convection diffusion pd
TRANSCRIPT
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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
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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
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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
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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.
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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 (?)
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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.
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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,
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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
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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.
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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
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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
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yxtyx
nn u
PPWW EE
ww ee yy
vJ
x
y
yyxx
SSxxz=1z=1
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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
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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]
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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
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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
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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
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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
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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
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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
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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)
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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?