Optimal Sub-parametric Finite Element Approach for a Darcy-Brinkman Fluid Flow Problem through a circular

channel using curved triangular elements.

V. Kesavulu Naidu 1, Dipayan Banerjee 2 ,K. V. Nagaraja 3 ,P.G. Siddheshwar 4 .1,3Department of Mathematics, Amrita School of Engineering, Bengaluru, Amrita Vishwa Vidyapeetham, Amrita University, India.

2Department of Mechanical Engineering, Amrita School of Engineering, Bengaluru, Amrita Vishwa Vidyapeetham, Amrita University, India.

4Department of Mathematics, Jnana Bharathi Campus, Bangalore University, Bangalore

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1

Flow of the presentation

1. Introduction and objectives

2. The use of parabolic arcs

3. FEM solution of Darcy-Brinkman equation

4. Conclusions

2

1. Oil recovery, Oil conveyance

2. Nuclear waste disposal using porous media

3. Soil Mechanics to understand Civil Engineering problems.

3

1. IntroductionMotivationSome examples from real life involving PDE’s

Solution of PDE’s

A. Analytical methods

1. Separation of variables2. Method of characteristics3. Integral transform4. Change of variables, etc.,

B. Numerical methods

1. Finite element method(FEM)2. Finite difference method(FDM), etc.,

4

Differences Between FEM and FDM FEM handles complicated geometries. FDM in its basic

form is restricted to handle rectangular shapes

Mathematical foundation of the FEM is more sound than FDM

The quality of a FEM approximation is often better than in the corresponding FDM approach

5

FEM Curved boundaries are often more accurately modeled by the fewer

curved triangular elements than by more straight edged triangular elements. The effort needed to obtain solutions is usually reduced

6

Experience gained from the study:

I. Computation using classical finite element method that makes use of straight edged triangular elementsInadequacy Greater input Requires more computation time Require more memory space

7

Summary of the present work The use of parabolic subparametric transformations for higher

order triangular elements is realized.

Got the feel of finite element method of solving of some PDE’s involving regular and irregular geometries

Experience gained from the study(Cont…):

II. Computation using non-classical finite element method that makes use of curved triangular elementsBenefits Lesser input Requires less computation time Require less memory space

8

Objectives of the present work

To solve Darcy-Brinkman(linear partial differential equation) over any curved domain using Galerkin finite element method and reduce computational effort & improve the accuracy.

9

2. “ The use of parabolic arcs in matching curved boundaries by point transformations for some

higher order triangular elements”

10

The curved triangular element was introduced into structural analysis by Ergatoudis, Irons and Zienkiewicz [1968]

The reference to the above can be found in

Felippa and Clough[1970]

Ciarlet and Raviart [1972]

Strang and Fix[1973]

Zlamal [1973]

11

2.1 Introduction

Mitchel [1976] describes three approaches to this problem

First Method: A transformation of the entire domain onto some standard shape which is not FEM

Second Method: The rational basis functions are constructed to match the curved boundaries

Third Method: The isoparametric method

12

13

Mapping of a 6-node quadratic curve triangle into standard triangle

Mapping of a 10-node cubic curve triangle into standard triangle

2.2 Point transformations for triangular elements with one curved boundaryThe transformation formulae

2

)2)(1(

1

)( ),(,),(

nn

ii

ni yxttNt

refers to the triangular element shape functions

Using the standard formulae to the straight sides 3-1 and 3-2 , then the above equation reduces to

14

),()( niN

)(

)()(1132

)(31

)(3

)()( )3()()()(),(jinji

jinij

nnnnn anHtattmttmtmtm ),(),7,6,5,4,3,2,1,1( yxtnnji

Subparametric transformation is obtained by systematically choosing the points on the curved boundary and interior points)

)7,6,5,4(,,)()(),( )(1132313 nyxtAtttttt n

)(

)(11)(

11 )( n

nn

ma

tA where

15

Subparametric transformation

Isoparametrictransformation

Explicit form of the Jacobians

yxyxyxJ

),(),(),( 210),( J

where))(())(( 313232210 yyxxyyxx

)()()()( )(1131

)(11311 xAyyyAxx nn

)()()()( )(1132

)(11322 yAxxxAyy nn

16

Sl. No.

