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7T H INDO GERMAN WINTER ACADEMY- 2008

Discretization of convection-diffusiontype equations by

Finite Volume Method

Ritika Tawani

Department of Chemical EngineeringIndian Institute of Technology, Bombay

Guides:Prof. Suman Chakraborty, IIT-Kharagpur

Prof. Vivek V. Buwa, IIT-Delhi


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Contents

The Convection Diffusion Equation Finite Volume Method

Four basic rules Central Differencing Scheme Upwind Differencing Scheme Exact Solution Exponential Scheme Hybrid Scheme Power Law Scheme Higher Order Differencing Schemes QUICK Scheme Discretization Equations for 2-D, 3-D

Handling the Source term Handling the Unsteady term

False Diffusion


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The Convection Diffusion Equation

The general differential equation, for the conservation of a physicalproperty,!

The 4 terms are: Unsteady term, Convection term, Diffusion term andSource term

In general,! =!(x, y, z, t)=!(x, y, z, t)(x, y, z, t)

! is the diffusion coefficient corresponding to the particular property!, S is the corresponding source term, S is the corresponding source term

As! takes different values we get conservation equations for differentquantities

eg:!=1: Mass conservation=1: Mass conservation

!=u: x-momentum conservation=u: x-momentum conservation

!=h: Energy conservation=h: Energy conservation


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Key concept: Integration of differential equation over Control Volume

For simplification, we first do finite volume formulation for 1-D steadystate equation(with no source term)

The flow field should also satisfy continuity equation


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Control Volume(CV) to be used:

Integration of transport equation for the shown CV gives

Derivatives for diffusion term are calculated assuming piecewise linearprofile of!

,


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Assuming , the integral of transport equation becomes,

where,

Also, from continuity equation, we have There are various methods to calculate the Convection term and will be

discussed after the four basic rules


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Four Basic Rules

For solutions to be: 1. Physically realistic2. Satisfy overall balance (conservative)

There are some basic rules that need to be satisfied by the discretizationequations

Standard form of discretization equations(1-D):

Rule 1: Flux consistency at CV facesWhen a face is common to two adjacent control volumes, flux across it

must be represented by the same expression in discretization equationsfor both the control volumes

Rule 2: Positive coefficients

All coefficients must always be of same sign because an increase inmust lead to increase in


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Four Basic Rules

Rule 3: Negative slope linearization of source term

If source term is dependent on!, it is linearized as:

This will then appear in along with other terms. To ensureremains positive, must be negative or zero

Rule 4: Sum of neighbour coefficients

If governing differential equation contains only derivatives of!, both

! and!+c will satisfy the equation. In this case,


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Central Differencing Scheme

The Convective term is evaluated using piecewise linear profile of!

Transport equation becomes,


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Discretization equation can be written aswhere

Assessment

Conservativeness : Uses consistent expressions to evaluate convectiveand diffusive fluxes at CV faces.

Unconditionally Conservative

Boundedness : will become negative ifScheme is conditionally bounded ( )


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Transportiveness : The CDS uses influence at node P from alldirections. Does not recognize direction of flow or strength ofconvection relative to diffusion

Does not possess Transportiveness at high Peclet Numbers

Accuracy : Second Order in terms of Taylor seriesStable and accurate only if

Now,

For stability and accuracy, either velocity should be very low or gridspacing should be small


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Upwind Differencing Scheme

The diffusion term is still discretized using piecewise linear profile of! For convection term,! at interface is equal to! at the grid point on

the upwind side

is defined similarly Define , then, upwind scheme gives

Discretization equation:


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Upwind Differencing Scheme

Assessment Conservativeness : It is conservative

Boundedness : When flow satisfies continuity equation, all coefficientsare positive. Also, which is desirable for stable iterativesolutions 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 falsediffusion, which will be discussed later


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Exact Solution

The governing transport equation:

If! = constant, the equation can be solved exactly

Boundary conditions: , Solution:

where,


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Exponential Scheme

Define Our transport equation becomes,

Integrating over CV,

The exact solution derived above can be used as profile assumptionwith

Substitution giveswhere


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Exponential Scheme

After substitution of similar expression for , equation in our standardform can be written as:

Merit: Guaranteed to produce exact solution for any Peclet number for1-D steady convection-diffusion

Demerits: 1. exponentials expensive to compute

2. not exact for 2-D, 3-D


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Hybrid Scheme

In exponential scheme,


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Hybrid Scheme

From Figure, we can see that1.

2.

3.

The 3 straight lines representing these limiting cases are shown in figure

The hybrid scheme is made up of these 3 straight lines,


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Hybrid Scheme

Standard Discretization equation

Significance of HDS:1. Combines advantages of both CDS and UDS

2. Identical to CDS for -2 2

3. Outside this range, it reduces to UDS with diffusion set equal to zero Disadvantage: First order accuracy in terms of Taylor Series


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Power Law Scheme

Similar to HDS but more accurate Diffusion is set equal to zero for >10 or < -10 Otherwise diffusion is calculated from a polynomial expression

Discretization equation


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Higher Order Differencing schemes

CDS has second order accuracy but does not posses transportivenessproperty.

Upwind, hybrid schemes are very stable and obey transportiveness butare first order in terms of Taylor series truncation error which makesthem prone to diffusion errors.

