4 The ﬁnite element method:discretization and application ... · vertical direction, caused by the temperature differences induced in the ﬂuid. Both structured and unstructured

Sunden CH004.tex 10/9/2010 15: 9 Page 129

4 The finite element method: discretizationand application to heat convection problems

Alessandro Mauro1,2, Perumal Nithiarasu1, Nicola Massarotti2,and Fausto Arpino3

1Civil and Computational Engineering Centre, School of Engineering,Swansea University, Swansea, U.K.2Dipartimento per le Tecnologie, Università degli Studi di Napoli“Parthenope”, Napoli, Italy.3Dipartimento di Meccanica, Structture, Ambiente e Territorio,Università di Cassino, Cassino, Italy.

Abstract

In this chapter, we give an overview of the finite element method and its applicationsto heat and fluid flow problems. An introduction to weighted residual approxima-tion and finite element method for heat and fluid flow equations are presented.The characteristic–based split (CBS) algorithm is also presented for solving theincompressible thermal flow equations. The algorithm is based on the temporal dis-cretization along the characteristics, and the high-order stabilization terms appearnaturally from this kind of discretization. The artificial compressibility (AC) and thesemi-implicit (SI) versions of the CBS scheme are presented and some examplesare also given to demonstrate the main features of both the schemes.

Keywords: Characteristic-based split, Finite elements, Porous media

4.1 Governing Equations

4.1.1 Non-dimensional form of fluid flow equations

In many heat transfer applications, it is often easy to generate data by non-dimensionalizing the equations, using appropriate non-dimensional scales. Toobtain a set of non-dimensional equations, let us consider five different cases ofconvection heat transfer. We start with the mixed convection problem followedby the natural and forced convection problems. The turbulent flow and convectiveflow through porous media are also presented. For each case, we discuss one set ofnon-dimensional scales.

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130 Computational Fluid Dynamics and Heat Transfer

Mixed convectionMixed convection involves features from both forced and natural flow conditionsthat are two limiting cases of mixed convection flow. The buoyancy effects becomecomparable to the forced flow effects at small and moderate Reynolds numbers.Since the flow is partly forced, a reference velocity value is normally known (exam-ple: velocity at the inlet of a channel). However, in mixed convection problems, thebuoyancy term needs to be added to the appropriate component of the momentumequation.

In incompressible mixed convection flow problems, the following non-dimensional scales are normally employed:

x∗i = xi

L, u∗

i = ui

uref, t∗ = t uref

L2 ,(1)

p∗ = p

ρu2ref

, T ∗ = T − Tref

Tw − Tref

where * indicates a non-dimensional quantity, L is a characteristic dimension, thesubscript ref indicates a constant reference value and Tw is a constant referencetemperature, for example, wall temperature. The density ρ and viscosity µ of thefluid are assumed to be constant everywhere and equal to the inlet value.

Using the above scales, the non-dimensional form of the incompressible mixedconvection equations, in indicial notation, are written as:

Continuity equation

∂u∗i

∂x∗i

= 0 (2)

Momentum equation

∂u∗i

∂t∗+ u∗

j∂u∗

i

∂x∗j

= −∂p∗

∂x∗i

+ 1

Re

∂

∂xj

(∂u∗

i

∂xj

)+ Gr

Re2 T ∗γi (3)

where γi is a unit vector in the gravity direction.

Energy equation

∂T ∗

∂t∗+ u∗

i∂T ∗

∂x∗i

= 1

Re Pr

∂

∂xi

(∂T ∗

∂xi

)(4)

where Re is the Reynolds number defined as:

Re = uref L

ν(5)

Pr is the Prandtl number given as:

Pr = ν

α(6)



Applications of finite element method to heat convection problems 131

Fluid circulation

Hot, vertical plate

g

x2

x1

Figure 4.1. Natural convective flow near a hot vertical plate.

and Gr is the Grashof number given as:

Gr = gβ�TL3

ν2 (7)

where ν=µ/ρ is the kinematic viscosity. Appropriate changes will be necessary ifan appreciable variation in these quantities occurs in a flow field.

Note that sometimes, a non-dimensional parameter referred to as the Richardsonnumber (Gr/Re2) is also used in the literature.

Natural convectionNatural convection is a limiting case of the more general case of mixed convec-tion. Natural convection is generated only by the density difference induced by thetemperature differences within a fluid system (Figure 4.1). Because of the smalldensity variations present in these types of flows, a general incompressible flowapproximation is adopted. In most buoyancy-driven convection problems, flow isgenerated either by a temperature variation or by a concentration variation in thefluid system, which leads to local density differences.

In practice, the following non-dimensional scales are adopted for naturalconvection in the absence of a real reference velocity value:

x∗i = xi

L, u∗

i = uiL

α, t∗ = t α

L2 ,(8)

p∗ = pL2

ρα2, T ∗ = T − Tref

Tw − Tref




Upon introducing the above non-dimensional scales into the governing equa-tions, and replacing Re with 1/Pr in the non-dimensional mixed convectionequations of the previous subsection, we obtain the non-dimensional equationsfor natural convection flows.

Often, another non-dimensional number called the Rayleigh number is used inthe calculations. This is given as:

Ra = Gr Pr = gβ�TL3

να(9)

Forced convectionForced convection is another limiting case of mixed convection. In forced convec-tion flow problems, the non-dimensional scales are the same as the mixed convectionproblem. The non-dimensional form of the forced convection governing equationsis obtained from that of mixed convection by neglecting the buoyancy term on theright hand side of the momentum equation.

Another non-dimensional number, which is often employed in forced convec-tion heat transfer calculations is the Peclet number and is given as Pe = RePr =uref L/α.

Some examplesThe following results are obtained solving the above equations for forced, naturaland mixed convective flows by using an artificial compressibility (AC) version anda semi-implicit (SI) version of the characteristic-based split (CBS) algorithm andfinite element discretization technique that will be explained later.

The first problem considered is the backward-facing step, that is a test case com-monly used to validate numerical solutions of the incompressible Navier–Stokesequations. In fact, several data are available in literature for this problem (Denhamand Patrik [1], Aung [2], Ichinose et al. [3], de Sampaio et al. [4], Massarotti et al.[5]). The computational domain and the boundary conditions used in this workare presented in Figure 4.2a. In particular, the velocity profile at the inlet is mea-sured experimental profile (Denham and Patrik [1]). The measurement section inthe experiments, as in the present calculations, was placed 1.3 times the step heightupstream of the step. All the horizontal walls and the step have been imposed withno-slip velocity boundary conditions.

For non-isothermal cases, in addition to the above boundary conditions (Aung[2]), a non-dimensional temperature equal to one is imposed along the bottom wall,while the temperature of the inlet flow is assumed to be equal to zero. All other wallsare assumed to be adiabatic. Figure 4.2 shows some details of the computationalgrids used for this problem.

Figure 4.3 shows the horizontal velocity distribution along different verticalsections of the channel, obtained by using the AC and the SI versions of the CBSalgorithm.

The temperature profiles at two different vertical cross sections downstream ofthe step are presented in Figure 4.4 for the case of a fully developed inlet profile with




p = 0

q = 0

T = 1v = 0,u = 0,

v = 0,

v = 0, q = 0,

u = 0,

u = 0,

10h8h6h4h2h0.8h0−1.3h

y

xh

2h

4h

u = u(y)v = 0T = 0

(a)

(b)

(c)

Figure 4.2. Backward-facing step: (a) computational domain and boundary condi-tions; (b) unstructured mesh (955 nodes and 1,704 elements); (c) detailof the structured mesh near the step (4,183 nodes, 8,092 elements).

v

0 2 4 6 8

SI-CBS

y

−1.3

AC-CBS Denham and Patrik [26]

Figure 4.3. Backward-facing step. Comparison of the horizontal velocity profiles,Re = 229.

