Chapter_5.pdf

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 Sφ
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.

2
Steady one-dimensional convection and diffusion
In 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 d
u
dx dx dx
φ
ρ φ = Γ (5.3)
dx dx dx
The flow must also satisfy continuity, so
( ) 0
d
u
dx
ρ =
Integrating Eqn. (5.3) over the CV
( ) ( )
e w
e w
uA uA A A
x x
φ φ
ρ φ ρ φ
∂ ∂
⎛ ⎞ ⎛ ⎞
− = Γ − Γ
⎜ ⎟ ⎜ ⎟
∂ ∂
⎝ ⎠ ⎝ ⎠
Integrating continuity Eqn.
( ) ( ) 0
e w
uA uA
ρ 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 e
F uA F uA
A A
ρ ρ
= =
Γ Γ
w w e e
w e
WP PE
A A
D D
x x
δ δ
Γ Γ
= =
Using central difference approach for the diffusion terms, Eqn (5.5)
becomes
( ) ( )
e e w w e E P w P W
F F D D
φ φ φ φ φ φ
− = − − −
C i i i b
(5.9)
0
e w
F 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 φ
φ φ φ
= +
= +
Substituting into eqn (5.9)
( ) ( ) ( ) ( )
2 2
e w
P E W P e E P w P W
F F
D D
φ φ φ φ φ φ φ φ
+ − + = − − −
Rearranging,
2 2 2 2
w e w e
w e P w W e E
F F F F
D D D D
F F F F
φ φ φ
⎡ ⎤
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
− + + = + + −
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎢ ⎥
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎣ ⎦
⎡ ⎤
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
( )
2 2 2 2
w e w e
w e e w P w W e E
F F F F
D D F F D D
φ φ φ
⎡ ⎤
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
+ + − + − = + + −
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎢ ⎥
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎣ ⎦
which is of the form
P P W W E E
a a a
φ φ φ
= +
where
(5.14)
( )
2 2
W E P
w e
w e W E e w
a a a
F F
D 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 difference
scheme, 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( / ) 1
o ux
φ φ ρ
− Γ −
=
exp( / ) 1
L o uL
φ φ ρ
− Γ −
(iii) Case 3: u = 2.5 m/s (20 CV’s)
The governing equation is: ( )
d d d
u
dx dx dx
φ
ρ φ
⎛ ⎞
= Γ
⎜ ⎟
⎝ ⎠
A B
1 2 3 4 5 6 7
φ = 1 W P E
e
w
φ = 0
P P W W E E u
a a a S
φ φ φ
= + +
where ( )
P W E e w P
a a a F F S
= + + − −
( ) ( )
w w e e
w w e e
F uA F uA
A A
D D
ρ ρ
= =
Γ Γ
For interior nodes:
For node 2: / 2
For node 6: / 2
WP PE
WP
PE
x x x
x x
x x
δ δ δ
δ δ
δ δ
= =
=
=
δx
δx/2 δxWP= δx δx/2
δxPE=δx
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 S
D F D F D F
D F D F
D F D F D F
φ
φ
− − + +
+ −
+ − − −
w w e e
w e
WP PE
D D
x x
δ δ
= =

5
The resulting system of equations are
2 2
2 2
P E
W p E
a a
Su
a a a Su
φ
φ
−
⎡ ⎤ −
⎡ ⎤ ⎡ ⎤
⎢ ⎥
⎢ ⎥ ⎢ ⎥
−
⎢ ⎥
⎢ ⎥ ⎢ ⎥
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 i
W p E
n n
W p E
Su
a a a Su
Su
a a a
Su
a a a
φ
φ
φ
φ
− − −
− −
⎢ ⎥ −
⎢ ⎥ ⎢ ⎥
⎢ ⎥
− ⎢ ⎥ ⎢ ⎥
−
⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥
⎢ ⎥ ⎢ ⎥
=
⎢ ⎥
⎢ ⎥ ⎢ ⎥
−
−
⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥
⎢ ⎥ ⎢ ⎥
−
⎢ ⎥
− ⎢ ⎥ ⎢ ⎥
⎢ ⎥
1 1
1 1
n n n
n n
n n
W p
Su
a 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
1
0.9421
φ
φ
⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
2
3
4
5
6
7
0.9421
0.8006
0.6276
0.4163
0.1573
0
φ
φ
φ
φ
φ
φ
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
=
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎣ ⎦
⎣ ⎦
7
φ ⎣ ⎦
⎣ ⎦
Exact solution is:
2.7183 exp( )
( )
1.7183
x
x
φ
−
=
Comparison of the numerical result with the
analytical solution.

