upwind2

Discretization of Convection
Diffusion type Equation
8th Indo-German Winter Academy 2009
Piyush Kumar Sao
Department of Electrical Engineering
Indian Institute of technology Madras,Chennai
Guides:
Prof. Suman Chakraborty, IIT-Kharagpur
Prof. Vivek V. Buwa, IIT-Delhi
Presentation Outline
o Introduction
o Finite Volume Method
o Spatial discretization
o Source term discretization
o Temporal discretization
o Convergance
o False diffusion
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PIYUSH KUMAR SAO IIT MADRAS
Introduction
General transport equation for the conservation of property φ :

∂ ( ρφ )
+ div( ρ u φ ) = div(Γ gradφ ) + S
∂t
rate of change
with time
convection
 Here φ is the general variable ,Γ is diffusion
diffusion
source term
constant corresponding to φ,S is
source term.

We shell look into different schemes for formulating set of linear equation to
solve it numerically ,numerical stability of formulation and false diffusion.
PIYUSH KUMAR SAO IIT MADRAS
Finite Volume Method
Basic methodology
Divide domain into control volumes
oIntegrate the differential equation over the control volume and apply divergence
theorem
oTo evaluate the derivative term , values at face of control volume are needed ,
assume a profile.
oResult is set of linear algebraic equation :one for each control volume
oSolve them iteratively or simultaneously
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Finite Volume Method
Basic methodology(1)
oControl volumes do not over lap
oNet flux through control volume is sum of flux through all four faces (6 in 3D)
oValues at faces are not known in general and found by interpolation.
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Finite Volume Method
Basic methodology(1)
So finally we get equation for cell P like
apφ = aWφ + aEφ + aNφ + aSφ + b
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PIYUSH KUMAR SAO IIT MADRAS
Finite Volume Method
4 basic rules
 For solutions to be physically realistic and satisfy overall
balance (conservative) There are some basic rules that need
to be satisfied by the discretization equations.
1. Flux consistency at CV faces
2. Positive coefficients
3. Negative slope linearization of source term
(
Sp should be negative)
4. Sum of neighbor coefficients
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Spatial discretization
(central difference and upwind)
To start with we shell consider simplest case 1D steady state
(with no source term)
It should also satisfy continuity equation
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Spatial discretization
(central difference and upwind)
 Consider the 1-dimensional grid system (∆y and ∆z = 1):
(δx) w
(δx) e
w
e
W
P
E
∆x
We integrate diffusion convection over control volume
 dφ   dφ 
( ρ uφ ) e − ( ρ uφ ) w =  Γ

 − Γ
 dx  e  dx  w
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PIYUSH KUMAR SAO IIT MADRAS
Spatial discretization
(central difference and upwind)
Central Differencing Scheme
If we assume linear profile
W
w
P
e
E
Γ (φ − φP ) Γw (φP − φW )
 dφ   dφ 
−
 = e E
 − Γ
Γ
dx
dx
(
δ
x
)
(δx) w
w
e 

e
( ρ u φ ) e − ( ρ uφ ) w =
10
1
1
( ρ u ) e (φE + φP ) − ( ρ u ) w (φP + φW )
2
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PIYUSH KUMAR SAO IIT MADRAS
Spatial discretization
(central difference and upwind)
Now we use following notation
Our equation becomes
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Spatial discretization
(central difference and upwind)
 Conservativeness : Uses consistent expressions to evaluate
convective and diffusive fluxes at CV faces. Hence
Unconditionally Conservative
 Boundedness :
will become negative if
Scheme is conditionally bounded for Pe < 2
 Transportiveness : The CDS uses influence at node P from all
directions. Does not recognize direction of flow or strength
of convection relative to diffusion Does not possess
Transportiveness at high Peclet Numbers
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PIYUSH KUMAR SAO IIT MADRAS
Spatial discritization
(central difference and upwind)
 Accuracy : Second Order in terms of Taylor series Stable and
accurate only if
Now
Hence For stability and accuracy, either velocity should be very
low or grid spacing should be small
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PIYUSH KUMAR SAO IIT MADRAS
Spatial discritization
(central difference and upwind)
 First order upwind scheme
Similarily for
If
φw
[[ A, B ]] = Max( A, B )
Then discretization equation
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PIYUSH KUMAR SAO IIT MADRAS
Spatial discretization
(central difference and upwind)
First order upwind scheme
 Conservativeness : It is conservative
 Boundedness : When flow satisfies continuity equation, all
coefficients are positive.
 Transportiveness : Direction of flow inbuilt in the
formulation, thus accounts for transportiveness.
 Accuracy : When flow is not aligned with the grid lines, it
produces false diffusion, which will form last part of our
discussion.
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PIYUSH KUMAR SAO IIT MADRAS
Spatial discretization
(exact solution)
 The governing transport equation:
d
d  dφ 
(ρ u φ ) =  Γ

