convection diffusion pd

7/28/2019 Convection Diffusion PD

1/23


2/23

assume a known flow field

xxxt jj

j

j

I II III IV

We know how to handle I, III and IV. Term II is new

Consider thatu is given, and satisfies the continuity equation as:

jju

xt


3/23

Steady1D convection and diffusion

convection- ddddiffusion

equation:

dxdx

u

dx

continuity constuu

dx or0

x)x)ww x)x)ee (Assume u is given)

PPWW EE

xxww ee

xx


4/23

Discretization using central differences`

we

wedx

d

dx

duu

.

diffusionnet

P

W

E (piece(piece--wise linear profile)wise linear profile)::

1

xxPPWW EE

ww ee WPw

PEe

212

WPwPEe

wWPePE uu

2121we


5/23

Discretization using centraldifferencesRe-arran in the discretization e uation

WWEEPP aaa

www

e

e

ee

E

Fu

FD

u

xa

22

weWEwwee

P

wW

FFaauu

a

xa

22

F = u represents the advection strength, while D = /x is the

di usion stren th.


6/23

Example: letue = uw = 4, e/x = w/x = 1

LetE=200, W=100, findP

ee u

321

2

wwW

E

ua

x

2 weWEP uuaaa

(?)50 P

P

Similarly, letE=100, W=200, then P= 250 (?)


7/23

The central difference scheme is found to be suitable for flows in

which diffusion strength is high compared to the advection

strength

The relevant non-dimensional number representing the relativestrength of the two transport mechanisms is called the Peclet

number (P = uL/), where L is a representative length scale and

is the diffusivity.

The central difference scheme assumes equal influence of the a ues o wo a acen gr po n s n e erm n ng e convec veflux at the interface of the control volume between those grid

points. This assumption will be valid only for low Peclet number

.

For very high Peclet number (Pe > 2)flows, the influence of theupstream node will be dominant at the interface, and this is the

basis of the U wind method.


8/23

In this scheme, the value ofat an interface is equal to that at the

node on the upstream side of the interface.

0if

0if

eEe

ePe

F

F

Accordingly, the advective flux at the interfaces e and w can be written as:

0,0, eEePe uMaxuMaxu 0,0, wPwWw uMaxuMaxu

e

WWEEPP aaa wwww

W

eeeE

FDux

a

uxa

0,0,

,,The discretized equation iswritten as

weWE

w

w

e

e

P

uuaa

u

x

u

x

a

0,0,


9/23

Ex onential scheme exact solution The Upwind scheme is ideal for high Peclet number flows (Pe >

2).

, ,

analytical solution for variation ofover the length of a one-dimensional control volume can be observed.

-

dx

d

dx

du

dx

d

x 0L

o

Lx

Solution:

1exp

1exp

P

LP

oL

o


10/23

Ex onential scheme contd In the exponential scheme, the solution above is used as the profile

assumption forbetween successive grid points.

respectively, in a local coordinate system where the origin corresponds to theface at which the flux of needs to be calculated, the discreization equation is:

www

ee

e

E

DFF

DFa

)/exp(

1)/exp(

eeee FxuP )()(

weWEPww

FFaaa

D

1)/exp( ee

volume length for any Peclet number.

However, approximate representations ofvariations are requiredin the context of a numerical solution for com utational efficienc

The Hybrid and the Power Law schemes are two such examples

described subsequently.


11/23

The H brid SchemeThe hybrid differencing scheme (Spalding, 1972) is based on a combination

of central and upwind differencing schemes. The central differencing

sc eme, w ic is suita e or sma Pec et num er, is emp oye or Pec et

numbers (Pe < 2). The upwind scheme, which is more suitable for higherPeclet number, (Pe 2). The hybrid scheme is essentially a piecewise linear

inter olation o the ex onential scheme. A summar o the scheme is:

,2For eEe PaP

21,22For e

e

E

e

e

P

D

aP

0,2For e

E

eD

aP


12/23

The Power-law differencing scheme of Patankar (1980) is a more accurate

approximation than the hybrid scheme. In the hybrid scheme, the influence of

diffusion is switched off for Pe>2, which may be too soon. In the power lawscheme, diffusion is set to zero when cell Pe exceeds 10. If 0 < Pe < 10, the

