Presentation by,

Raghavi Rao K

Department of Chemical Engineering

IIT Madras

Discretization of Convection- Diffusion Type Equations Computational Methods and Techniques,

IEWA 2014

Under the guidance of, Prof. Vivek V Buwa, IIT Delhi

Outline

1. Discussion of physical situations involving Convection and Diffusion and need for numerical

schemes

2. Finite volume discretization of diffusion eqn.

3. Finite volume discretization of 1D convection-diffusion eqn. (CDS method)

4. Properties of Discretization schemes: Consistency, Boundedness, Transportiveness and Accuracy

5. Other schemes Upwinding, Hybrid, Power law and Exponential

6. Higher order schemes QUICK

7. Discretization of unsteady equations

8. Summary

9. References

Physical Situations modeled by Convection-Diffusion equations

Need for Numerical Solutions

1

Physical Interpretation

Generalized transport equation

c could stand for mass, momentum, energy, species concentration

Control volume interpretation Types of transport

. - diffusion contribution

. - convection contribution

R is source/sink term

Applications

Applications exist in a wide variety of areas

Consider a falling liquid film with mass diffusion and energy transfer taking place. An engineering application of this theory is in ammonia- water refrigeration systems where water film absorbs ammonia. This is 2D convection and diffusion problem. The performance is analyzed by numerically solving these equations:

+

=

+

=

+

Falling liquid film absorption - schematic

Numerical Solutions

Analytical solutions not always possible in the case of pdes and odes with non linear terms

Numerical methods can give a very good approximation of the analytical solution.

Discretization methods: Finite Difference, Finite Volume and Finite Element methods

Well explore the more general Finite Volume method for convection diffusion equation in detail

Finite Volume Method

2

Finite Volume Method

Generic scalar transport equation, when integrated with respect to control volume can be written as follows

= . +

= . . +

= . +

The above equation can be applied over the entire domain discretized into control volumes.

Choice of construction of control volumes node centered CV and vice-versa

1D steady state diffusion:

1. Grid Generation

Each node is surrounded by a control

Volume or cell. And the west and the east

faces are marked by w and e notation

2. Discretization

+ S = 0

Integrating this over the control volume,

dV +

V = 0

Assuming constant area of cross-section,

-

+ = 0

= +

1D steady state diffusion(contd):

Assuming linear profiles of , and averaging S

+

+ = 0

It can be written in the form of

= + + b

where, =

, =

, = + , b =

= + b (General discretized form of equation)

3. Solution of the equations

Equations need to be solved simultaneously for all control volumes

() =

We have seen Finite Volume method applied to systems involving Diffusion and Source terms.

We may try to apply the same discretization scheme to convection terms also

Diffusion process affects the distribution of a transported quantity along its gradient in all directions, where as convection spreads its influence only in the flow direction. This crucial difference manifests itself in a stringent upper limit to the grid size, that is dependent on the relative strength of convection and diffusion, for stable convection - diffusion calculations

Could we apply the same technique to Convection Diffusion equations?

Central Differencing Scheme - 1d C-D equation

Applying the Central Differencing Scheme to a convection-diffusion case

Steady 1D convection and diffusion

() =

And equation of continuity

= 0

Integrating and applying the boundary conditions, (const. A)

() () = (

)

and () = 0

Let, = D/x and F = and = (+)/2, (

) =

= +

+2

2

+ +( )*

CDS scheme ( Contd. ) The above discretization holds for interior points

At the end points, we apply the boundary conditions

= at the west node and = at the east node

This results in equations of the form

= + +

= + + ( - ) -

With = , and = we have, for the corner nodes

Left node 0 /2 -(2 + ) (2 + )

Right node + /2

0 -(2 )

(2 )

Example

u = 0.1 m/s, 5 nodes

u =2.5 m/s, 5 nodes

u = 2.5 m/s, 20 nodes

Clearly, the CDS scheme is limited in agreement, at least for higher us and less number of nodes

Properties of Discretization

Conservativeness, Boundedness, Transportiveness, Accuracy

3

Conservativeness

Conservativeness

To ensure global conservation of flux, the value of flux used at the boundary of a control volume should match that

of its neighboring ones. This depends on the flux interpolation equation used

Consistent representation of fluxes will eliminate any spurious source terms

Consistency of CDS scheme

Linear interpolation of fluxes is only employed. Hence, consistency is preserved.

Boundedness To solve the equations resulting from discretization, we would require

Scarborough criterion to e satisfied.

||

|| 1, for all grid points

||

||< 1, at least for one node

Leads to Diagonally Dominant form of the matrix that is necessary ( but not sufficient) for the convergence of Gauss Siedel method

Another criterion to be satisfied is that all the coefficients (s) are positive and is negative

=

This qualitatively indicates that should be bounded by values at its neighboring points

Transportiveness A cell Peclet number can be defined as a measure of convection versus

diffusion in the control volume cell

=

()

Convection affects the relative influence of neighboring points over the node in question

interpolation in any discretization scheme should take the direction of flow into account

As Pe increases, the contours of become more ellipitical, decreasing the influence of the downstream points on the current node

Accuracy

Accuracy of any discretization scheme is of the order of truncation error in Taylor series used in interpolation of