T

y/δt

0 0.25 0.5 0.75 1T

0 0.25 0.5 0.75 10

0.5

1

1.5

2

2.5

3AC-CBSSI-CBSAung [2]

AC-CBSSI-CBSAung [2]

x/Xr = 0.482 x/Xr = 0.946

y/δt

0

0.5

1

1.5

2

2.5

3

Figure 4.4. Backward-facing step. Temperature profiles at two different sectionsdownstream of the step (Re = 233).




u1 = u2 = 0

u1 = u2 = 0

T = 1 T = 0

Insulated

Insulated

u 1 =

u2

= 0

u 1 =

u2

= 0

Figure 4.5. Natural convection in a square cavity. Computational domain andboundary conditions.

Re = 233. The profiles are practically the same for both the schemes and they showa very good agreement with the experimental data available from the literature forthis problem (Aung [2]). It should be observed that the position along the verticalsection is related to the thickness of the thermal boundary layer in the channel.

Another benchmark problem that is commonly employed for code validation is awell known natural convection problem in which the fluid-filled square cavity withthe vertical walls subjected to different temperatures is used. The two horizontalwalls are assumed to be adiabatic as shown in Figure 4.5. No-slip velocity conditionsare applied on all walls. The flow is driven by the buoyancy forces acting in thevertical direction, caused by the temperature differences induced in the fluid. Bothstructured and unstructured meshes are used (Figure 4.6).

Figures 4.7–4.9 show the temperature contours and streamlines obtained withthe AC-CBS algorithm at different Rayleigh numbers.

The transient flow over a circular cylinder is a popular test case for validatingtemporal discretization of numerical schemes. The inlet flow is uniform and thecylinder is placed at a centreline between two slip walls. The distance from theinlet to the centre of the cylinder is 4D, where D is the diameter of the cylinder.Total length of the domain is 16D. No slip condition is applied on the cylindersurface. The problem has been solved both on two and three dimensions. Figure4.10 shows the mesh used for two-dimension case. As seen, the mesh close tothe cylinder is very fine in order to capture the boundary layer and separation.Figure 4.11 shows the contours of the horizontal and vertical velocity componentsat a real non-dimensional time of 98.02. The time history of the vertical velocitycomponent at an exit central point is plotted with respect to time and is shown inFigure 4.12.

In order to show that the method works also on three dimensions, a coarsemesh with 17,382 nodes and 69,948 tetrahedral elements was used to recomputethe solution. This mesh was generated using the PSUE software available within the




(a) (b)

(d)(c)

Figure 4.6. Natural convection in a square cavity. (a) Mesh 1: 1,251 nodes and 2,888elements; (b) Mesh 2: 2,601 nodes and 5,000 elements; (c) Mesh 3:1,338 nodes and 2,474 elements; (d) Mesh 4: 5,515 nodes and 10,596elements.

Figure 4.7. Natural convection in a square cavity. Streamlines and isotherms atRa = 105.

School of Engineering, Swansea University (Morgan et al. [6]). The finite elementmesh and the contours of velocity components are shown in Figures 4.13 and 4.14,respectively, at a real non-dimensional time of 83.33. As seen, the solution obtainedis very similar to the one shown in Figure 4.11 for a two-dimensional case.






Figure 4.10. Flow past a circular cylinder. Finite element mesh, 7,129 nodes and13,990 elements.




(a)

(b)

(c)

Figure 4.11. Flow past a circular cylinder, Re = 100. (a) Horizontal velocitycontours; (b) vertical velocity contours; (c) temperature contours.

Adimensional time

Ver

tical

vel

ocity

0.6

0.4

0.2

0 15 30 45 60

0

−0.2

−0.4

−0.6

AC-CBSde Sampaio et al. [4]

Figure 4.12. Flow past a circular cylinder. The vertical velocity distribution at anexit point of the domain with respect to real time. Comparison withde Sampaio et al. [4].




Figure 4.13. Three-dimensional flow past a circular cylinder. Finite element mesh:17,382 nodes, 69,948 elements.

Figure 4.14. Three-dimensional flow past a circular cylinder, Re = 100; u1 velocitycontours (left); u3 velocity contours (right).

4.1.2 Non-dimensional form of turbulent flow equations

For turbulent flow computations, Reynolds averaged Navier–Stokes equations ofmotion are written in conservation form as follows:

Mean continuity equation

1

β2

∂p

∂t+ ∂(ρui)

∂xi= 0 (10)




Mean momentum equation

∂ui

∂t+ ∂

∂xj(ujui) = − ∂p

∂xi+ ∂τij

∂xj+ ∂τR

ij

∂xj(11)

where β is an AC parameter, which will be well explained in Section 2.6, ui are themean velocity components, p is the pressure, ρ is the density and τij is the laminarshear stress tensor given as:

τij = 1

Re

(∂ui

∂xj+ ∂uj

∂xi− 2

3

∂uk

∂xkδij

)(12)

The Reynolds stress tensor, τRij , is introduced by Boussinesq assumption as:

τRij = νT

Re

(∂ui

∂xj+ ∂uj

∂xi− 2

3

∂uk

∂xkδij

)− 2

3kδij (13)

In the above equations, ν is the kinematic viscosity of the fluid, νT is the turbulenteddy viscosity and δij is the Kroneker delta. The following non-dimensional scalesare used to derive the above equations:

x∗i = xi

L, u∗

i = ui

uref, t∗ = t uref

L2 , p∗ = p

ρu2ref

,

k∗ = k

u2ref

, ε∗ = εL

u3ref

, v∗T = vT

vref, v∗ = v

uref(14)

where L is a characteristic dimension and the subscript ref indicates a referencevalue. The turbulent flow solution is obtained by solving equations (10) and (11)with appropriate boundary conditions and a turbulence model, as the Spalart–Allmaras model (Spalart and Allmaras [7], Nithiarasu and Liu [8], Nithiarasu et al.[9]). The Spalart–Allmaras (SA) model was first introduced for aerospace appli-cations and currently being adopted for incompressible flow calculations. The SAmodel is another one-equation model, which employs a single scalar equation andseveral constants to model turbulence. The scalar equation is:

∂v

∂t+ ∂(uj v)

∂xj= cb1Sv+ 1

Reσ

[∂

∂xi

{(1 + v) ∂v

∂xi

}+ cb2

(∂v

∂xi

)2]

− cw1fwRe

[v

y

]2

(15)where

S = S + 1

Re(v/k2y2)fv2 (16)

and

fv2 = 1 − X /(1 + Xfv1) (17)




In equation (16) S is the magnitude of vorticity. The eddy viscosity iscalculated as:

vT = vfv1 (18)

where

fv1 = X 3/(X 3 + c3v1) (19)

and

X = v/v (20)

The parameter fw is given as:

fw = g

[1 + c6

w3

g6 + c6w3

]1/6

(21)

where

g = r + cw2(r6 − r) (22)

and

r = 1

Re

v

Sk2y2(23)

The constants are cb1 = 0.1355, σ= 2/3, cb2 = 0.622, k = 0.41, cw1 = cb1/k2 +(1 + cb2)/σ, cw2 = 0.3, cw3 = 2 and cv1 = 7.1. From the above equations it is clearthat the turbulent kinetic energy is not calculated here. Thus, the last term of equation(13) is dropped when SA model is employed.

Some examplesThe following results are obtained by solving the above equations using an AC CBSalgorithm and finite element discretization technique that will be described later inthis chapter.