. ,

,10For eEe PaP

1.01,010For 5 eee

E

e

e

PPD

aP

1.01,100For 5 ee

E

e

a

PDaP

,or e

eD


13/23

yxtyx

nn u

PPWW EE

ww ee yy

vJ

x

y

yyxx

SSxxz=1z=1


14/23

Discretized unstead 2-D convection-diffusion equation (fully implicit)

oldoldold yxc baaaaTa SSNNWWEEPP

PSNWEP

ppCp

x

yxSaaaaa

t

etc.

e c.

nn

ee

n

n

w

w

e

e

FP

FP

xxx

.assumptionprofileondependssne

PA


15/23

Scheme Formula for

Central difference 1- 0.5P S

PA

pw n

Hybrid 0, 1- 0.5P

Power law 0, (1- 0.1P)5

Exponential P/ [exp(P)-1]


16/23

False diffusion Commonly misinterpreted and controversial topic

Common view: - Central diff. is second order accurate

- Upwind causes severe false diffusion

Ta lor series central di . : Truncation error TE x 2

Upwind: can be shown that TE(x)

,

to forward or backward difference)

ommon notion: Upwin ess accurate. However, since

~ x variation is exponential, Taylor series approx. is good

only for smallx (not good for high Peclet number)

Upwind better in these circumstances


17/23

False diffusion (continued) Compare coefficients:

Central diff. Upwind

2

e

eE

F

Da 0,eeE FDa , e . e e

in Upwind i.e. replaced by2

xu

Flaw in argument: we have assumed that central diff.

scheme is the most perfect

Conclusion:Small Peclet No.: central accurate; UPWIND over-predicts

us on; power aw, y r , exponen a en o cen ra

Large Peclet No.: central fails, others are correct


18/23

False diffusion (continued)

Make = 0. Any effect of diffusion will be false diffusion.

100 C, Hot 100 C, Hot

0 C, Cold 0 C, Cold

0 = 0


19/23

Uniform flow in x directionNN

Hot

PPWW EE

ColdSS

UPWIND SCHEME

Since = 0, and Fn = Fs = 0 aN= aS= 0.

lso, a = 0 UPWIND

aP= aW P= W

,

NO FALSE DIFFUSION


20/23

Uniform flow at 45 to the grid lines

NNUPWIND (or high Peclet No.)

PPWW EE

,

N, E downstream neighbours

SSHot

Cold

Since = 0, and aE= aN= 0.

Also, aW= aS P= 0.5 W+ 0.5 S

ARTIFICIAL (FALSE) DIFFUSION OCCURED


21/23

Quadratic Upstream Interpolation for

Higher order differencing scheme: more accurate

minimise false diffusion: can take into account corner nodes

3 - point quadratic interpolation

E EEPW

WW

WWWW

ww

PPWW EE EEEE

ee

at interface is obtained from a quadratic function passing

Control volume Control volume boundary

through the two surrounding nodes (one on each side of the

face) and a node on the upstream side


22/23

Let ue > 0, uw >0

= =for uniform grid, quadratic interpolation gives:

w W P- WW

e= (6/8)P+ (3/8) E- (1/8) W

For the diffusion terms, use gradient of the appropriate parabola

aaaa

aaa 1163

weWWWEP

wewwee

FFaaaa 8888

Stability: Conditionally stable (high Pe, negative coeff. possible)


23/23

MODULE 4 Review Questions

Define Peclet number in a convection-diffusion problem. Why is the significance of Peclet

number in a convection-diffusion problem? Why does the Central difference scheme fail for

high Peclet number problems?

Show that the ex onential scheme is a ro riate from a h sical reasonin stand oint.

Show that the upwind scheme can introduce false diffusion in low Peclet number problems.

The power law scheme is a close appropriation of the exponential scheme, and is therefore

regarded as a very accurate scheme. However, the exponential scheme is a result of analytical

.

problems is you use power law scheme.

What is false diffusion? Can false diffusion occur because of the gridding system? What are

the remedies for false diffusion? For various Peclet number values (Pe = -20, -7, 0, 5, 10), calculate the aE/De from the power

law scheme and from the exponential scheme, and express the difference as a percentage of

the exact aE/De value. Although you might get a rather large percentage difference for some

values of Pe, the power law scheme is expected to give good results. Why?

Why is the QUICK scheme more accurate than conventional schemes? Does it alwaysconverge?

convection diffusion pd

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