6
The solution for case 2: (u = 2.5 m/s, 5 CV’s)
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)
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
• Transportiveness
1. Conservativeness
To 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 face
adjacent 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.

8
Example of consistent specification of diffusive fluxes
2 2 1
( )
w
x
φ φ
δ
Γ − 2 3 2
( )
e
x
φ φ
δ
Γ −
Flux entering CV 2 Flux leaving CV 2
An overall flux balance may be obtained by summing the net flux through each CV
3 2
2 1 2 1
1 2 2
( )
( ) ( )
e A e w
q
x x x
φ φ
φ φ φ φ
δ δ δ
−
− −
⎡ ⎤
⎡ ⎤
Γ − + Γ − Γ
⎢ ⎥
⎢ ⎥
⎣ ⎦ ⎣ ⎦
4 3 3 2 4 3
3 3 4
( ) ( ) ( )
e w B w B A
q q q
x x x
φ φ φ φ φ φ
δ 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.
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
nb
a
a
≤
⎧
⎨
′ ⎩
∑
P P p
a a S
′ = − (5.22)
< 1 at one node at least
P
a ⎩
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 boundedness
criterion.
This states that in the absence of sources the internal nodal values of φ
should 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 contain
the 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
2 2
e e e e e
E e
PE
F A u A
a D
x
ρ
δ
Γ
= − = −
2
3
4
5
6

10
3) Transportiveness
The transportiveness property of a fluid flow can be illustrated by
considering a constant source of φ at a point P
cell Peclet number
/
F u
Pe
D x
ρ
δ
= =
Γ
Distribution of φ in the vicinity
of a source at different Peclet
numbers.
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
Conservativeness
The central differencing scheme uses
consistent expressions to evaluate convective
and diffusive fluxes at the CV faces.
The scheme is conservative.

11
Boundedness
(i) The internal coefficients of discretised scalar transport equation
(5.14) are
W E P
a a a
( )
2 2
w e
w e W E e w
F F
D D a a F F
+ − + + −
(Fe – Fw) = 0 from continuity → aP = aW + aE
Thus, convergence criteria (5.22) is satisfied by the central difference
scheme
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 e
F
a D
> → <
For 0
2
or 2 to have positive .
E e
e
e E
e
a D
F
Pe a
D
> → <
= <
If Pe > 2 → CD scheme violates boundedness
→ gives physically unrealistic solutions.

12
Transportiveness
The 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 at
Thus, it does not posses the transportiveness property at
high Pe.
Accuracy
The CD scheme is stable and accurate only if Pe = F/D <
2.
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 scheme
The scheme takes into account the flow direction, φ at cell face = φ at upstream
node formulation is used
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 W
F F D D
φ φ φ φ φ φ
− = − − − (5.26)
( ) ( )
P W E
D D F D F D
φ φ φ
+ + = + +
to give
When the flow is in the negative direction, uw<0, ue<0 (Fw<0, Fe<0), the
scheme takes
and
w P e E
φ φ φ φ
= = (5.28)
( ) ( )
w e e P w w W e E
D D F D F D
φ φ φ
+ + + +
(5.27)
( ) ( ) ( )
w w e e w P w w W e E
D F D F F D F D
φ φ φ
⎡ ⎤
+ + + − = + +
⎣ ⎦
Now the discretised euqation is
or
( ) ( )
e E w P e E P w P W
F F D D
φ φ φ φ φ φ
− = − − −
( ) ( ) ( )
w e e e w P w W e e E
D D F F F D D F
φ φ φ
⎡ ⎤
+ − + − = + −
⎣ ⎦
(5.29)
(5.30)
the equations (5.27) and (5.30) can be written in the usual general
form
with central coefficient
P P W W E E
a a a
φ φ φ
= + (5.31)
and neighbour coeffcients
( )
P W E e w
a a a F F
= + + −
Fw>0, Fe>0 Dw + Fw De
F <0 F <0 D D - F
w
a e
a
A form of notation for neighbour coefficients of the upwind
differencing method that covers both flow directions is:
e
a
Fw<0, Fe<0 Dw De Fe
Dw + max(Fw,0) De + max(0, – Fe)
w
a