A standard test case commonly employed for turbulent incompressible flowmodels at moderate Reynolds number is the recirculating flow past a backward-facing step. The definition of the problem is shown in Figure 4.15. The characteristicdimension of the problem is the step height. All other dimensions are defined withrespect to the characteristic dimension. The step is located at a distance of fourtimes the step height from the step. The inlet channel height is two times the stepheight. The total length of the channel is 40 times the step height. The inlet velocityprofile is obtained from the experimental data reported by Denham et al. [10]. Noslip conditions apply on the solid walls. A fixed value of 0.05 for the turbulent scalarvariable of the SA model at inlet was prescribed. The scalar variable was assumedto be zero on the walls. Both structured and unstructured meshes were employedin the calculation (Figures 4.16 and 4.17).




Parabolic u1 and u2 = 0

2L

4L 36L

p = 0

L

u1 = u2 = 0

Figure 4.15. Turbulent flow past a two-dimensional backward-facing step. Problemdefinition.

Figure 4.16. Detail of the structured mesh (8,092 elements and 4,183 nodes) nearthe step.

Figure 4.17. Detail of the unstructured mesh (8,662 elements and 4,656 nodes)near the step.

Figure 4.18 shows the comparison of velocity profiles against the experimentaldata of Denham et al. [10]. The SA model predicts accurately the recirculationregion.

The other turbulence problem considered is a complex three-dimensional modelof flow through an upper human airway (Nithiarasu and Liu [8], Nithiarasu et al.[9]). The geometry used is a reconstruction of an upper human airway employedin the spray dynamics studies (Gemci et al. [11]). It is apparent from the availablestudies on particle movement in the upper human airways that this problem isimportant (Li et al. [9, 10], Martonen et al. [11]). To understand the mechanismbehind many upper human airway-related problems including “sleep apnoea” andvocal cord-related problems. Most of the reported studies on upper airway fluiddynamics use either structured or semi-structured meshes in the calculations. Here,the results obtained with an unstructured mesh are presented. The surface meshof the grid used in the present study is shown in Figure 4.19. This mesh containsjust under a million tetrahedral elements. The mesh is generated using the PSUEcode (Morgan et al. [6]). The Reynolds number of the flow is defined based on the




Exp.SA model

Re = 3,0253

2.5

1.5V

ertic

al d

ista

nce

Horizontal velocity

0.5

1

00 2 4 6 8 10 12 14

2

Figure 4.18. Incompressible turbulent flow past a backward-facing step. Velocityprofiles at various downstream sections at Re = 3,025.

Figure 4.19. Incompressible turbulent flow through a model upper human airway.Surface mesh.

diameter of the narrow portion close to the epiglottis. The total non-dimensionallength of the domain in the horizontal direction is 29.68, and in the vertical directionit is 23.05. The diameter at the inlet of the geometry (at the top) is 4.91.

A uniform velocity is assumed at the inlet of the geometry in the negativedirection perpendicular to the inlet surface in the downward direction. No slipconditions are assumed on the solid walls. The turbulent scalar variable value at theinlet is fixed at 0.05 and assumed to be equal to zero on the walls.

Figure 4.20 shows the contours of velocity components and pressure within asection along the axis in the middle of the geometry. It is apparent that the majorityof the activities are taking place near the narrow portion close to the epiglottis ofthe upper human airway. The flow is accelerated as it passes through the narrowportion of the airway. As seen, the pressure contours are clustered close to thenarrow portion representing a very high gradient region. Figure 4.21 shows thevelocity vectors within the section. The velocity vectors close to the narrow portionare also shown in this figure. As seen, the recirculation region is clearly predictedby the AC-CBS method.




(a) (b)

(c)

Figure 4.20. Incompressible turbulent flow through a model upper human airway.(a) u1 contours, (b) u3 contours, (c) pressure contours.

(a) (b)

Figure 4.21. Incompressible turbulent flow through a model upper human airway.(a) Velocity vectors and (b) velocity vectors near the narrow portion.

4.1.3 Porous media flow: the generalized model equations

The general form of the equations for a porous medium can be derived by averagingthe Navier–Stokes equations over a representative elementary volume (REV; Figure4.22), using the well-known volume averaging procedure (Whitaker [15], Vafai andTien [16], Hsu and Cheng [17]).




Fluid

REV

Figure 4.22. Representative elementary volume.

The non-dimensional form of the generalized model for the description of flowthrough a fluid-saturated porous medium can be written as:

Continuity equation

∂u∗i

∂x∗i

= 0 (24)

Momentum equation

1

ε

∂u∗i

∂t∗+ 1

ε2 u∗j∂u∗

i

∂x∗j

= −∂p∗f

∂x∗i

+ J

εRe

∂

∂x∗j

(∂u∗

i

∂x∗j

)(25)

− 1

Re Dau∗

i − F√Da

|u∗|u∗i + Ra

Pr Re2 T ∗γi

Energy equation

σ∂T ∗

∂t∗+ u∗

i∂T ∗

∂x∗i

= λ∗

Pr Re

∂

∂x∗i

(∂T ∗

∂x∗i

)(26)

where Da is the Darcy number and F is the Forchheimer coefficient. The buoyancyeffects are incorporated by invoking the Boussinesq approximation:

g(ρf − ρref ) = ρref gβ(Tref − T ) (27)

where the subscript f refers to the fluid that saturates the porous medium. The scalesand the parameters used to derive the above non-dimensional equations for mixedconvection through a saturated porous medium are the same as those shown at thebeginning of the chapter. The new parameters introduced here are:

J = µeff

µf, σ = ε(ρcp)f + (1 − ε)(ρcp)s

(ρcp)f, λ∗ = λeff

λf, Da = κ

L2 (28)

where J is the ratio between the effective and the fluid viscosity, σ is the heat capac-ity ratio, λ* is the ratio between the effective and the fluid thermal conductivity, Da is




v = parab.

v = 0u = 0q = 1

v = 0u = 0q = 1

x

y

u = 0T = 0

p = 0

2

(a) (b)

Figure 4.23. Mixed convection in porous vertical channel. (a) Computationaldomain and boundary conditions; (b) detail of the structured com-putational grid near the entrance (8,601 nodes and 16,800 elements).

the Darcy number, κ and ε are the permeability and porosity of the medium, respec-tively, the subscripts p and eff refer to the porous medium and to the effective values,respectively. As mentioned, equations (24)–(26) are derived for mixed convectionproblems, therefore this set of PDEs describes both natural and forced convection.When forced convection dominates the problem (Ra/PrRe2< 1), the buoyancy termon the right hand side of the momentum conservation can be neglected.

The generalized model equations introduced above reduce to the Navier–Stokesequations when the solid matrix in the porous medium disappears, that is whenε→ 1 and Da → ∞, while, as the porosity ε→ 0 and Da → 0, the equations rep-resent a solid. Therefore, the procedure can be used to describe interface problemsin which a saturated porous medium interacts with a single-phase fluid. In thesecases, it is possible to use a single domain approach, by changing the property ofthe medium accordingly.

Some examplesThe following results are obtained by solving the above equations for mixed con-vection flows by using an AC version of the CBS algorithm and finite elementdiscretization technique that will be explained later.

The example showed here is the fully developed mixed convection in a regionfilled with a fluid-saturated porous medium, confined between two vertical walls.The computational domain and the boundary conditions are shown in Figure 4.23a.




x

u/u

mea

n

−1 0 10

1

2 Chen et al. [18]PresentAC-CBS

Ra = 0Ra = 104

Ra = 2x104

Ra = 5x104

Figure 4.24. Mixed convection in porous vertical channel. Non-dimensionalvertical velocity at different Ra for aiding flow, at Da = 10−4.

The flow enters the domain from the bottom at a non-dimensional temperatureof T = 0 with a fully developed parabolic velocity profile. The uniform heat fluxcondition is symmetrically imposed on both walls. Figure 4.23b shows the detailsof the computational grid near the entrance. The mesh employed has 8,601 nodesand 16,800 elements and it is refined near the walls and at the inlet region.

