cobem 1385 kledson santiago

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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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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Introduction

Introduction




centered schemes.





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Introduction

Introduction




centered schemes.





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Introduction

Introduction




centered schemes.






22 d I t ti l C f M h i l E i i 2/15
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Introduction

Introduction




centered schemes.






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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)

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Mathematical Model

Order Analysis


x

fg

x

i

ni

fi

mi

(g)

x2 +O(xm1,xn) (1)


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)

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Mathematical Model

Order Analysis


x

fg

x

i

ni

fi

mi

(g)

x2 +O(xm1,xn) (1)


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)

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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)

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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)

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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)



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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)



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Metodology

Spatial Resolution


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



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Results


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

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Results


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.

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Results


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.

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Results


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.

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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


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.


Conclusions
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Conclusions

Conclusions








Conclusions
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Conclusions

Conclusions






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