Advantages of subparametric point

transformation

Disadvantages of isoparametric point transformation

1 Jacobian is a bivariate linear polynomial for all cases

Jacobian is 2nd order bivariate polynomial for cubic order triangular elements

2 The transformation t = x, y is always a polynomial of order 2 for all cases

The transformation t = x, y is a polynomial of order 3rd for cubic order triangular elements

4 The computational effort and time is reduced

The computational effort and time is more

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3. “Finite element solution of Darcy-Brinkman equation (Linear PDE) for some

flow channel using triangular elements”

18

19

Flow through porous media

Introduction:

The accuracy and limitations of the Darcy–Brinkman model literature can be found in Lage [1], and Nield and Bejan [2]. Forced convection in porous media of different channels are reported in the recent works of Nakayama et al. [3], Narasimhan and Lage [4], Haji-Sheikh and Vafai [5], Nield and Kuznetsov [6], and Hooman and coworkers [7,8,9,10,11] Pantokratoras and Magyari [12]. Restricting our attention to uni-directional flows, we call attention to the work of Hooman and Merrikh [8] that deals with hyperporous medium rectangular channels for which the analytical results of heat transfer are investigated by applying Darcy–Brinkman model.

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Introduction( Continued):

A steady, uni-dimensional, linear, non-Darcy flow is assumed in the present work. In spite of there being works reporting solution of the Darcy–Brinkman equation for fully developed three-dimensional rectangular channel flows, we venture to revisit the problem due to the need to consider irregular cross-section porous channels in many applications. Curved triangular elements are used to match the irregular geometry flow problems than by straight sided triangular elements. The detailed application of the method is reported in our recent work Nagaraja et. al. [13].

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Partial differential equations: Flow through the porous media is important in numerous engineering applications including geothermal energy, underground coal gasification and gas drainage, petroleum reservoirs, nuclear reactors, environmental pollution, fuel cells and nano-material processing, etc.,

K. Hooman, A. A. Merrikh, Analytical Solution of Forced Convection in a Duct of Rectangular Cross Section Saturated by a Porous Medium, J. Heat Transfer, June 2006, Volume 128, Issue-6, 596 .

D. A. Nield, Convection in porous media, 2nd ed.,

Springer-Verlag, New York, 1999.

22

2323

Fig. 1a: Schematic of physical configuration.

Fig. 1b: Cross section of Fig.1a.

2424

Slow velocity :The Darcy-Brinkman momentum equation for the case of unidirectional (fully developed) steady state flow in the z- direction in a flow channel (Fig.1a) occupied by a porous medium with velocity u(x,y) is

dzdpu

yu

xu

2

2

2

2

boundarytheon0u

viscosityactualμviscositydynamicμ

tyPermeabiliK

25

Non-dimensionalization

)(,

)(,,

dXdP

hRupP

dXdP

RuuUh

yYhxX

Fig.1b)in shown as channel theofsection (Cross

22

22

2

U

YU

XU

OnU 0

Number) (Brinkmann'

number)(Darcy

Kh

25

Brinkman number is dimensionless number related to heat conduction from wall to a flowing viscous fluid

Darcy number is related to permeability

The Lagrange interpolant is

where

The element geometry is also expressed

Jacobian

NP

i

ei

ni UNU

1

)( ),(

)()()(),( )(1132313 tAtttttt n

210 J26

cubicforquadraticfor

NP106

Using the Galerkin weighted residual FE procedure, we obtain the following FE equations:

Wherejiyy

jixx

A

jijiji KKdxdy

yN

yN

xN

xNK ,

,,,,

ddyNyNyNyNJ

K jjiijixx

1

0

1

0

,,

1

ddxNxNxNxNJ

K jjiijiyy

1

0

1

0

,,

1

BGUKU

2115106,1 orororNPji

ddJNdxdyNB iA

iei

1

0

1

0

****

K and G are 6×6(quadratic), 10×10 (cubic) and B are 6×1(quadratic), 10×1 (cubic), column matrices are evaluated for each element

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ddJNNdXdYNNG e

jiji

e

jie

1

0

1

0

22

,

For single cubic element

is as shown below:

where H= K+G

BUH

10

9

8

7

6

5

4

3

2

1

10

9

8

7

6

5

4

3

2

1

10,109,108,107,106,105,104,103,102,101,10

10,9999897969594939291

10,8898887868584838281

10,7797877767574737271

10,6696867666564636261

10,5595857565554535251

10,4494847464544434241

10,3393837363534333231

10,2292827262524232221

10,1191817161514131211

BBBBBBBBBB

UUUUUUUUUU

HHHHHHHHHH

HHHHHHHHHH

HHHHHHHHHH

HHHHHHHHHH

HHHHHHHHHH

HHHHHHHHHH

HHHHHHHHHH

HHHHHHHHHH

HHHHHHHHHH

HHHHHHHHHH

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The FEM calculation process:

(a) Discretization of solution domain into a finite element mesh

(b) For each element obtain the components of Ki,j and Fi

(c) Obtain the global FE equations for the whole system by assembling element equations

(d) Impose boundary conditions(e) Solve for the unknown velocity vector of the whole

system

6- node(quadratic) curved triangular element

8 elements

There are 25×25(quadratic), 49×49(cubic) components in the global stiffness matrix K and 25×1 (quadratic), 49×1 (cubic) components in the global column matrix G, H and B for eight elements