Figure 4.24 shows the dimensionless vertical velocity profile at a Darcy numberequal to 10−4 and four different Rayleigh numbers. The present results are comparedwith the numerical results of Chen et al. [18]. For pure forced convection (Ra = 0),the velocity profile is flat in the central part of the domain and sharply goes tozero at the walls of the channel. In the case of aiding flow, if the Rayleigh numberincreases, the buoyancy forces result in an increase in the fluid velocity near thewalls. As the flow rate entering the channel is the same for any Rayleigh numbersconsidered, the increase in fluid velocity in the near-wall regions will result in adecreased velocity at the centre region of the channel. When Rayleigh number isincreased to 5 × 104, the maximum velocity in the near-wall region is significantlyhigher than the fluid velocity at the centre of the channel.

4.2 The Finite Element Method

We provide here an introduction to weighted residual approximation and finiteelement method for self-adjoint equations and for convection–diffusion equations.

4.2.1 Strong and weak forms

The Laplace equation is a very convenient example for the start of numerical approx-imations. We shall generalize slightly and discuss in some detail the quasi-harmonic




(Poisson) equation:

− ∂

∂xi

(k∂φ

∂xi

)+ Q = 0 (29)

where k and Q are specified functions. These equations together with appropriateboundary conditions define the problem uniquely. The boundary conditions can beof Dirichlet type:

φ = φ, on �φ (30)

or that of Neumann type:

qn = −k∂φ

∂n= qn, on �q (31)

where a bar denotes a specified quantity. Equations (29)–(31) are known as thestrong form of the problem.

We note that direct use of equation (29) requires computation of second deriva-tives to solve a problem using approximate techniques. This requirement may beweakened by considering an integral expression for equation (29) written as:∫

�

v

[− ∂

∂xi

(k∂φ

∂xi

)+ Q

]d� = 0 (32)

in which v is an arbitrary function.If we assume equation (29) is not zero at some point xi in� then we can also let

v be a positive parameter times the same value resulting in a positive result for theintegral equation (32). Since this violates the equality we conclude that equation(29) must be zero for every xi in �, hence proving its equality with equation (32).

We may integrate by parts the second derivative terms in equation (32) to obtain:∫�

∂v

∂xi

(k∂φ

∂xi

)d�+

∫�

vQd�−∫�

vni

(k∂φ

∂xi

)d� = 0 (33)

We now split the boundary into two parts, �φ and �q, with �=�φ+�q, anduse equation (31) in equation (33) to give:∫

�

∂v

∂xi

(k∂φ

∂xi

)d�+

∫�

vQd�−∫�q

vqnd� = 0 (34)

which is valid only if v vanishes on �φ. Hence we must impose equation (30) forequivalence.

Equation (34) is known as the weak form of the problem since only first deriva-tives are necessary in constructing a solution. Such forms are the basis for obtainingthe finite element solutions.




4.2.2 Weighted residual approximation

In a weighted residual scheme, an approximation to the independent variable φ iswritten as a sum of known trial functions (basis functions) Na(xi) and unknownparameters φa. Thus we can always write:

φ ≈ φ = N1(xi)φ1 + N2(xi)φ2 + . . .(35)

=n∑

a=1

Na(xi)φa = N(xi)ϕ

where

N = [N1, N2, . . .Nn] (36)

and

φ = [φ1, φ2, . . . , φn]T

(37)

In a similar way we can express the arbitrary variable v as:

v ≈ v = W1(xi)v1 + W2(xi)v2 + . . .

=n∑

a=1

Wa(xi)va = W(xi)v (38)

in which Wa are test functions and va arbitrary parameters. Using this form ofapproximation will convert equation (34) to a set of algebraic equations.

In the finite element method and indeed in all other numerical procedures forwhich a computer-based solution can be used, the test and trial functions willgenerally be defined in a local manner. It is convenient to consider each of thetests and basis functions to be defined in partitions �e of the total domain �. Thisdivision is denoted by:

� ≈ �h =⋃�e (39)

and in a finite element method �e are known as elements. The very simplest useslines in one dimension, triangles in two dimensions and tetrahedra in three dimen-sions in which the basis functions are usually linear polynomials in each elementand the unknown parameters are nodal values of φ. In Figure 4.25 we show a typicalset of such linear functions defined in two dimensions.

In a weighted residual procedure, we first insert the approximate function φinto the governing differential equation creating a residual, R(xi), which of courseshould be zero at the exact solution. In the present case for the quasi-harmonicequation we obtain:

R = − ∂

∂xi

(k∑

a

∂Na

∂xiφa

)+ Q (40)




a

y

x

Na∼

Figure 4.25. Basis function in linear polynomials for a patch of triangular elements.

and we now seek the best values of the parameter set φa, which ensures that:∫�

WbRd� = 0, b = 1, 2, . . . , n (41)

Note that this is the term multiplying the arbitrary parameter vb. As notedpreviously, integration by parts is used to avoid higher-order derivatives (i.e. thosegreater than or equal to two) and therefore reduce the constraints on choosing thebasis functions to permit integration over individual elements using equation (39).In the present case, for instance, the weighted residual after integration by partsand introducing the natural boundary condition becomes:

∫�

∂Wb

∂xi

(k∑

a

∂Na

∂xiφ

)d�+

∫�

WbQd�+∫�q

Wbqnd� = 0 (42)

4.2.3 The Galerkin, finite element, method

In the Galerkin method we simply take Wb = Nb, which gives the assembled systemof equations:

n∑a=1

Kbaφa + fb = 0, b = 1, 2, . . . , n − r (43)

where r is the number of nodes appearing in the approximation to the Dirich-let boundary condition (i.e., equation (30)) and Kba is assembled from elementcontributions Ke

ba with:

Keba =

∫�e

∂Nb

∂xik∂Na

∂xid� (44)




Similarly, fb is computed from the element as:

f eb =

∫�e

NbQd�+∫�eq

Nbqnd� (45)

To impose the Dirichlet boundary condition we replace φa by φa for the rboundary nodes.

It is evident in this example that the Galerkin method results in a symmetric set ofalgebraic equations (e.g. Kba = Kab). However, this only happens if the differentialequations are self-adjoint. Indeed the existence of symmetry provides a test forself-adjointness and also for existence of a variational principle whose stationarityis sought.

It is necessary to remark here that if we were considering a pure convectionequation:

ui∂φ

∂xi+ Q = 0 (46)

symmetry would not exist and such equations can often become unstable if theGalerkin method is used.

4.2.4 Characteristic Galerkin scheme for convection–diffusion equation

Unlike a simple conduction equation (as the Laplace equation), a numerical solutionfor the convection equation has to deal with the convection part of the governingequation in addition to diffusion. For most conduction equations, the finite elementsolution is straightforward. However, if a Galerkin type approximation was usedin the solution of convection equations, the results will be marked with spuriousoscillations in space if certain parameters exceed a critical value (element Pecletnumber). This problem is not unique to finite elements as all other spatial dis-cretization techniques have the same difficulties. A very well-known method usedin finite elements approximation to reduce these oscillations is the CharacteristicGalerkin (CG) scheme (Lewis et al. [19], Zienkiewicz et al. [20]). Here, we followthe Characteristic Galerkin (CG) approach to deal with spatial oscillations due tothe discretization of the convection transport terms.

