎟−⎝ ⎠

QUICK scheme may be written as

(5.73)

(5.74)

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Q y

1 13 ( )2 4

1 = 3 for QUICK scheme4

P We P E P

E P

P W

E P

φ φφ φ φ φφ φ

φ φψφ φ

⎡ ⎤⎛ ⎞−= + + −⎢ ⎥⎜ ⎟−⎝ ⎠⎣ ⎦

⎛ ⎞−→ +⎜ ⎟−⎝ ⎠

(5.75)

(5.76)

Generalisation of Upwind-biased Schemes

ψ is a function of r: ψ = ψ(r)

P W

E P

r φ φφ φ

−=

−let

(5.77)

ψ ψ ψ( )

A higher order convection scheme can be written as1 ( )( )2e P E P

rφ φ ψ φ φ= + −

ψ = 0 for UD scheme

ψ = 1 for CD scheme

(5.78)

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ψ

1= 3 for QUICK scheme4

P W

E P

φ φψφ φ

⎛ ⎞−+⎜ ⎟−⎝ ⎠

= for LUD schemeP WE P

φ φψφ φ

⎛ ⎞−⎜ ⎟−⎝ ⎠

• 35

All of the above expressions assume that the flow direction is positive (from west to east).Similar expressions exist for negative flow direction.p gIn that case, r is still the ratio of upwind-side gradient to downwind-side gradient.

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Total Variation and TVD SchemesUD scheme is the most stable scheme (no wiggles)CD and QUICK have higher order accuracy but give rise to wiggles under certain conditions.Our aim is to find a convection scheme with a higher-Our aim is to find a convection scheme with a higher-order accuracy but without wiggles.The desirable property for a stable, non-oscillatory, higher order scheme is monotonicity preserving.For a scheme to preserve to preserve monotonicity:

1. It must not create local extrema

I. Sezai - Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 70

2. The value of an existing local minimum must be non-decreasing and that of a local maximum must be non-increasing.

Monotonicity preserving schemes do not create new undershoots and overshoots.

• 36

Total Variation and TVD SchemesConsider the discrete data set shown in the figure.

The total variation of this data set is2 1 3 2 4 3 5 4

3 1 5 3

( )

TV φ φ φ φ φ φ φ φ φ

φ φ φ φ

= − + − + − + −

= − + −(5.79)

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For monotonicity, this TV must not increase with time.

3 1 5 3φ φ φ φ

Total Variation and TVD SchemesIn other words TV must diminish with time.Hence, the term total variation diminishing or TVD.Originally TVD was developed for time-dependent flows.For TVD: TV(φn+1) ≤ TV(φn) where n refer to time step.

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In the next section we show how TVD is also linked to desirable behaviour of discretisation schemes for steady convection-diffusion problems.

• 37

Criteria for TVD SchemesNecessary and sufficient conditions for a scheme to be TVD1) For 0 < r < 1 → ψ(r) ≤ 2r2) For r ≥ 1 → ψ(r) ≤ 2

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UD scheme is TVDLUD scheme is not TVD for r > 2CD scheme is not TVD for r < 0.5QUICK scheme is not TVD for r < 3/7 and r > 5

Requirement for Second Order AccuracyFor second order accuracy, the flux limiter function ψ should pass through the point (1, 1) in the r– ψ diagram.

Range of possible second-order schemes is bounded by the CD and LUD schemes:LUD schemes:

For 0 < r < 1 for TVD to be second order r ≤ ψ(r) ≤ 1 For r ≥ 1 for TVD to be second order 1 ≤ ψ(r) ≤ r

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Region for a second-order TVD scheme

• 38

Symmetry Property for Limiter FunctionsSymmetry Property for limiter functions:

( ) (1/ )r rr

ψ ψ= (5.80)

A limiter function that satisfies the symmetry property ensures that backward and forward-facing gradients are treated in the same fashion without the need for special coding.

r

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Flux Limiter Functions

2

Van Leer Van Leer (1974)1

V Alb d V Alb d (1982)

r rr

r r l

+++

Name Limiter function Source

Min-Mod ( ) Roe (1985)0 if 0

SUPERBEE max[0,min(2 ,1),min( , 2)] Roe

et alr

r rr

rr r

ψ

+>⎧

= ⎨ ≤⎩ (1985)

Sweby max[0,min( ,1),min( , )] Sweby (1984)QUICK max[0,min(2 , (3 ) / 4, 2)] Leonard (1988)

S [0 i (2 (1 3 ) / 4 (3 ) / 4 2)] i d h i (1993)

r rr rβ β

+

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UMIST max[0,min(2 , (1 3 ) / 4, (3 ) / 4, 2)] Lien and Leschziner (1993)r r r+ +

0 ≤ β ≤ 2β = 1 → Min-Mod Limiterβ = 2 → SUPERBEE Limiter of Roe

• 39

Flux Limiter Functions

MIN‐MOD

MIN‐MOD

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All Limiter functions in a r–ψ diagram

All limiter functions are symmetric except QUICK limiter.

UMIST limiter function is a symmetric version of the QUICK limiter.

Implementation of TVD Schemes

The coefficients of the discretized equation are written in the deferred correction h I thi h th ffi i t th f UD h

For the one dimensional convection diffusion equation

( )d d dudx dx dx

φρ φ ⎛ ⎞= Γ⎜ ⎟⎝ ⎠ (5.81)

approach. In this approach, the aE, aW, aP coefficients are the same as of UD scheme. The extra terms resulting from the application of a limiter function is added to the source term Sdc.

1 ( )( )21 ( )( )

e P e E Pr

r

φ φ ψ φ φ

φ φ ψ φ φ

+

+

= + −

+

The face values are:

P We

E P

W WW

r

r

φ φφ φ

φ φ

+

+

⎛ ⎞−= ⎜ ⎟−⎝ ⎠⎛ ⎞−

= ⎜ ⎟

For u > 0(5.82)

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( )( )2w W w P W

rφ φ ψ φ φ= + − wP W

rφ φ

= ⎜ ⎟−⎝ ⎠

1 ( )( )21 ( )( )2

e E e P E

w P w W P

r

r

φ φ ψ φ φ

φ φ ψ φ φ

= + −

= + −

For u < 0 EE EeE P

E Pw

P W

r

r

φ φφ φ

φ φφ φ

⎛ ⎞−= ⎜ ⎟−⎝ ⎠⎛ ⎞−

= ⎜ ⎟−⎝ ⎠

(5.83)

• 40

Implementation of TVD SchemesThe discretisation equation takes the form

The central coefficient isP P W W E E dca a a Sφ φ φ= + + (5.84)

(5 85)

where( )P W E e wa a a F F= + + −

max[ ,0]( ) max[ ,0]( )max( ,0) max(0, )

max[ ,0]( ) max[ ,0]( )

w e dc

w w W w w Pw w e e

e e E e e P

a a SF F

D F D FF F

φ φ φ φφ φ φ φ− − − −

+ + −+ − − − −

(5.85)

(5.86)

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φe and φw are as defined in Eqs. (5.82) and (5.83)Note that Sdc is the same as defined in Eq. (5.65). Note also that aw and ae are the same as that of the upwind method.The advantage of this approach is that the coefficients are always positive and now satisfy the requirements for conservativeness, boundedness and transportiveness

( )

Evaluation of TVD Schemes

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Comparison of TVD schemes for pure convection flowing 45o to the grid direction.

TVD schemes does not give unphysical overshoots or undershoots.

However, TVD schemes require about 15% more CPU time.