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.
The governing equation is: ( )
d d d
u
dx dx dx
φ
ρ φ
⎛ ⎞
= Γ
⎜ ⎟
⎝ ⎠
A B
1 2 3 4 5 6 7
φ = 1 W P E
e
w
φ = 0
P P W W E E u
a a a S
φ φ φ
= + +
where ( )
P W E e w P
a a a F F S
= + + − −
( ) ( )
w w e e
w w e e
F uA F uA
A A
D D
ρ ρ
= =
Γ Γ
For interior nodes:
For node 2: / 2
For node 6: / 2
WP PE
WP
PE
x x x
x x
x x
δ δ δ
δ δ
δ δ
= =
=
=
δx
δxPE=δx
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 S
D F D F D F
D F D F
D F D F D F
φ
φ
+ − − + +
+ + −
+ − + − + −
w w e e
w e
WP PE
D D
x x
δ δ
= =

15
u = 0.l m/s:
u = 2.5 m/s
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 conservative
Boundedness
the coefficients of the discretised equation are always
positive and satisfy the requirements for boundedness
Fe – Fw = 0 → aP = aW + aE Stable iterative solution
All coefficients are positive No wiggles in
Coefficient matrix is diagonally dominant
Transportiveness
The scheme accounts for the direction of the flow so
transportiveness is build into the formulation.
solution
Accuracy
the 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 g
φ is smeared Æ error has a diffusion-like appearanceÆ false diffusion

17
Consider pure convection without diffusion and no source term.
the true solution is:
all nodes above diagonal should be 100
all nodes below diagonal should be 0
Upwind method is not suitable for accurate flow calcualtions
5.7 The hybrid differencing scheme
Central differencing scheme: accurate to second order → Not transportive
Upwind differencing scheme: accurate to first order → is transportive
Hybrid difference scheme uses:
central difference scheme for Pe < 2
upwind 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 w
w
w w WP
u
F
Pe
D x
ρ
δ
= =
Γ (5.35)
1 2 1 2
1 1 2 2
2 2
2
2
w w W P w
w w
w w W w
w w P w
q F for Pe
Pe Pe
q F for Pe
q F for Pe
φ φ
φ
φ
⎡ ⎤
⎛ ⎞ ⎛ ⎞
= + + − − < <
⎢ ⎥
⎜ ⎟ ⎜ ⎟
⎢ ⎥
⎝ ⎠ ⎝ ⎠
⎣ ⎦
= ≥
= ≤ −
(5.36)

18
The general form of the discretised equation is
The central coefficient is given by
P P W W E E
a a a
φ 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 w
a a a F F
= + + −
max , ,0 max , ,0
2 2
W E
w e
w w e e
a a
F F
F 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

19
Comparison with the analytical solution
The numerical results are compared with the analytical solution in table 5.9
5.7.1 Assessment of the hybrid differencing scheme
Is fully conservative
Is unconditionally bounded (since the coefficients
are always positive)
Satisfies the transportiveness property
Produces physical realistic solutions
Highly stable compared with higher order scheme
Is only first order accurate

20
Hybrid differencing scheme for multi-dimensional
convection-diffusion
The discretised equation that covers all cases is given by
P P W W E E S S N N B B T T
a a a a a a a
φ φ φ φ φ φ φ
= + + + + +
P W E S N B T
a a a a a a a F
= + + + + + + Δ
The coefficient of this equation for the hybrid differencing scheme
are 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]
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 b
Face 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
The Power Law Scheme
Is a more accurate approximation to the 1-D exact
solution
Produces better results than the hybrid scheme
y
for Pe > 10 Æ diffusion is set to zero
for 0 < Pe < 10 Æ the flux is evaluated by a
polynomial expression

