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8/12/2019 COBEM 1385 Kledson Santiago
1/23
22nd International Congress of Mechanical Engineering 1/15
ANALYSIS OF THE SPATIAL RESOLUTION OFDIFFUSIVE TERMS
Universidade Federal Fluminense UFFLaboratory of Theoretical and Applied Mechanics LMTA
COBEM 2013
Authors: Kledson Flavio Silveira SantiagoLeonardo S. de B. Alves
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22nd International Congress of Mechanical Engineering 2/15
Introduction
Introduction
This study presents an extensive numerical investigation on the
treatment of diffusive terms with variable properties.
Spatial resolution tests are restricted to second order accurate
centered schemes.
For this formulation, conservative and non-conservative finite
difference methods were employed.
They are compared to different finite difference methods.
A new formulation is proposed in order to overcome the
difficulties associated with each studied scheme.
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8/12/2019 COBEM 1385 Kledson Santiago
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22nd International Congress of Mechanical Engineering 2/15
Introduction
Introduction
This study presents an extensive numerical investigation on the
treatment of diffusive terms with variable properties.
Spatial resolution tests are restricted to second order accurate
centered schemes.
For this formulation, conservative and non-conservative finite
difference methods were employed.
They are compared to different finite difference methods.
A new formulation is proposed in order to overcome the
difficulties associated with each studied scheme.
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8/12/2019 COBEM 1385 Kledson Santiago
4/23
22nd International Congress of Mechanical Engineering 2/15
Introduction
Introduction
This study presents an extensive numerical investigation on the
treatment of diffusive terms with variable properties.
Spatial resolution tests are restricted to second order accurate
centered schemes.
For this formulation, conservative and non-conservative finite
difference methods were employed.
They are compared to different finite difference methods.
A new formulation is proposed in order to overcome the
difficulties associated with each studied scheme.
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8/12/2019 COBEM 1385 Kledson Santiago
5/23
22nd International Congress of Mechanical Engineering 2/15
Introduction
Introduction
This study presents an extensive numerical investigation on the
treatment of diffusive terms with variable properties.
Spatial resolution tests are restricted to second order accurate
centered schemes.
For this formulation, conservative and non-conservative finite
difference methods were employed.
They are compared to different finite difference methods.
A new formulation is proposed in order to overcome the
difficulties associated with each studied scheme.
22 d I t ti l C f M h i l E i i 2/15
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8/12/2019 COBEM 1385 Kledson Santiago
6/23
22nd International Congress of Mechanical Engineering 2/15
Introduction
Introduction
This study presents an extensive numerical investigation on the
treatment of diffusive terms with variable properties.
Spatial resolution tests are restricted to second order accurate
centered schemes.
For this formulation, conservative and non-conservative finite
difference methods were employed.
They are compared to different finite difference methods.
A new formulation is proposed in order to overcome the
difficulties associated with each studied scheme.
22nd International Congress of Mechanical Engineering 3/15
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8/12/2019 COBEM 1385 Kledson Santiago
7/23
22nd International Congress of Mechanical Engineering 3/15
Mathematical Model
Order Analysis
Finite Difference - Conservative
x
fg
x
i
ni
fi
mi
(g)
x2 +O(xm1,xn) (1)
Finite Difference - Non-Conservativef
x
g
x+f
2g
x2
i
mi
(f) ni(g)
x2 +fi
oi
(g)
x2+O(xm1, xn1, xo
(2)
Finite Volume
h
x
i
mi
ni(f) o
i(g)
x2
+O(xn2, xo1, xm) (3)
22nd International Congress of Mechanical Engineering 3/15
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22nd International Congress of Mechanical Engineering 3/15
Mathematical Model
Order Analysis
Finite Difference - Conservative
x
fg
x
i
ni
fi
mi
(g)
x2 +O(xm1,xn) (1)
Finite Difference - Non-Conservativef
x
g
x+f
2g
x2
i
mi
(f) ni(g)
x2 +fi
oi
(g)
x2+O(xm1, xn1, xo
(2)
Finite Volume
h
x
i
mi
ni(f) o
i(g)
x2
+O(xn2, xo1, xm) (3)
22nd International Congress of Mechanical Engineering 3/15
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22nd International Congress of Mechanical Engineering 3/15
Mathematical Model
Order Analysis
Finite Difference - Conservative
x
fg
x
i
ni
fi
mi
(g)
x2 +O(xm1,xn) (1)
Finite Difference - Non-Conservativef
x
g
x+f
2g
x2
i
mi
(f) ni(g)
x2 +fi
oi
(g)
x2+O(xm1, xn1, xo
(2)
Finite Volume
h
x
i
mi
ni(f) o
i(g)
x2
+O(xn2, xo1, xm) (3)
22nd International Congress of Mechanical Engineering 4/15
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8/12/2019 COBEM 1385 Kledson Santiago
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22nd International Congress of Mechanical Engineering 4/15
Mathematical Model
Heat Equation
CPT
t =
x
kT
x
+q (4)
Boundary condition thermal insulation
T
x
x=0
= 0 (5)
and heat transfer by convection
kT
x
x=L
+h T(x=L, t) =h TL (6)
Initial condition
T(x, t= 0) =T0 (7)
22nd International Congress of Mechanical Engineering 5/15
