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Discretization of Convection

Diffusion type equation

By,

Rajesh Sridhar,Indian Institute of Technology Madras

Guides:

Prof. Vivek V. Buwa Prof. Suman Chakraborty

IIT Delhi IIT Kharagpur

10th Indo German Winter Academy 2011


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Agenda:

Introduction

Finite Volume Discretization

Introduction

Spatial Discretization

Spatial Discretization for 2-D and 3-D problems Source Term Discretization

Unsteady Term Discretization

False Diffusion


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Introduction:

General transport equation for the conservation of property is,

Unsteady

term

Convection

term

Diffusion

term

Source

term

is the diffusion coefficient corresponding to and S is the source term.

The above equation transforms into certain famous equations for different

s.

= 1 Mass Conservation Equation

= v Momentum Conservation Equation

= e Energy Conservation Equation


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Finite Volume Method:

Key concept in FVM:

Integration of the differential equation over the Control

Volume.

Solution Procedure:

Grid Generation. Discretization of differential equation into algebraic equations.

Simultaneous solution of the algebraic equations.


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Grid Generation:

Divide the domain into discrete control volumes.

Place a number of nodes between the system boundaries.

Boundaries of control volumes are placed midway between the

adjacent nodes.


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Discretization:

aPP = anbnb + b

The general form of discretized equation is,

where, indicates summation over the neighboring nodes,

anb is the neighboring node coefficient,nb is the value of property at the neighboring node,

b is a constant.


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Discretization Schemes:

For the solutions to be physically realistic, a discretization schemeshould satisfy the following requirements:

1. Conservativeness

2. Boundedness

3. Transportiveness

Conservativeness:Flux consistency at the CV faces. i.e.

Flux of leaving a control volume = Flux of entering the adjacent control

volume through the same face


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Boundedness:

The discretized equations are solved by iterative methods.

Sufficient condition for a convergent iterative solution is,

|anb||aP|< 1

where,

aP is the net coefficient of the central node. i.e. aP SP

Negative Slope Linearization of Source term:

If the source term depends on , it can be linearizedas,

S = SC + SPP


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|anb|

|aP|

This constraint can be satisfied if aP is made as large as possible. Thiscan be achieved if the scheme can ensure,

i. SP is always negative.

All coefficients should be of the same sign, preferably positive. i.e. an

increase in at one node should result in an increase in atneighbouring nodes.

If SP becomes zero, then both and + c will satisfy the equation.

< 1

Boundedness:


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Transportiveness:

A non-dimensional Peclet number is defined to measure the relative

strengths of convection and diffusion.

Pe = =

where,

F = convective mass flux per unit area D = diffusion conductance

= fluid density u = fluid flow velocity = diffusion coefficient x = characteristic length

DF u

/x

Primary Objective: The relationship between the magnitude of Peclet

Number and the directionality of influencing .i.e. Transportiveness should

be borne out of the discretization scheme.


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Consider the general contours of constant for a constant source at P for different values of Pe depicted below.

For Pe = 0 .i.e pure diffusion, the contours are concentric circles for

constant values of because diffusion tends to spread out evenly in all

directions.

As Pe increases, the curve becomes more elliptic and the value of at E

node is more influenced by the upstream node.

At Pe = .i.e pure convection, the contours are completely outstretched

and value of at E is affected only by upstream conditions.


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Spatial Discretization:

Consider the steady convection and diffusion of a property in a one-dimensional flow field u.

The flow must also satisfy continuity equation.

Integration of the transport equation over the control volume

yields,

Integration of the continuity equation yields,


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Spatial Discretization:

Assuming Ae = Aw = A and employing central differencing approach to

represent the diffusion term, we get

The integrated continuity equation,

where,

Schemes for calculating the value of at the cell faces will be discussednow.


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Spatial Discretization Schemes:

Central Differencing Scheme.

First Order Upwind Scheme

Exact Solution

Exponential Scheme

Hybrid Scheme

Power Law Scheme

QUICK Scheme


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Central Differencing Scheme:

Here, we use linear interpolation for computing the cell face values.

W Ew eP

W

wP

e E

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

2 2

Transport equation becomes,


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The final discretized equation can be written as,

where,

Assessment:

Conservativeness:

Uses consistent expressions to evaluate convective and

diffusive fluxes at CV faces. Unconditionally Conservative.