In order to demonstrate the CG method, let us consider the simple convection–diffusion equation in one dimension, namely:

∂φ

∂t+ u1

∂φ

∂x1− ∂

∂x1

(k∂φ

∂x1

)= 0 (47)

Let us consider a characteristic of the flow as shown in Figure 4.26 in the time–space domain. The incremental time period covered by the flow is �t from the nthtime level to the n + 1th time level and the incremental distance covered during this




Characteristic

n + 1

n

x1φ n + 1

x1φ n

∆t

x1− ∆x1 ∆x1x1

x1− ∆x1φ n

Figure 4.26. Characteristic in a space–time domain.

time period is�x1, that is, from (x1 −�x1) to x1. If a moving coordinate is assumedalong the path of the characteristic wave with a speed of u1, the convection termsof equation (47) disappear (as in a Lagrangian fluid dynamics approach). Althoughthis approach eliminates the convection term responsible for spatial oscillationwhen discretized in space, the complication of a moving coordinate system x′

1 isintroduced, that is, equation (47) becomes:

∂φ

∂t(x′

1, t) − ∂

∂x′1

(k∂φ

∂x′1

)= 0 (48)

The semi-discrete form of the above equation can be written as:

φn+1|x1

− φn|x1−�x1

�t− ∂

∂x′1

(k∂φ

∂x′1

)n

|x1−�x1

= 0 (49)

Note that the diffusion term is treated explicitly. It is possible to solve the aboveequation by adapting a moving coordinate strategy. However, a simple spatialTaylorseries expansion in space avoids such a moving coordinate approach. With referenceto Figure 4.26, we can write using a Taylor series expansion:

φn|x1−�x1= φn|x1

− ∂φn

∂x1

�x1

1! + ∂2φn

∂x21

�x21

2! − . . . (50)

Similarly, the diffusion term is expanded as:

∂

∂x′1

(k∂φ

∂x′1

)n

|x1−�x1

= ∂

∂x1

(k∂φ

∂x1

)n

|x1

− ∂

∂x1

[∂

∂x1

(k∂φ

∂x1

)n]�x (51)




ji

l

Figure 4.27. One-dimensional linear element.

On substituting equations (50) and (51) into equation (49), we obtain (higher-order terms being neglected) the following expression:

φn+1 − φn

�t= −�x

�t

∂φ

∂x1

n

+ �x2

2�t

∂2φ

∂x21

n

+ ∂

∂x1

(k∂φ

∂x1

)n

(52)

In this case, all the terms are evaluated at position x1, and not at two positionsas in equation (49). If the flow velocity is u1, we can write�x = u1�t. Substitutinginto equation (52), we obtain the semi-discrete form as:

φn+1 − φn

�t= −u1

∂φ

∂x1

n

+ u21�t

2

∂2φ

∂x21

n

+ ∂

∂x1

(k∂φ

∂x1

)n

(53)

By carrying out a Taylor series expansion (Figure 4.26), the convection termreappears in the equation along with an additional second-order term. This second-order term acts as a smoothing operator that reduces the oscillations arising fromthe spatial discretization of the convection terms. The equation is now ready forspatial approximation.

The following linear spatial approximation of the scalar variable ϕ in space isused to approximate equation (53):

φ = Niφi + Njφj = [N]{ϕ} (54)

where [N] are the shape functions and subscripts i and j indicate the nodes of alinear element as shown in Figure 4.27.

On employing the Galerkin weighting to equation (53), we obtain:∫�

[N]T φn+1 − φn

�td�+

∫�

[N]T u1∂φ

∂x1

n

d�

(55)

− �t

2

∫�

[N]T u21∂2φ

∂x21

n

d�−∫�

[N]T ∂

∂x1

(k∂φ

∂x1

)n

= 0

The above equation is equal to zero only if all the element contributions areassembled. For a domain with only one element, we can substitute:

[N]T =[

Ni

Nj

](56)




On substituting a linear spatial approximation for the variable φ, over elementsas typified in Figure 4.27, into equation (55), we get:∫�

[N]T [N]{φn+1 − φn}

�td� = − u1

∫�

[N]T ∂

∂x1([N]{φ})nd�

(57)

+ �t

2u2

1

∫�

[N]T ∂2

∂x21

([N]{φ})nd�+∫�

[N]T k∂2

∂x21

([N]{φ})nd�

Before utilizing the linear integration formulae, we apply Green’s lemma to thesecond-order terms of equation (57), we obtain:∫�

[N]T [N]{φn+1 − φn}

�td� = −u1

∫�

[N]T ∂[N]

∂x1{φ}nd�

− �t

2u2

1

∫�

∂[N]T

∂x1

∂[N]

∂x1{φ}nd�+ �t

2u2

1

∫�

[N]T ∂[N]

∂x1{φ}nn1d� (58)

−∫�

∂[N]T

∂x1k∂[N]

∂x1{φ}nd�+

∫�

[N]T k∂[N]

∂x1{φ}nn1d�

where n1 and n2 are the direction cosines of the outward normal n,� is the domainand � is the domain boundary. The first-order convection term can be integratedeither directly or via Green’s lemma. Here, the convection term is integrated directlywithout applying Green’s lemma. However, integration of the first derivatives byparts is useful for problems in which the traction is prescribed. Using the integrationformulae (Lewis et al. [19]), it is possible to derive the element matrices for all theterms in equation (58). The term on the left-hand side for a single element is:

∫�

[N]T [N]{φn+1 − φn}

�td� = l

6

[2 11 2

]⎧⎨⎩φn+1

i −φni

�t

φn+1j −φn

j

�t

⎫⎬⎭ = [Me]�{ϕ}�t

(59)

where [Me] is the mass matrix for a single element. The above mass matrix for asingle element will have to be utilized in an assembly procedure for a fluid domaincontaining many elements.

In a similar fashion, all other terms can be integrated; for example, theconvection term is given by:

u1

∫�

[N]T ∂[N]

∂x1{φ}nd� = u1

2

[−1 1−1 1

]{φi

φj

}n

= [Ce]{ϕ}n (60)

where [Ce] is the elemental convection matrix. The values of the derivatives of theshape functions are substituted in order to derive the above matrix.




The diffusion term within the domain is integrated as:∫�

∂[N]T

∂x1k∂[N]

∂x1{φ}nd� = k

l

[1 −1

−1 1

]{φi

φj

}n

= [Ke]{ϕ}n (61)

where [Ke] is the elemental diffusion matrix. The characteristic Galerkin termwithin the domain is integrated as:

�t

2u2

1

∫�

∂[N]T

∂x1

∂[N]

∂x1{φ}nd� = u2

1�t

2

1

l

[1 −1

−1 1

]{φi

φj

}n

= [Kse]{ϕ}n (62)

where [Kse] is the elemental stabilization matrix.The boundary term from the diffusion operator is integrated by assuming that i

is a boundary node, as follows:

∫�

[N]T k∂[N]

∂x1{φ}nn1d� = k

⎧⎨⎩−φi

l+ φj

l0

⎫⎬⎭n

n1 = [fe] (63)

where {f e} is the forcing vector due to the diffusion term.The boundary integral from the characteristic Galerkin term is integrated, again

by assuming that i is a boundary node, as:

�t

2u2

1

∫�

[N]T ∂[N]

∂x1{φ}nn1d� = �t

2u2

1

⎧⎨⎩−φi

l+ φj

l0

⎫⎬⎭n

n1 = [fse] (64)

where {f se} is the forcing vector due to the stabilization term.For a one-dimensional domain with more than one element, all the matrices and

vectors need to be assembled in order to obtain the global matrices. Once assembled,the discretized one-dimensional equation becomes:

[M]�{ϕ}�t

= −[C]{φ}n − [K]{φ}n − [Ks]{φ}n + {f}n + {fs}n (65)

Let us now consider a simple one-dimensional convection problem, as given inFigure 4.28, to demonstrate the effect of a discretization with and without the CGscheme.