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Convergence experiment Convergence experiment of finite element solution for

the problem is anlysed by discretizing the domain into 8, 12 and 20 triangular elements for the Fig. 1b

We have used quadratic and cubic elements in all the

cases

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0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0y

x

0

0.01525

0.03050

0.04575

0.06100

0.07625

0.09150

0.1068

0.1220

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

y

x

0

0.01575

0.03150

0.04725

0.06300

0.07875

0.09450

0.1103

0.1260

4 element quadratic plot 4 element cubic

plot

5,1 2

Results Analysis

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0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

y

x

0

0.03125

0.06250

0.09375

0.1250

0.1563

0.1875

0.2188

0.2500

0,1 2

.

Figure 5b. Contour plot for cubic 8 elements

Results Analysis

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0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0y

x

0

0.01575

0.03150

0.04725

0.06300

0.07875

0.09450

0.1103

0.1260

5,1 2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0y

A

0

0.01028

0.02055

0.03083

0.04110

0.05138

0.06165

0.07193

0.08220

10,1 2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0y

A

0

0.02056

0.04113

0.06169

0.08225

0.1028

0.1234

0.1439

0.1645

5,2 2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0y

A

0

0.01181

0.02363

0.03544

0.04725

0.05906

0.07088

0.08269

0.09450

10,2 2

Conclusions

A finite element approach for solving the linear partial differential equation using triangular elements

The convergence of the solution is taken care by taking the staggered mesh along the curved domain

The results at common nodal points are analysed which are converging very well for the higher order triangular elements

The corners of the domain are very well captured

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Conclusions(Cont…)

The velocity increases as the Brinkman number increases for fixed value of Darcy number

Similarly, the velocity decreases as the Darcy number increases for fixed values of Brinkman number

It may be concluded that the results depicted by the figures are in tune with the observation of Givler and Altobelli

Our results show that the method can be used for more complicated domains

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REFERENCES [1] A. Haji-Sheikh, K. Vafai, Analysis of flow and heat transfer in porous

media imbedded inside various-shaped ducts, International Journal Of Heat Mass Transfer. 47. (2004),1889-1905.

[2] D. A. Nield, A. V. Kuznetsov, Forced convection in porous media: transverse heterogeneity effects and thermal development, in: K. Vafai (Ed.),Handbook of Porous Media. 2nd ed. New York: Taylor and Francis.(2005),143-193.

[3] K. Hooman. Fully developed temperature distribution in a porous saturated duct of elliptical cross section, with viscous dissipation effects and entropy generation analysis Heat Transfer –Jpn Res.36.(2005), 237-245.

[4] A. Narasimhan, J. L. Lage, Forced convection of a fluid with temperature-dependent viscosity flowing through a porus medium channel, Numerical Heat Transfer: Part A: Applications. 40. (2001), 801-820.

[5] R. A. Greenkorn. Steady Flow Through Porous Media. AIChE Journal.27.(1981),529-545.

[6] I. Ergatoudis, B. M. Irons, O. C. Zienkiewicz, Curved isoparametric quadrilateral finite element analysis, International Journal Solids Structure. 4.(1968),31–42.

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REFERENCES(Cont…)

[7] M. A. Bhatti. Fundamental Finite Element Analysis and Applications: John Wiley & Sons, Inc. (2005).

[8] H. T. Rathod, K. V. Nagaraja, V. Kesavulu Naidu, B. Vekatesudu. The use of parabolic arcs in matching curved boundaries by point transformations for some higher order triangular elements, Finite Elements in Analysis and Design.44.(2008),920-932.

[9] K. V. Nagaraja, V Kesavulu Naidu, P. G. Siddheshwar. Optimal subparametric finite elements for elliptic partial differential equations using higher-order curved triangular elements, International Journal For Computational Methods in Engineering Science and Mechanics. 15(2014),83-100.

[10] H. T. Rathod, K. V. Nagaraja, B. Vekatesudu, N. L. Ramesh. Gauss Legendre quadrature over a triangle, Journal Of Indian Institute Of Sciences.84 (2004),183-188.

[11] V. Kesavulu Naidu, P. G. Siddheshwar, K. V. Nagaraja. Finite Element Solution of Darcy-Brinkman Equation for Irregular Cross-Section Flow Channel Using Curved Triangular Elements. International Conference On computational Heat and Mass Transfer, Procedia Engineering .127 (2015), 301-308.

[12] R. C. Givler, S. A. Altobelli. A determination of the effective viscosity for the Brinkman-Forchheimer flow model, Journal. Fluid Mechanics.258 (1994), 355-370.

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Thank you

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