dx
dx  dx 
 This can be exactly solved if
 Boundary conditions:

Where
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Γ
is constant
Spatial discretization
Example
 Consider following convection diffusion equation
 Suppose f=0; then exact solution is
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Spatial discretization
Example
Exact solutions for w = 1, 5, 10, 20
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Spatial discretization
Solution by CDS
 Suppose we choose N=4 on interval [0,1] then
x0 = 0, x1 =
 Here
1
∆x =
4
1
2
3
, x2 = , x3 = , x4 = 1
4
4
4
Hence
 Hence we get 3 equation as
w
1
Fe = Fw = ; De = Dw = 2
∆x
∆x
w 
w 
 1   1
 1
ui +1 = 0; i = 1,2,3
ui −i +  2 ui +  − 2 +
− 2 −
 ∆x   ∆x 2∆x 
 ∆x 2∆x 
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PIYUSH KUMAR SAO IIT MADRAS
Spatial discretization
Solution by CDS
 Further we need boundary condition , in this case it is
u0 = 1; u4 = 0
− 16 + 2w
0   u1  16 + 2w 
 32

  

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− 16 + 2w   u2  =  0 
 − 16 − 2w

u   0 
w
−
16
−
2
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
 3  

 Now we solve it to get values of u1,u2 and u3
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Solution by CDS
Exact and numerical solution for u(x),w=40,N=5,10,15 and 20
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Solution by First order upwind
 Here coefficients are different from CDS
 2 w  1 
 1 w
 − 2 − ui −i +  2 + ui +  − 2 ui +1 = 0; i = 1,2,3
 ∆x ∆x   ∆x 
 ∆x ∆x 
 Boundary condition
u0 = 1; u4 = 0
− 16
0   u1  16 + 4w 
 32 + 4w

  

− 16   u2  =  0 
 − 16 − 4w 32 + 4w

u   0 
w
w
−
16
−
4
32
+
4

 3  

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PIYUSH KUMAR SAO
IIT MADRAS
Spatial discretization
Solution by first order upwind
Exact and numerical solution for u(x),w=40,N=5,10,15 and 20
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PIYUSH KUMAR SAO IIT MADRAS
Exponential scheme
(exponential)
 Define
 Our transport equation becomes
 integrating over CV
 The exact solution derived above can be used as profile
assumption with
 Substitution gives
 Where
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Exponential scheme
(exponential)
 After substitution of similar expression for
standard form can be written as:
J w , equation in our
 Merit: Guaranteed to produce exact solution for any Peclet
number for 1-D steady convection-diffusion
 Demerits: 1. exponentials expensive to compute
2. not exact for 2-D, 3-D
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PIYUSH KUMAR SAO IIT MADRAS
Hybrid Scheme
 In exponential scheme
P
aE
= Pe e
De
e −1
 Hence
 Hybrid scheme is piecewise linear approximation of
exponential scheme
Pe < −2
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P
aE
a
a
= − Pe ;−2 < Pe < 2 E = 1 − e ; Pe > 2 E = 0
2
De
De
De
PIYUSH KUMAR SAO IIT MADRAS
Power Law scheme
 Another approximation to exponential scheme
 Diffusion is set equal to zero for
 Otherwise calculated as follows