The scalar variable value at the inlet is ϕ= 0, and at the exit its value is 1. Thisscalar variable is transported in the direction of the velocity as shown in Figure4.28. Note that the convection velocity u1 is constant. The element Peclet numberfor this problem is defined as:

Pe = u1h

2k(66)

where h is the element size in the flow direction, which, in one dimension is thelocal element length. Figure 4.29 shows the comparison between a solution with




L

φ = 0Inlet

u1 = constant φ = 1Exit

Figure 4.28. One-dimensional convection–diffusion problems.

0.8

1

0.6

Fun

ctio

n

0.4

0.2

0

0 0.2 0.4Horizontal distance

(a) Pe = 1.0

0.6 0.8 1

0.8

1

0.6

Fun

ctio

n

0.4

0.2

0

0 0.2 0.4Horizontal distance

(b) Pe = 1.5

0.6 0.8

Standard GalerkinCharacteristic Galerkin

Exact solution

1

Standard GalerkinCharacteristic Galerkin

Exact solution

Figure 4.29. Spatial variation of a function, φ, in one-dimensional space fordifferent element Peclet numbers.

the CG discretization scheme and one without it. Only two Peclet numbers areshown in these diagrams to demonstrate the spatial oscillations without the CGdiscretization. As seen, both discretizations give no spatial oscillations at a Pevalue of unity. However, at a Pe value of 1.5, the CG discretization is accurate andstable, while the discretization without the CG term becomes oscillatory. The exactsolution to this problem is given as follows (Brooks and Hughes [21]):

φ = 1 − eu1x1

k

1 − eu1L

k

(67)

In this equation, L is the total length of the domain and x1 is the local length ofthe domain.

The extension of the characteristic Galerkin scheme to a multi-dimensionalscalar convection–diffusion equation is straightforward and follows the previousprocedure as discussed for a one-dimensional case.

4.2.5 Stability conditions

The stability conditions for a given time discretization may be derived usinga Von Neumann or Fourier analysis for either the convection equation or theconvection–diffusion equations. However, for more complicated equations such




A3

l3

l2

l1

l4

l5

Node

A2

A1

A4

A5

Figure 4.30. Two-dimensional linear triangular element.

as the Navier–Stokes equations, the derivation of the stability limit is not straight-forward. A detailed discussion on stability criteria is not given here and readersare asked to refer to the relevant text books and papers for details (Hirsch [22],Zienkiewicz and Codina [23]). A stability analysis will give some idea about thetime-step restrictions of any numerical scheme. In general, for fluid dynamics prob-lems, the time-step magnitude is controlled by two wave speeds. The first one isdue to the convection velocity and the second due to the real diffusion introducedby the equations. In the case of a convection–diffusion equation, the convection

velocity is√

uiui, which is, in two dimensional problems,√

u21 + u2

2 = |u|. The dif-fusion velocity is 2k/h where h is the local element size. The time-step restrictionsare calculated as the ratio of the local element size and the local wave speed. It istherefore correct to write that the time step is calculated as:

�t = min (�tc,�td ) (68)

where �tc and �td are the convection and diffusion time-step limits, respectively,which are:

�tc = h

|u| ,�td = h2

2k(69)

Often, it may be necessary to multiply the time-step �t by a safety factor dueto different methods of element size calculations. A simple procedure to calculatethe element size in two dimensions is:

h = min(

2Areai

li

), i = 1, number of elements connected to the node (70)

where Areai are the area of the elements connected to the node and li are the lengthof the opposite sides as shown in Figure 4.30. For the node shown in this figure, thelocal element size is calculated as:

h = min (A1/l1, A2/l2, A3/l3, A4/l4, A5/l5) (71)




In three dimensions, the term 2Areai is replaced by 3Volumei and li is replacedby the area opposite the node in question.

4.2.6 Characteristic-based split scheme

It is essential to understand the characteristic Galerkin procedure, discussed inSection 2.4 for the convection–diffusion equation, in order to apply the concept tosolve the real convection equations. Unlike the convection–diffusion equation, themomentum equation, which is part of a set of heat convection equations, is a vectorequation. A direct extension of the CG scheme to solve the momentum equationis difficult. In order to apply the CG approach to the momentum equations, wehave to introduce two steps. In the first step, the pressure term from the momentumequation will be dropped and an intermediate velocity field will be calculated.In the second step, the intermediate velocities will be corrected. This two-stepprocedure for the treatment of the momentum equations has two advantages. Thefirst advantage is that without the pressure terms, each component of the momentumequation is similar to that of a convection–diffusion equation and the CG procedurecan be readily applied. The second advantage is that removing the pressure termfrom the momentum equations enhances the pressure stability and allows the useof arbitrary interpolation functions for both velocity and pressure. In other words,the well-known Babuska–Brezzi condition is satisfied (Babuska [24], Brezzi andFortin [25], Chung [26]). Owing to the split introduced in the equations, the methodis referred to as the CBS scheme.

The CG procedure may be applied to the individual momentum componentswithout removing the pressure term, provided the pressure term is treated as asource term. However, such a procedure will lose the advantages mentioned inthe previous paragraph. For more mathematical details, please refer to Zienkiewiczet al. [20], Zienkiewicz and Codina [23], Nithiarasu [27] and Zienkiewicz et al. [28].

In order to apply the CG procedure, we can refer to the general case of governinggeneralized porous medium flow and heat transfer equations in non-dimensionalform and indicial notation, for mixed convection, that have been presented inSection 1.3 (see equations (24)–(26)).

From the governing equations, it is obvious that the application of the CGscheme is not straightforward. However, by implementing the following procedure,it is possible to obtain a solution to the convection heat transfer porous mediumequations. The solution of the free fluid flow equations is obtained by applying thesame procedure.

Temporal discretizationFor the sake of simplicity, the asterisks are omitted and the Darcy and Forchheimerterms in equation (25) are grouped as a “porous” term to obtain:

P =(

1

ReDa+ F√

Da|u|

)(72)




Temporal discretization along the characteristics of continuity, momentum andenergy conservation equations results in the following set of equations:

∂un+ϑ1i

∂xi= 0 (73)

un+1i (1 +�tεPϑ3) − un

i (1 −�tεP(1 − ϑ3))

= −ε�t

(ϑ2∂pn+1

f

∂xi+ (1 − ϑ2)

∂pnf

∂xi

)−

[�t

εuj∂ui

∂xj

]n

+[�t

J

Re

∂2ui

∂x2i

]n

+ ε�t2

2

[1

ε2 uk∂

∂xk

(uj∂ui

∂xj

)+ uk

∂

∂xk

∂pf

∂xi

]n

+ ε�tRa

Pr Re2 T nγi (74)

T n+1 − T n

�t= −

[ui∂T

∂xi

]n

+[

1

σRe Pr

∂2T

∂x2i

]n

+ �t

2σ

[uk∂

∂xk

(ui∂T

∂xi

)]n

(75)

In equations (74) and (75), additional stabilization terms appear naturally fromthe discretization along the characteristics (Zienkiewicz et al. [20]), while the char-acteristic parameters ϑ1, ϑ2 and ϑ3 (0.5 ≤ϑ1 ≤ 1 and 0 ≤ϑi ≤ 1, with i = 2, 3) aredefined according to the following:

∂un+ϑ1i

∂xi= ϑ1

∂un+1i

∂xi+ (1 − ϑ1)

∂uni

∂xi,

∂pn+ϑ2f

∂xi= ϑ2

∂pn+1f

∂xi+ (1 − ϑ2)

∂pnf

∂xi, (76)

[Pui]n+ϑ3 = ϑ3[Pui]n+1 + (1 − ϑ3)[Pui]n.