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Boundedness:

aW or aE will become negative if Pe > 2, depending on the flow

direction.Scheme is conditionally bounded ( 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.

Fe2

< 0 Pe > 2aE = De -


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Accuracy:

Second Order in terms of Taylor series.

Stable and accurate only if Pe < 2.

For a low value of Pe, either velocity should be less or

the grid spacing should be small.


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First Order Upwind Scheme:

Here,

e = P

e = E

if, Fe > 0

if, Fe < 0 W Ew eP

W wP e E

w is defined similarly.

Now, we can define [[ ]] operator as,

[[ A , B ]] = MAX (A,B).

The final discretization equation is,

where,


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First Order Upwind Scheme:

Assessment:

Conservativeness:

Utilises consistent expressions to calculate through cell faces.

Hence, conservative.

Boundedness:When flow satisfies continuity equation, all coefficients

are positive. Also, aP = aE + aW which is desirable for stable iterative

solutions of linear equations.

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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Exact Solution:

The transport equation,

If is constant, the above equation can be solved exactly for theboundary conditions,

= 0 x = 0

= L x = L

Solution:

where, P =uL


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Exponential Scheme:

Here, we define a variable J such that J = u d

Now, our transport equation can be expressed as,

dJ

Integrating over the CV, we get

Je = JwThe exact solution derived in the previous slide can be used as a profile

assumption between P and E with 0 being replaced by P ,L by E ,L by xe

dx

dx= 0

resulting in

where,


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Exponential Scheme:

Similar substitution for Jw results in the standard discretization equation,

with the values of coefficients,

Merits:

Guaranteed to produce exact solution for any Pecletnumber for 1-D steady convection-diffusion.

Demerits:

Exponentials are expensive to calculate.

Not exact for 2-D and 3-D cases.


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Hybrid Differencing Scheme:

Combination of Upwind and Central Differencing Scheme.

Central Differencing Scheme (2nd order accurate) for small Pe

(Pe < 2).

Upwind Scheme (transportive) for large Pe (Pe > 2).

Simplification of the exponential scheme for easier computation.

-2 2 Pe

aEDe

Exponential schemeHybrid Scheme


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aEDe

=

-Pe

1 -Pe2

-2 < Pe < 2

Pe > 2

Pe < 2

0

So, we can define a hybrid piece-wise linear function as,

From exponential scheme,


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The final discretization equation along with the value of the coefficientsis given below:

Assessment:

Combines advantages of both Upwind and CDS schemes.

Conservative and unconditionally bounded.

Satisfies transportiveness by using Upwind scheme for large Pe.

Only disadvantage is accuracy in terms of Taylor series expansion

which is first order.


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Power Law Scheme: Similar to Hybrid Differencing Scheme, but more accurate and

produces better results.

Diffusion is set equal to zero for |Pe| > 10.


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Higher Order Differencing Schemes:

Need for Higher Order Schemes:

Accuracy of Upwind and Hybrid schemes is only first order in

terms of Taylor Series truncation error, leading to numerical diffusion

errors.

Central differencing scheme is second order accurate but doesnot

possess transportiveness property, hence unstable.

Errors reduced by bringing in a wider influence involving more

neighbour points.

Formulations that do not take into account the flow direction are

unstable and, therefore, more accurate higher order schemes, which

preserve upwinding for stability and sensitivity to flow direction, are

needed.


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QUICK Scheme:

QUICK stands for Quadratic Upwind Interpolation for Convective Kinetics.

3 point upstream-weighted quadratic interpolation used for cell facevalues

WW W P E EE

w e

uw ue

WW

W wP e E

EE

When Fe > 0, e = P+ E - W

68

38

1

8

When Fw > 0,

w = W+ P - WW6

838

1

8


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QUICK Scheme:

General expressions valid for both positive and negative flow directions

is given below:

with central coefficient ,

and neighbouring coefficients,


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Assessment of QUICK Scheme:Conservativeness:

Uses consistent quadratic profiles. Hence, conservative.

Boundedness:

Conditionally stable. For uw > 0 and ue > 0, aWW is

negative and aE can become negative for Pe > 8/3.