22
For example, the net flux per unit area at the west control volume
face is evaluated using
( )
( )
5
for 0 10
where 1 0 1
w w W w P W
q F Pe
Pe Pe
φ β φ φ
β
⎡ ⎤
= − − < <
⎣ ⎦
= −
(5.44a)
The coefficients of the one-dimensional descretised equation
utilising the power-law scheme for steady one-dimensional
convection-diffusion are given by
Central coefficient:
( )
where 1 0.1
and for 10
w w w
w w W w
Pe Pe
q F Pe
β
φ
=
= >
(5.44b)
( )
a a a F F
= + + −
Central coefficient:
and
W
a
( )
P W E e w
a a a F F
= + +
E
a
( ) [ ]
5
max 0, 1 0.1 max ,0
e e e
D Pe F
⎡ ⎤
− + −
⎢ ⎥
⎣ ⎦
( ) [ ]
5
max 0, 1 0.1 max ,0
w w w
D Pe F
⎡ ⎤
− +
⎢ ⎥
⎣ ⎦
5.9 Higher order differencing schemes for convection-
diffusion problems
Hybrid and Upwind schemes are
Stable
Obey the transportiveness requirement
But have first order accuracy
Æ Are prone to numerical diffusion errors
Such errors can be minimized by employing higher order
discretisations.
CentralDifference scheme is second order accurate but is not
stable.
Formulations that do not take into account the flow direction are
unstable
For 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)
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 1
8 8 8
face 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 1
8 8 8
w W P WW
φ φ φ φ
= + − (5.46)
6 3 1
φ φ φ φ
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 8
e P E W
φ 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 W
F F D D
φ φ φ φ φ φ
− = − − −
(5.9)
( ) ( )
6 3 1 6 3 1
F F D D
φ 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 8
e P E W w W P WW e E P w P W
F F D D
φ φ φ φ φ φ φ φ φ φ
⎡ ⎤
⎛ ⎞ ⎛ ⎞
+ − − + − = − − −
⎜ ⎟ ⎜ ⎟
⎢ ⎥
⎝ ⎠ ⎝ ⎠
⎣ ⎦
3 6 6 1 3 1
8 8 8 8 8 8
w w e e P w w e W e e E w WW
D F D F D F F D F F
φ φ φ φ
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
− + + = + + + − −
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
(5.48)
a a a a
φ φ φ φ
+ + (5 49)
where
P P W W E E WW WW
a a a a
φ 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 1
8 8 8
w 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 1
8 8 8
e E P EE
φ φ φ φ
= + −
(5.50)

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 EE
a a a a a
φ φ φ φ φ
= + + + (5.51)
( )
P W E WW EE e w
a a a a a F F
= + + + + −
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
δx
1 2 3 4 5 6 7
φ = 1 W P E
e
w
φ = 0
δxPE=δx
y
Consider node 2. φw = φA
To calculate φe : φw is needed. But there is no φw Æ 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 P
φ φ φ
= −
Node 2
Mirror Node Domain boundary
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 2
2
8 8 8 8 8 8
e 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.

27
The discretised equations for nodes 2, 3 and 6 are now written to fit into the
standard form to give:
with
P P WW WW W W E E u
a a a a S
φ φ φ φ
= + + + (5.59)
( )
F F S
The solution is
( )
P W E WW EE e w P
a a a a a F F S
= + + + + − −
2
3
4
0.9648
0.8707
0.7309
φ
φ
φ
⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
=
⎢ ⎥ ⎢ ⎥ (5 60)
5
6
0.5226
0.2123
φ
φ
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎣ ⎦
⎣ ⎦
(5.60)

28
5.9.2 Assessment of the QUICK scheme
The scheme:
Uses consistent quadratic profiles Æ is conservative
Is based on a quadratic functionÆ has 3rd order truncation error
Is based on 2 upstream and 1 downstream nodes Æ has
Is based on 2 upstream and 1 downstream nodes Æ has
transportiveness
aP = Σ anb if flow field satisfies continuity Æ desirable for
boundedness
aE and aW may not be positive Æ aWW and aEE are negative
If uw > 0 and ue > 0 :
8
F
Æ Then aE = De – 3 / 8 Fe becomes negative for
Æ Gives rise to stability problems and unbounded solutions.
Æ QUICK scheme is conditionally stable
Involves φWW and φEE which are not immediate-neighbour nodes
8
3
e
e
e
F
Pe
D
= >
5.9.3 Stability problems of the QUICK scheme and remedies
May have negative main coefficients Æ can be unstable
Also other higher order schemes may be oscillatory and unstable
under certain conditions
In this case use:
Method of deferred correction
In 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 o
f f f
o o o
f f f
φ φ φ
φ φ φ
= + Δ
Δ = −
where:
value to be computed by 1st order upwind method
value computed by high order scheme from previous old values
value computed by 1st order upwind method from previous old values
f f f
φ φ φ
u
f f
φ φ
=
0HO
f f
φ φ
=
0u
f f
φ φ
=