Different versions of the CBS scheme can be obtained depending on the valueof the above parameters. In particular an SI and an AC version of the CBS schemecan be obtained by varying the parameter ϑ2. For ϑ2 between 0.5 and 1, the SI-CBSis obtained, while for ϑ2 equal to 0, the AC-CBS scheme is derived. Moreover, forϑ3 between 0.5 and 1, an implicit treatment for the porous term is obtained, whilefor ϑ3 equal to 0, an explicit one is derived.

The splitting in the CBS scheme consists of solving the above equations in anumber of subsequent steps. In the first step, the pressure term is removed fromequation (74) and the intermediate velocity components ui are obtained from:

Step 1: Intermediate velocity calculation

ui(1 +�tεPϑ3) − uni (1 +�tεP (ϑ3 − 1)) = −

[�t

εuj∂ui

∂xj

]n

+[�t

J

Re

∂2ui

∂x2i

]n

+ ε�t2

2

[1

ε2 uk∂

∂xk

(uj∂ui

∂xj

)]n

+ ε�tRa

Pr Re2 T nγi (77)




Removing the pressure term from the momentum equation, the pressure stabilityis enhanced and the use of arbitrary interpolation functions for both velocity andpressure is allowed. In other words, the well-known Babuska–Brezzi condition issatisfied (Babuska [24], Brezzi and Fortin [25], Chung [26]). The correct velocitiescan be determined, once the pressure field is known, using the equation:

Step 3: Velocity correction

un+1i − ui = − 1( 1

ε�t + Pϑ3)[ϑ2

∂pn+1f

∂xi+ (1 − ϑ2)

∂pitart

∂xi+

(�t

2uk∂

∂xk

∂pf

∂xi

)n]

(78)

The solution of equation (78) is the third step of the algorithm.The second step consists in the pressure calculation through the continuity equa-

tion. This second step is where the SI and AC versions of the scheme differ. Inparticular, this second step is obtained in the SI version of the scheme by derivingequation (78) with respect to xi and imposing equation (73), obtaining the followingPoisson type of equation:

Step 2 SI-CBS: Pressure calculation

0 = −�t1

ε

(ϑ1∂ui

∂xi+ (1 − ϑ1)

∂uni

∂xi

)+�t2

∂2pn+1f

∂x2i

(79)

Therefore, in this case, the incompressibility constraint is satisfied at each iter-ation, and the pressure, evaluated through equation (79), represents the actualpressure. However, the solution of equation (79) needs a matrix inversion. Further-more, the SI version of the scheme uses a global time step, which is the minimumvalue of the time step limit over the entire domain.

Alternatively, the AC scheme can be derived when the left hand side of equa-tion (79) is obtained from a mass conservation equation retaining the transientdensity term:

∂ρf

∂t+ 1

ε

∂(ρf ui)

∂xi= 0 (80)

In general, it is possible to relate the density time variation to the pressure timevariation, through the speed of sound, as follows:

∂ρf

∂t= 1

c2

∂pf

∂t(81)

where the real compressibility parameter, c (compressible wave speed), approachesinfinity for many incompressible flow problems and the solution scheme becomesstiff and imposes severe time step restrictions. However, this parameter can bereplaced locally by an appropriate artificial value, β, of finite value, employing the




AC method, that was first introduced by Chorin [29] and then further developed[30–42]:

∂ρf

∂t= 1

β2

∂part

∂t⇒ 1

β2

pit+1art − pit

art

�t= −ρf

ε

∂uit+ϑ1i

∂xi(82)

In equation (82), the pressure is an artificial pressure, because the incompressibilityconstraint is not achieved at each time step, but only when steady-state convergenceis reached. The superscripts it + 1 and it are referred to the iterative procedure anddo not refer to real time levels. The second step of the AC scheme is carried out byimposing the equation (82) as:

Step 2 AC-CBS: Pressure calculation

1

β2 (pit+1art − pit

art) = −�t1

ε

(ϑ1∂ui

∂xi+ (1 − ϑ1)

∂uiti

∂xi

)+�t2 ∂

2pitart

∂x2i

(83)

The local value of β is calculated through the following procedure:

β = max (0.5, uconv, udiff , uther) (84)

The local convective, diffusive and thermal velocities can be calculated throughthe following non-dimensional relations:

uconv = √uiui , udiff = 2

hRe, uther = 2

hRe Pr. (85)

The diffusive time step limitation for the AC method may be written as h2/Re/2,while the convective time step limitation may be written as:

�tconv = h

|u| + β (86)

The above relation includes the viscous effect via the artificial parameter.Depending on the problem of interest, it is possible to consider other equations

coupled to the above set, such as species concentration for multi-component flows.The fourth step, for non-isothermal problems, is represented by the energy con-

servation equation (75) that allows calculating the temperature at every iteration forproblems with a coupling between momentum and energy conservation equationsoccurs.

Spatial discretization and solution methodologyThe spatial discretization of the governing equations is obtained through Galerkinfinite element procedure and triangular elements. Within an element, each variableis calculated through linear approximation on the basis of nodal values, accordingto the following equation:

φ =3∑

n=1

Nnφn (87)




where Nn is the shape function at node n and φn is the value of the generic variableφ at node n. Applying the standard Galerkin procedure to the set of equationspresented in the previous section, the following four steps, expressed in a matrixform, are obtained:

Step 1: Intermediate velocity calculation

Mui = (1 −�tεP)Muni − �t

ε[Cui]n −�t

J

Re[Kdui]n − 1

2

�t2

ε[Kuui]n

(88)

+ε�t

[Ra

Pr Re2 MTγi

]n

+�tJ

Re[fd]n + 1

2

�t2

ε[fu]n

Step 2 SI-CBS: Pressure calculation

[Kppf ]n+1 = − 1

ε�t[Dui] (89)

Step 2 AC-CBS: Pressure calculation

1

β2 M(pit+1art − pit

art) = −�t

ε[Dui] −�t2[Kppart]it (90)

Step 3: Velocity correction

M(un+1i − ui) = −ε�t

[ϑ2Dpn+1

f + (1 − ϑ2)Dpitart − �t

2Kupn

f

](91)

Step 4: Temperature calculation

MT n+1 = MT n −�t[CT ]n − �tλ

Re Pr Rk[KdT ]n

(92)

− �t2

2Rk[KuT ]n + �tλ

Re Pr Rk[ft]n + �t2

2Rk[fut]n

where M is the mass matrix, C is the convection matrix, Kd is the diffusion matrix,Ku is the stabilization matrix obtained from higher-order terms, fd and fu are theboundary vectors from the momentum equation, Kp is the stiffness matrix, D is thegradient matrix, f t and fut are the boundary vectors from the temperature equation.Details of all the terms presented in equations (88)–(92) are given in Lewis et al.[19] and Zienkiewicz et al. [20].

In the present procedure, the mass matrix is lumped using a standard row-summing approach, inverted and stored in an array during the pre-processing.Therefore, in the AC formulation, the calculation of the artificial pressure at it + 1iteration, through equation (90), does not need a matrix inversion and the result isa matrix inversion-free procedure. On the other hand, in the second step of the SIsolution procedure, the stiffness matrix needs to be inverted at each iteration.




u = 0, v = 0, q = 1

u = 0, v = 0, q = 1

u = 1T = 0

y

x

(b)

(a)

Figure 4.31. Forced convection in a porous channel. (a) Computational domain andboundary condition; (b) structured computational grid (3,321 nodes,6,400 elements).

Some examplesBoth the schemes presented above can be used to solve many engineering problems,such as mixed, natural or forced convection both in fluid-saturated porous mediaand in partly porous domains, or simply in free fluid flow cases.

As mentioned before, changing the properties such as porosity and thus per-meability, it is possible to handle porous medium-free fluid interface problems asa single problem with different properties. The following limits are used for theporous medium part and for the free fluid part:

ε < 1Da = finite

⇒ porous mediumε = 1

Da → ∞ ⇒ free fluid (93)

A suitable set of matching conditions, to connect the porous medium and thefree fluid domains, is needed (Massarotti et al. [43]).