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PIYUSH KUMAR SAO IIT MADRAS
Second order upwind scheme
We determine the value of φ
from the cell values in the
two cells upstream of the
face.
More accurate than the first
order upwind scheme, but in
regions with strong
gradients it can result in face
values that are outside of the
range of cell values. It is then
necessary to apply limiters
to the predicted face values.
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PIYUSH KUMAR SAO IIT MADRAS
φW
φ(x)
interpolated
value
φP
φe
W
P
Flow direction
e
φE
E
Quick Scheme
QUICK stands for Quadratic
Upwind Interpolation for
Convective Kinetics.
A quadratic curve is fitted
through two upstream nodes
and one downstream node.
This is a very accurate scheme,
but in regions with strong
gradients, overshoots and
undershoots can occur. This
can lead to stability
problems in the calculation.
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PIYUSH KUMAR SAO IIT MADRAS
φW
φ(x)
interpolated
value
φP
W
P
Flow direction
φe
φE
e
E
Accuracy of different scheme
o
Each of the previously discussed numerical schemes assumes some shape of the function
φ. These functions can be approximated by Taylor series polynomials:
n
( xP )
φ ' ' ( xP )
φ
φ ' ( xP )
( xe − x P ) +
( xe − xP ) 2 + .... +
φ ( xe ) = φ ( x P ) +
( xe − xP ) n + ...
2!
n!
1!



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The first order upwind scheme only uses the constant and ignores the first derivative
and consecutive terms. This scheme is therefore considered first order accurate.
For high Peclet numbers the power law scheme reduces to the first order upwind
scheme, so it is also considered first order accurate.
The central differencing scheme and second order upwind scheme do include the first
order derivative, but ignore the second order derivative. These schemes are therefore
considered second order accurate. QUICK does take the second order derivative into
account, but ignores the third order derivative. This is then considered third order
accurate.
PIYUSH KUMAR SAO IIT MADRAS
Spatial Discretization 2-D and 3-D
Discretization in 2D
Discretization in 3D
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Spatial Discretization 2-D and 3-D
Coefficients for 2-D, 3-D(HDS)
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Source term discretization
 For 1-D, Discretization equation simply becomes,
aPφP = aEφE + aW φW + b
 If the source term is a constant then
 If Source term is dependent of
s.t.
 Hence
b = S∆x
φthen linearize S about P
 And all other coefficients remain the same
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PIYUSH KUMAR SAO IIT MADRAS
Temporal discretization
For simplicity we consider time-dependent heat conduction:
ρc
∂  ∂T 
∂T
=
k

∂t ∂x  ∂x 
We integrate in time (from t to t+∆t) and space:
e t + ∆t
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ρ c∫
∫
w
t
∂T
dt dx =
∂t
W
PIYUSH KUMAR SAO IIT MADRAS
t + ∆t e
∫∫
t
w
∂  ∂T 
k
 dx dt
∂t  ∂x 
(δx) w
(δx) e
w
e
P
∆x
E
Temporal discretization
If we assume the value of T prevails over the entire volume:
e t + ∆t
ρ c∫
∫
w
t
∂T
dt dx = ρ c∆x (TP1 − TP0 )
∂t
For the diffusion term:
ρ c∆x (T − T ) =
1
P
0
P
t + ∆t
∫
t
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PIYUSH KUMAR SAO IIT MADRAS
 k e (TE − TP ) k w (TP − TW ) 
dt
−
 (δx )