29
Let us apply the deferred correction method to QUICK scheme.
For uw > 0 QUICK scheme is
6 3 1
w W P WW
φ φ φ φ
= + −
This can be written as
Similarly:
u
f
φ
8 8 8
w W P WW
φ φ φ φ
[ ]
1
3 2 0
8
w W P W WW w
For F
φ φ φ φ φ
= + − − >
0
is added to source term
f u
S
φ
Δ
[ ]
1
3 2 0
F F
φ φ φ φ φ
+ >
(5.62)
[ ]
[ ]
[ ]
3 2 0
8
1
3 2 0
8
1
3 2 0
8
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 is
P P W W E E u
a a a S
φ φ φ
= + +
(5.63)
( )
P W E e w
a 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 P
w w e e
e e E e e P
a a S
F F
D F D F
F F
φ φ φ φ
φ φ φ φ
− − − −
+ + −
+ − − − −
( )
(5.65)
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 scheme
Has greater accuracy than central, upwind or power schemes
Retains the upwind weighted characteristics
Resultant false diffusion is small
Resultant false diffusion is small
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
φf
certain values (ULTRA SHARP)
2. Total variation diminishing schemes (TVD)

31
Homework
y
H
•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
2
1 1 and 0
u y
v
u H
⎛ ⎞
= − − =
⎜ ⎟
⎝ ⎠
where umax = 1.5umean. The energy equation is
( ) ( )
p p
uT vT k T k T
x 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 Schemes
For 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)
EE
E
P
W
WW
e
w
φe
φP
ve
UPWIND
φe = φP
φe = φP
The face value of φ is taken to be equal to the value of the upstream
node;
(5.66)

32
φe
φP
φW
φe = φP +(φP – φW) / 2
2) Linear Upwind Differencing Scheme (LUD) also called the second
order upwind differencing scheme (SOU)
E EE
P
W
WW
e
w
ve
SOU (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 x
φ φ δ
φ φ
−
( )
2
1
( )
2
P W
e P
P P W
x
φ φ
φ φ
δ
φ φ φ
= +
= + −
The term ½(φP – φW) can be thought as a second order correction to
the standard upwind scheme.
(5.67)
3) Central Differencing Scheme (CD)
φe
φP
φE
φe = (φP + φW) / 2
EE
E
P
W
WW
e
w
ue
CENTRAL
The value of φ is assumed to vary linearly between the two nodes
straddling the face, that is;
( )
P E
φ φ
φ
+ (5 68)
( )
2
or
1
( )
2
P E
e
e P E P
φ φ
φ
φ φ φ φ
=
= + −
(5.68)
(5.69)

33
4) QUICK Scheme
φe
φP
φE
φW
φe = 6/8φP + 3/8φE – 1/8φW)
E EE
P
W
WW
e
w
ue
QUICK
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
downstream node E.
6 3 1
8 8 8
e P E W
φ φ φ φ
= + −
1
(3 2 )
8
e P E P W
φ φ φ φ φ
= + − −
or
(5.70)
(5.71)
All higher order schemes can be expressed in the form:
1
( )
2
e P E P
φ φ ψ φ φ
= + −
ψ = an appropriate function
(5.72)
ψ pp p
Convective flux at face e is Feφe
For a higher order scheme convective flux consist of two parts:
1) Upwind flux, FeφP
2) Additional flux, Feψ(φE – φP)/2
E P
Additional flux is connected to the gradient of φ at face e, as
indicated by (φE – φP)