Transportiveness:

Built-in because the quadratic function is based on 2

upstream and 1 downstream node.

Accuracy:Third order in terms of Taylor series truncation error on

a uniform mesh.

Note: Discretization equations not only involve immediate

neighbour nodes but also nodes further away, thus TDMA methods

are not directly applicable.


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QUICK Summary:

Has greater formal accuracy than central differencing or hybrid

schemes and it retains upwind weighted characteristics.

But, can sometimes give minor undershoots and overshoots.

Other higher order schemes: Use increases accuracy. Implementation of Boundary Conditions can be problematic.

Computation costs also need to be considered.

To avoid undershoots and overshoots (get oscillation free solution),

class of TVD (Total variation diminishing) schemes have been

formulated.


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Spatial Discretization for 2-D and 3-D:

Discretization in 2D:

Discretization in 3D:


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Spatial Discretization for 2-D and 3-D:

Coefficients for HDS in 2D and 3D cases:


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Spatial Discretization Summary:


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Source Term Discretization:

In a one-dimensional case, the discretization equation simply becomes,

If the source term is a constant S = SC then all other coefficients

remain same and,

If source term is dependent on , linearization is done as:

In this case, b and aP become,

All other coefficients remain same.

In a similar way, Source term can be incorporated in 2-D and 3-D.

aPP = aEE + aWW + b

b = SCx

S = SC + SPP


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Consider a 1-D unsteady diffusion equation.

Integration over the control volume gives,

Unsteady Term Discretization:

w

e

= eE -P

xe

- wP -W

xw

t tt+tt+t


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Now we need an assumption for with t, we assume

t t+t

f =1 Implicitf = 0.5 Crank-Nicholsonf = 0 Explicit

where 0< f


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The final discretization equation becomes,

aPP1 = aE{fP

1+(1-f)E0} + aW {fW

1 +(1-f)W0} + {aP

0-(1-f)aE (1-f) aW} P0

where,


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Asssesment:

Explicit Scheme:

aPP1 = aEE

0 + aWW0 + {aP

0- aE - aW} P0

The coefficient of P0 becomes negative if aP

0 exceeds aE + aW.

For uniform conductivity and equal grid spacing, scheme is stable if

Crank Nicholson Scheme:

Coefficient of P0 is


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Even in Crank-Nicholson Scheme, if the time step is not sufficientlysmall, the coefficient of P

0 will become negative.

Crank Nicholson Scheme is also conditionally stable.

Implicit Scheme:

Only in this case, the coefficient of P0 is always positive. Thus,

fully implicit scheme satisfies requirements of simplicity and physically

realistic behaviour.

However, at small time steps, Crank Nicholson scheme is more

accurate than fully implicit scheme.


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False Diffusion

Usually it is stated that the Central difference scheme is superior to theUpwind scheme because it is second-order accurate whereas the Upwind

scheme is only first-order accurate.

But when we compare the Central difference and Upwind schemes:

This added diffusion is considered bad at small Peclet numbers, however it

actually corrects the solution at large Peclet number.


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False Diffusion

False diffusion is a numerically introduced diffusion and arises inconvection dominated flows, i.e. high Pe number flows.

False diffusion is a multidimensional phenomenon and occurs when the

flow is perpendicular to the grid lines.

Cold FluidT = 0C

Hot Fluid

T = 100C

Consider the following

example:

If there was no false

diffusion, then the temperature

will be exactly 100 Ceverywhere above the diagonal

and exactly 0 C everywhere

below the diagonal.

X

X


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False Diffusion


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False Diffusion Upwind Scheme:


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False Diffusion QUICK Scheme:

It diverges!


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False Diffusion - Summary:

Occurs when flow is oblique to grid lines and nonzero gradient existsin direction normal to flow.

False diffusion reduction: Use smaller x and y, align grid lines more

in the direction of flow.

Enough to make false diffusion


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References

Patankar S.V. Numerical Heat Transfer and Fluid Flow

Versteeg H. K. and Malalasekera W. An introduction to computational

fluid Dynamics: The finite volume method

Lecture Notes by Dr. Shamit Bakshi, IIT Madras.


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Thank You!

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