(δx ) w
e


Temporal discretization
We need an assumption for the variation of T in time (between t and t+∆t):
t + ∆t
∫
[
]
TP dt = f TP1 + (1 − f ) TP0 ∆t
t
where 0 < f < 1
f =0:
Explicit
scheme (first-order
accurate)
f = 0.5: CrankNicolcon (second-order
accurate)
f = 1:
Implicit (firstorder accurate)
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Explicit
TP0
Crank-Nicolson
TP1
PIYUSH KUMAR SAO IIT MADRAS
Implicit
Temporal discretization
For stability, the Explicit scheme requires (on uniform grid):
∆t <
ρ ( ∆x ) 2
2Γ
To give realistic results, the Crank-Nicolson scheme requires:
∆t <
ρ ( ∆x ) 2
Γ
The Implicit scheme is always stable (but still only first-order)!
Temporal and spatial discretization are strongly coupled
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PIYUSH KUMAR SAO IIT MADRAS
Temporal Discretization
 Schemes that are higher-order accurate in time exist, e.g.:
∂T
1
=
(3 T n +1 − 4 T n + T n −1 )
∂t 2∆ t
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PIYUSH KUMAR SAO IIT MADRAS
Iteration and Convergence
 At each iteration, at each cell, a new value for variable φ in cell P
can then be calculated from that equation.
 It is common to apply relaxation as follows:
φPnew, used =
φPold + U (φPnew, predicted − φPold )
Here U is the relaxation factor:
– U < 1 is underrelaxation. This may slow down speed of
convergence but increases the stability of the calculation, i.e. it
decreases the possibility of divergence or oscillations in the
solutions.
– U = 1 corresponds to no relaxation. One uses the predicted value
of the variable.
– U > 1 is overrelaxation. It can sometimes be used to accelerate
convergence but will decrease the stability of the calculation.
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PIYUSH KUMAR SAO IIT MADRAS
Convergence
o The iterative process is repeated until the change in the variable from
one iteration to the next becomes so small that the solution can be
considered converged.
o At convergence:
– All discrete conservation equations (momentum, energy, etc.) are obeyed
in all cells to a specified tolerance.
– The solution no longer changes much with additional iterations.
Residuals measure imbalance (or error) in conservation equations.
The absolute residual at point P is defined as:
RP = a Pφ P − ∑ nb anbφ nb − b
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PIYUSH KUMAR SAO IIT MADRAS
Convergance
 Always ensure proper convergence before using a solution:
unconverged solutions can be misleading!!
 Solutions are converged when the flow field and scalar fields
are no longer changing.
 Determining when this is the case can be difficult.
 It is most common to monitor the residuals.
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PIYUSH KUMAR SAO IIT MADRAS
False Diffusion
 Often it is stated that the Central difference scheme is
superior to the Upwind scheme because it is secondorder accurate whereas the Upwind scheme is only
first-order accurate.
If we compare the Central difference and Upwind schemes:
ΓUpwind = Γ +
ρ uδx
2
This added diffusion is considered bad, however it actually
corrects the solution at large Peclet number (large cells)!
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PIYUSH KUMAR SAO IIT MADRAS
False Diffusion
 False diffusion is numerically introduced diffusion and arises
in convection dominated flows, i.e. high Pe number flows.
 False diffusion is a multidimensional phenomenon and
occures when the flow is NOT perpendicular to the grid
lines!
Consider If there is no false
diffusion, the temperature will
be exactly 100 ºC everywhere
above the diagonal and exactly 0
ºC everywhere below the
diagonal.
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PIYUSH KUMAR SAO IIT MADRAS
Hot fluid
T = 100ºC
Diffusion set to zero
k=0
Cold fluid
T = 0ºC
False diffusion
First-order Upwind
8x8
64 x 64
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PIYUSH KUMAR SAO IIT MADRAS
Second-order Upwind
False diffusion
 An approximate expression for false diffusion is given by
Γ false
ρu∆x∆y sin 2θ
=
4(∆y sin 3 θ + ∆x cos 3 θ )
 False diffusion reduction: Use smaller ∆x and ∆y, allign grid
lines more in direction of flow, Enough to make false
diffusion << real diffusion
 CDS is no remedy for false diffusion. At high Pe, it produces
unrealistic results
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Summary
 We saw different scheme of discretizing convection diffusion
equation,and considered their numerical accuracy stability
and looked into false diffusion
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References
 Numerical heat transfer ,S.Pathnkar
 Computational Science , Gilbert Strang
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