Such errors minimized by employing higher order discretisation. Higher order schemes involve more neighbour points and reduce

discretization errors by bringing wider influence.

Formulations that do not take into account the flow direction areunstable and, therefore, more accurate higher order schemes, whichpreserve upwinding for stability and sensitivity to flow direction, are

needed.


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Quadratic upwind differencing scheme(QUICK)

Quadratic upstream interpolation for convective kinetics(QUICK) 3 point upstream-weighted quadratic interpolation used for cell face

values

For,


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QUICK Scheme

Diffusion terms are evaluated using gradient of the appropriateparabola (For uniform grid, gives same results as CDS for diffusion) Discretized convection diffusion transport equation:

Standard form of discretized equation

Similarly, coefficients can be obtained for


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QUICK Scheme

Assessment Conservativeness : Ensured Boundedness : For , is always negative, can

become negative for , thus the scheme isconditionally stable.

Transportiveness : Built in because the quadratic function is based on 2upstream and 1 downstream node

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

Another feature : Discretization equations not only involve immediateneighbour nodes but also nodes further away, thus TDMA methods are

not applicable


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QUICK Scheme

QUICK scheme above can be unstable due to negative coefficients Reformulated in different ways- Formulations involve placing -ve

coefficients in source term to retain +ve main coefficients

The Hayse et el(1990) QUICK scheme is summarized as:

Discretization equation:


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QUICK Scheme

Summarizing: Has greater formal accuracy than central differencing or hybrid

schemes and it retains upwind weighted characteristics

But, can sometimes give minor undershoots and overshoots(examplegiven later)

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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Discretization Equations for 2-D, 3-D

Discretization Equation for 2-D

Discretization Equation for 3-D

The coefficients for 2-D, 3-D for hybrid differencing scheme are shown onnext page


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Coefficients for 2-D, 3-D(HDS)


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Summary


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Handling the Source term

For 1-D, Discretization equation simply becomes, If the source term is a constant , then all other coefficients

remain same and,

If source term is dependent on!, linearization is done as:In this case, b and become,

All other coefficients remain same

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


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Handling the Unsteady term

Density remains constant(from continuity equation)

Now we need an assumption for with t, We assume

We use similar formulas for and


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Final Discretization Equation:

where,


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If f=0: Scheme is explicit If f=0.5: Crank Nicholson Scheme If f=1: Implicit Scheme Variation of Temperature with time for the three schemes is :


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Analysis: Explicit Scheme:

The coefficient of becomes negative if exceeds

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

Crank Nicholson Scheme: Coefficient of is


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Even in Crank- Nicholson Scheme, if the time step is not sufficientlysmall, the coefficient of will become negative Crank Nicholson Scheme is also conditionally stable

Implicit Scheme: Only in this case, the coefficient of is alwayspositive. Thus, fully implicit scheme satisfies requirements of simlicityand physically realistic behavior.

However, at small time steps, Crank Nicholson scheme is more accuratethan fully implicit scheme

Reason: Temperature time curve is nearly linear for small timeintervals which is exactly what we assumed in Crank Nicholson scheme


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False Diffusion - Common View

CDS has 2

nd

order accuracy while UDS has 1st

order accuracy : FromTaylor series expansion

UDS causes severe false diffusion : UDS is equivalent to replacing! inthe CDS by!+!u!x/2

CDS is better than UDS (misleading, true only for small Pe)

Problem with this view:

Truncated taylor series ceases to be a good representation(except forsmall!x or small Pe), since!~x variation is exponential

We assumed CDS as standard, then compared diffusion coefficient ofUDS with that of CDS

The so called false diffusion coefficient!u!x/2 is indeed desirable atlarge Peclet numbers


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False Diffusion - Proper View

Important only for large Pe(for small Pe, real diffusion is large enough) Multidimensional phenomena Consider example: 2 parallel streams with equal velocity, nonequal

Temperature contacted

If! 0, mixing layer forms where T changes from higher to lower value If!=0, T discontinuity persists in streamwise direction=0, T discontinuity persists in streamwise directionTo observe false diffusion: set!=0, If numerical solution producessmeared T profile(characteristic of!0), it entails false diffusion


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CDS: For!=0, it gives non unique or unrealistic solutions UDS:

1. Uniform flow in x-direction:!=0 and y-direction velocity = 0

Thus, given upstream value

on each horizontal line gets

established at all points on

that line

No false diffusion


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2. Uniform flow at 45togrid lines(say, x=y)

Results obtained are shown in

adjacent figure

Thus, false diffusion is

observed

For no false diffusion:

!=100 above the diagonal

!=0 below the diagonal


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The above problem solved for different grid sizes gives the followingresults


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Conclusions

Occurs when flow is oblique to grid lines and nonzero gradient exists indirection normal to flow

False diffusion reduction: Use smaller x and y, allign grid lines morein direction of flow

Enough to make false diffusion


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QUICK Scheme (cont)

The above problem if solved on a 50*50 grid using Upwind and QUICKschemes gives the following results.

Notice the undershootsand overshoots by the

QUICK scheme


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References

Ferziger J. H. and Peric M. Computational Methods for FluidDynamics

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


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