34
LUD scheme may be written as
ψ = 0 for UD scheme
ψ = 1 for CD scheme
LUD scheme may be written as
1
( )
2
= for LUD scheme
P W
e P E P
E P
P W
E P
φ φ
φ φ φ φ
φ φ
φ φ
ψ
φ φ
⎛ ⎞
−
= + −
⎜ ⎟
−
⎝ ⎠
⎛ ⎞
−
→ ⎜ ⎟
−
⎝ ⎠
QUICK scheme may be written as
(5.73)
(5.74)
Q y
1 1
3 ( )
2 4
1
= 3 for QUICK scheme
4
P W
e P E P
E P
P W
E P
φ φ
φ φ φ φ
φ φ
φ φ
ψ
φ φ
⎡ ⎤
⎛ ⎞
−
= + + −
⎢ ⎥
⎜ ⎟
−
⎝ ⎠
⎣ ⎦
⎛ ⎞
−
→ +
⎜ ⎟
−
⎝ ⎠
(5.75)
(5.76)
ψ is a function of r: ψ = ψ(r)
P W
E P
r
φ φ
φ φ
−
=
−
let
r = ratio of upwind-side gradient to downwind-side gradient
(5.77)
ψ ψ ψ( )
A higher order convection scheme can be written as
1
( )( )
2
e P E P
r
φ φ ψ φ φ
= + −
ψ = 0 for UD scheme
ψ = 1 for CD scheme
(5.78)
ψ
1
= 3 for QUICK scheme
4
P W
E P
φ φ
ψ
φ φ
⎛ ⎞
−
+
⎜ ⎟
−
⎝ ⎠
= for LUD scheme
P W
E P
φ φ
ψ
φ 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 g
In that case, r is still the ratio of upwind-side
gradient to downwind-side gradient.
Total Variation and TVD Schemes
UD 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
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
Consider the discrete data set shown in the figure.
The total variation of this data set is
2 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)
For monotonicity, this TV must not increase with
time.
3 1 5 3
φ φ φ φ
In 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.
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 Schemes
Necessary and sufficient conditions for a scheme to be TVD
1) For 0 < r < 1 → ψ(r) ≤ 2r
2) For r ≥ 1 → ψ(r) ≤ 2
UD scheme is TVD
LUD scheme is not TVD for r > 2
CD scheme is not TVD for r < 0.5
QUICK scheme is not TVD for r < 3/7 and r > 5
Requirement for Second Order Accuracy
For 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
Region for a second-order TVD scheme

38
Symmetry Property for Limiter Functions
Symmetry Property for limiter functions:
( )
(1/ )
r
r
r
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
Flux Limiter Functions
2
Van Leer Van Leer (1974)
1
V Alb d V Alb d (1982)
r r
r
r r
l
+
+
+
Name Limiter function Source
2
Van Albada Van Albada . (1982)
1
min( ,1) if 0
Min-Mod ( ) Roe (1985)
0 if 0
SUPERBEE max[0,min(2 ,1),min( ,2)] Roe
et al
r
r r
r
r
r 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 r
r r
β β
+
UMIST max[0,min(2 ,(1 3 )/ 4,(3 )/ 4,2)] Lien and Leschziner (1993)
r r r
+ +
0 ≤ β ≤ 2
β = 1 → Min-Mod Limiter
β = 2 → SUPERBEE Limiter of Roe

39
Flux Limiter Functions
MIN‐MOD
MIN‐MOD
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 d
u
dx dx dx
φ
ρ 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
( )( )
2
1
( )( )
e P e E P
r
r
φ φ ψ φ φ
φ φ ψ φ φ
+
+
= + −
+
The face values are:
P W
e
E P
W WW
r
r
φ φ
φ φ
φ φ
+
+
⎛ ⎞
−
= ⎜ ⎟
−
⎝ ⎠
⎛ ⎞
−
= ⎜ ⎟
For u > 0
(5.82)
( )( )
2
w W w P W
r
φ φ ψ φ φ
= + − w
P W
r
φ φ
= ⎜ ⎟
−
⎝ ⎠
1
( )( )
2
1
( )( )
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 E
e
E P
E P
w
P W
r
r
φ φ
φ φ
φ φ
φ φ
−
−
⎛ ⎞
−
= ⎜ ⎟
−
⎝ ⎠
⎛ ⎞
−
= ⎜ ⎟
−
⎝ ⎠
(5.83)

40
Implementation of TVD Schemes
The discretisation equation takes the form
The central coefficient is
P P W W E E dc
a a a S
φ φ φ
= + + (5.84)
(5 85)
where
( )
P W E e w
a a a F F
= + + −
max[ ,0]( ) max[ ,0]( )
max( ,0) max(0, )
max[ ,0]( ) max[ ,0]( )
w e dc
w w W w w P
w w e e
e e E e e P
a a S
F F
D F D F
F F
φ φ φ φ
φ φ φ φ
− − − −
+ + −
+ − − − −
(5.85)
(5.86)
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
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.

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