For example, CDS has second order accuracy

= +

+2

2!2

2+ H.O.T (1)

=

+2

2!2

2 + H.O.T (2)

Summing (1) and (2), we have = (+)/2 + (second order terms)

Hence, second order accuracy

However, higher order schemes do not necessarily mean better solutions

Assessment of CDS

We have seen earlier that the coefficients at all the internal nodes are as below:

= +2

, = 2

, = + +( )

Boundedness Scarboroughs criterion are satisfied as = 0 , when continuity

equation is satisfied

The rule for all coefficients being positive and being negative is satisfied only

when 2> 0 , which implies that Pe < 2

So, CDS is not suitable under large velocities. However, the grid can be refined to bring Pe below 2

Transportiveness Direction of convection is not brought about in the discretization as it includes

influence of all neighboring points over P

Upwind, Hybrid, Exponential and Power Law schemes

False Diffusion

4

Upwind scheme

Upwind scheme brings this directionality into the discretization by these approximations

The form of equations remaining same, the coefficients are as below ( when u > 0 )

Conservativeness, Boundedness and Transportiveness satisfied

Boundary points Interior points

0/ +F +F

/0

-(2+F)/-2 0

(2+F)/ 2 0

False Diffusion

Consider this situation where a hot and a cold stream are flowing one over the other, without any diffusion at the boundary. One would expect the equations to suggest a discontinuity at the boundary

But, by the upwinding scheme, we get

that = 0.5 + 0.5

This equations result in a smeared profile like in the grid shown beside

Accuracy of the upwind scheme False diffusion

Upwind scheme uses backward/ forward differencing, resulting in first order truncation errors. This is lower than that of CDS!

= +

+2

2!2

2+ H.O.T

When we substitute = , the error of convected flux looks diffusive flux, with a particular

Upwind scheme also introduces False Diffusion when there is no diffusion

At low Pes, this terms contribution is insignificant in comparison to the actual diffusion. Grid Pe should at least be greater 10, to curb the error due to false diffusion

Exact solution

Fortunately, the exact solution of

() =

can be derived

For boundary conditions,

= 0 , = 0

= 1, =

The solution is,

00= (

.

1)/( 1)

with =

The above solution can be used for another

type of interpolation

Hybrid Differencing Scheme

When the exponential scheme is implemented, the coefficients are obtained as below:

Hybrid Differencing scheme uses a piecewise formula depending on the grid Pe

The general form of the equations are this-

= + with = + + ( - )

- Max[, +

2, 0]

- Max[ ,

2, 0]

=

exp 1

Power Law scheme

Deviation from exact solution at Pe = 2, the deviation is quite large

Diffusion effects are immediately set to 0, once Pe exceeds 2. This is not physically appropriate

A smoother and an easy to compute fit is obtained in the Power Law scheme

General form of the coefficient can be given as

= 0, 1

0.1

5+[0, ]

This is computationally more expensive than the simpler methods, but is lesser than the exponential scheme

QUICK scheme

5

Higher-order discretization schemes

The accuracy of the Hybrid and Upwind schemes is only first order. This lack of accuracy causes false diffusion

Higher order discretization takes more neighboring points into the scheme and brings wider influence and can reduce the discretization error

Quadratic Upwind differencing, QUICK scheme, is an often used higher order discretization scheme. QUICK expands to Quadratic Upstream Interpolation for Convective Kinetics

QUICK scheme

This scheme uses a three point upstream-weighted quadratic interpolation function between two bracketing nodes and an upstream node to evaluate flux at the faces.

If the two boundary nodes are i,i-1 and the upstream node is i-2, then flux at cell face is given by the formula-

For 1D convection diffusion problem, (>0)

=6

81 +

3

8 1

82

= + + + = + + + + ( )

+6

8 +

1

8 +

3

8(1 )

1

8

3

8

6

8(1 )

1

8(1 )

1

8(1 )

Characteristics of QUICK scheme

Consistent flux values are used from face to face, ensuring Conservativeness of the method

Transportiveness is guaranteed because of the consideration of upstream points

Accuracy is third order, in terms of Taylors series truncation error

One of the conditions for Boundedness, however, is not satisfied, in terms of all coefficients being positive.

The coefficients and are negative and the main coefficients and are not guaranteed to be positive always. ( When >0,

become negative for >

8

3

Different reformulations of the QUICK scheme have been proposed to alleviate this problem*

Discretization in unsteady flows

6

Unsteady Transport Equation

Consider,

+ . = . +

Integrating this over volume and time,

+ .

+

= +

. + +

+

+

The unsteady term is discretized as ( 0) , assuming is constant through the control volume

While the discretization with respect to space is same as in the steady state case, integration with respect to time can take different forms

Time discretization

=

(1

)

Integrating with respect to time and volume and simplifying,

(0) = (

+

)

+

= + 1 0

Therefore, the equation becomes

(0) = (

) + (1 ) (

00

00

)

f=0,explicit scheme f=0.5, Crank-Nicholson scheme f=1, implicit scheme

Time discretization

The above equation can be written as

= + 1

0 + + 1 0

+ [0 (1 ) (1 )]

0

= /, = /, 0 =

=

0 + +

When explicit (f=0), the following condition needs to be satisfied