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22nd International Congress of Mechanical Engineering 5/15
Mathematical Model
Manufactured Solution
Manufatured Solution:
T(x) = 300
1
2tanh
x
L
1
2
+
1
2
sin
axL
+ cos
axL
+ 500 cosxL12 12tanhx
L 12
(8)Equation:
CPT
t =0
=
x kT
x+q (9)Substituting eq.8into eq.9we obtain the source term:
q=
x kT
x (10)
22nd International Congress of Mechanical Engineering 6/15
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g g g
Metodology
Spatial Resolution
Finite Difference Schemes
Non-Conservative Formulation
T = t
Cp
k
x
n
i
Tni+1 T
ni1
2x
+knit
Cp
Tni+1 2T
ni +T
ni1
x2
+qnit
Cp(11)
Conservative FormulationE
x
n
i
=(kni+1(T
ni+2 T
ni ) k
ni1(T
ni T
ni2))
4x2 (12)
22nd International Congress of Mechanical Engineering 7/15
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8/12/2019 COBEM 1385 Kledson Santiago
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g g g
Metodology
Spatial Resolution
Finite Volume Schemes
Traditional Finite Volume Formulation
T
x n
i=
(Tni1
Tni )(kn
i1+kn
i) + (Tn
i+1 Tni )(kn
i +ki+1)
2x2
(13)
New Finite Volume Formuation - 0
T
x
ni
= (kni1+kni)(Tni2 27Tni1+ 27Tni Tni+1)
48x2
+ (kn
i +kn
i+1)(Tn
i1 27Tn
i + 27Tn
i+1 Tni+2
)
48x2 (14)
22nd International Congress of Mechanical Engineering 8/15
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8/12/2019 COBEM 1385 Kledson Santiago
14/23
Metodology
Spatial Resolution
Finite Volume Schemes
New Finite Volume Formuation - 1
T
x n
i
= (kn
i2 9kn
i1 9kn
i +kn
i+1)(Tn
i Tn
i1)
16x2
+ (kn
i1 9kn
i 9kn
i+1+kn
i+2)(Tn
i Tn
i+1)
16x2 (15)
New Finite Volume Formulation - 2
T
x
n
i
= (kni2 9k
ni1 9k
ni +k
ni+1)(T
ni2 27T
ni1+ 27T
ni T
ni+1)
384x2
(kni1 9kni 9k
ni+1+k
ni+2)(T
ni1 27T
ni + 27T
ni+1 T
ni+2)
384x2
22nd International Congress of Mechanical Engineering 9/15
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8/12/2019 COBEM 1385 Kledson Santiago
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Results
Finite Difference Schemes
Table:Average and minimum error order. Analytical formulation.
NON-CONSERVATIVE ANALYTICAL
Values of Minimum Order Average Order
10 1.97566 1.99991
20 1.99523 2.0027
30 1.99737 2.00629
40 1.99889 2.0114250 2.00452 2.01995
60 2.03583 2.04693
22nd International Congress of Mechanical Engineering 10/15
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Results
Finite Difference Schemes
Table: Average and minimum error order. Numerical formulation.
NON-CONSERVATIVE NUMERICAL
Values of Minimum Order Average Order
10 1.9945 1.99697
20 1.98401 1.98804
30 1.96724 1.97305
40 1.94532 1.9520950 1.91584 1.92586
60 1.88857 1.90021
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Results
Conservative Formulation
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Order
X
= 10.0
= 20.0= 30.0
Figure:Numerical Order. Finite Difference Formulation.
22nd International Congress of Mechanical Engineering 12/15
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Results
Finite Volume Schemes
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Order
X
= 30.0
= 40.0
= 50.0
(a) Traditional Finite Volume
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Order
X
= 30.0
= 40.0
= 50.0
(b) New Finite Volume Formulation - 0
Figure:Numerical Order. Finite Volume Formulation.
22nd International Congress of Mechanical Engineering 13/15
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19/23
Results
Finite Volume Schemes
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Order
X
= 30.0
= 40.0
= 50.0
(a) New Finite Volume Formulation - 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Order
X
= 30.0
= 40.0
= 50.0
(b) New Finite Volume Formulation - 2
Figure:Numerical Order. Finite Volume Formulation.
22nd International Congress of Mechanical Engineering 14/15
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Results
Spectral Analysis
0
2
4
6
8
0 0.5 1 1.5 2 2.5 3
ModifiedWaveNumber
Wave Number
K2
NonConservativeFinite Diference Conservative
Finite Volume ConservativeNew Finite Volume Conservative 2(0)New Finite Volume Conservative 2(1)New Finite Volume Conservative 2(2)
Figure:Modified wave number described as wave number, for second
derivative
22nd International Congress of Mechanical Engineering 15/15
C l i
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Conclusions
Conclusions
We note that the conservative method of finite differences
can be clearly ruled out.The new finite volumes formulations showed some stability
in the calculation of numerical order, except for some
cases in the region variable property.
The new finite volume formulation - 2 presented better
modified wave numbers.
22nd International Congress of Mechanical Engineering 15/15
Conclusions
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Conclusions
Conclusions
We note that the conservative method of finite differences
can be clearly ruled out.The new finite volumes formulations showed some stability
in the calculation of numerical order, except for some
cases in the region variable property.
The new finite volume formulation - 2 presented better
modified wave numbers.
22nd International Congress of Mechanical Engineering 15/15
Conclusions
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Conclusions
Conclusions
We note that the conservative method of finite differences
can be clearly ruled out.The new finite volumes formulations showed some stability
in the calculation of numerical order, except for some
cases in the region variable property.
The new finite volume formulation - 2 presented better
modified wave numbers.
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