The first example is the forced convection heat transfer inside a uniform porouschannel with constant wall heat flux. The computational domain, together with theboundary conditions employed, is sketched in Figure 4.31a, while Figure 4.31bshows the computational grid employed. The flow enters the domain from the leftwith a constant velocity and at a non-dimensional temperature T = 0. The channelwalls are heated by a constant heat flux. The numerical results have been obtainedusing a non-uniform grid with 6,400 elements and 3,321 nodes.

Figure 4.32 shows the non-dimensional velocity and temperature profiles, fordifferent Darcy numbers, varying from 10−4 to 1.0 at a section, where the flowand the thermal field are hydrodynamically and thermally developed. The resultshave been compared with the analytical solutions presented by Nield et al. [44] andLauriat and Vafai [45]. The dimensionless temperature is evaluated using:

Tmix = 1

umeanSx

∫Sx

uiTidS (94)




y

u/u

mea

n

−1 −0.5 0

(a) (b)

0.5 10

0.5

1

1.5

Analytical Nieldet al. [44]AC-CBSSI-CBS

Da = 10−4

Da = 10−2Da = 5x10−2 Da =10−1

Da = 1

y

T�

Tw

all/

Tm

ix�

Tw

all

−1 −0.5 0 0.5 10

0.5

1

1.5

Analytical Lauriatand Vafai [45]AC-CBSSI-CBS

Da = 10−3

Da = 10−1

Da = 1

Da = 10−4

Figure 4.32. Forced convection in a porous channel: (a) non-dimensional velocityas function of transverse distance; (b) non-dimensional temperatureas function of transverse distance.

where umean is the mean velocity in the section considered Sx.For a small Darcy number (10−4), the velocity profile is nearly independent of

the transverse distance and slip flow occurs at the walls (Figure 4.32). An increasein Darcy number leads to a non-linear distribution of the velocity. When the Darcynumber is equal to unity, the velocity profile approaches a profile similar to that offree fluid flow.

The non-dimensional temperature increases as the Darcy number decreases, asshown in Figure 4.32. A decrease in the Darcy number corresponds to a decreasein the fluid velocity in the middle of the channel and therefore the conduction heattransfer is dominating over the effect of convection.

Figure 4.33 shows the convergence histories to steady state for the AC-CBS andSI-CBS schemes obtained by using the following L2 norm for velocities:

Velocity residual =√√√√nodes∑

i=1

( |ui|n+1 − |ui|n�ti

)2/

nodes∑i=1

( |ui|n+1

�ti

)2

(95)

This figure shows that the SI version of the scheme converges faster as the Darcynumber decreases. Instead, theAC version of the scheme has an opposite behaviour.Table 4.1 reports the CPU time needed by both the procedures to reach steady statesolution, on a machine with 4 Gb of RAM and 2.4 GHz CPU speed. These resultsconfirm the behaviour shown in Figure 4.33. In particular, Table 4.1 shows thatthe AC-CBS scheme converges faster than the SI-CBS scheme when Da ≥ 10−2. Itwas noticed that the SI-CBS scheme needs more time than the AC-CBS scheme pertime iteration. In correspondence of smaller Darcy numbers (Da ≤ 10−3), the AC-CBS scheme takes more time to reach the steady state. This is due to the time step




Iterations

Velo

city

resi

dual

100

100 101 102 103 104 105

10−1

10−2

10−3

10−4

10−5

10−6

AC-CBSSI-CBS

(a)

Iterations

Velo

city

resi

dual

100

100 101 102 103 104 105

10−1

10−2

10−3

10−4

10−5

10−6

AC-CBSSI-CBS

(b)

Figure 4.33. Convergence histories for SI and AC scheme. (a) Da = 1;(b) Da = 10−4.

Table 4.1. CPU time (s) for forced convection in a porous channel at different Darcynumbers

Da 1 10−1 5 × 10−2 10−2 10−3 10−4

AC-CBS 35 87 137 491 4,341 6,400

SI-CBS 14,730 10,250 8,055 3,357 1,107 250

calculation procedure used. Essentially, the AC scheme becomes slower in reachingthe steady state when diffusion is overriding the effect of convection (Da ≤ 10−3)and the local time step approaches the global time step, losing the advantage ofusing a higher time step in the convective zones.

The second example is the natural convection in a cavity heated uniformly fromthe bottom side. The computational domain, together with the boundary conditionsemployed, is sketched in Figure 4.34a, while Figure 4.34b shows the computationalgrid employed, composed of 7,200 triangular elements and 3,721 nodes.

Figure 4.35 shows the dimensionless temperature contours for a Prandtl numberequal to 0.71, and for two different Rayleigh numbers (106 and 7 × 104) and twodifferent Darcy numbers (10−3 and 10−4). The results obtained have been comparedwith the numerical solution presented by Basak et al. [46].

In general, the fluid circulation is strongly dependent on the Darcy number.In fact, when smaller Rayleigh and Darcy numbers are considered, the flow isvery weak and the temperature distribution is similar to that of stationary fluid(Figure 4.35a). As the Darcy number increases, the role of convection becomes




1

(a) (b)

u=v=0

T=0

u=v=0, T=0

u=v=0

y

xT=0

u=v=0, =0∂T

∂y

Figure 4.34. Natural convection in a porous cavity heated from the bottom side:(a) computational domain and boundary conditions; (b) structuredcomputational grid (3,721 nodes, 7,200 elements).

0.4

0.1

0.4

0.3

0.1

0.30.1

0.1

0.80.70.5

0.4

0.1

0.3

0.2

0.90.8

0.7

0.50.4

0.6

0.3

0.1

0.1 0.1

0.40.20.3

0.5

0.1

0.40.30.2

0.1

0.40.9

0.2

0.2

0.1

0.6

0.70.2 0.6

0.50.8 0.v 0.6

0.8 0.7

0.2

0.30.40.5

0.7

0.6

0.40.5

0.5

0.5

0.4 0.

3

0.3

0.4

0.3

0.9

0.5

0.2

0.2

0.5

0.2

(a) (b)

Figure 4.35. Temperature contours for natural convection in a porous cavity heatedfrom the bottom side. (a) Ra = 7 × 104 and Da = 10−4; (b) Ra = 106

and Da = 10−3.

more significant and the fluid rises up strongly from the middle portion of thebottom wall, as depicted in Figure 4.35b.

Figure 4.36 shows the steady-state convergence histories for the AC-CBS andthe SI-CBS schemes. This figure shows a different behaviour from that of the caseof forced convection flow problem. When natural convection occurs, the AC-CBSscheme converges faster than the SI-CBS scheme for any Rayleigh or Darcy number,both in terms of number of iterations (Figure 4.36) and CPU time to reach the steadystate (Table 4.2).

The SI-CBS algorithm takes more time to reach the steady state because of thecoupling between momentum and energy conservation equations present in naturalconvection problems. The coupling is due to the presence of a generation term on




Vel

ocity

res

idua

l

10−6

10−5

105

10−4

10−2

10−3

103

10−1

101

100

Vel

ocity

res

idua

l

10−6

10−5

10−4

104

10−2

102

10−3

10−1

100

100

Iterations105103101 104102100

Iterations

AC-CBSSI-CBS

AC-CBSSI-CBS

(a) (b)

Figure 4.36. Convergence histories for SI andAC. (a) Ra = 7 × 104 and Da = 10−4;(b) Ra = 106 and Da = 10−3.

Table 4.2. CPU time (s) for natural convection in a porous cavity heated from thebottom at different Rayleigh and Darcy numbers

Ra 106 7 × 104

Da 10−3 10−4 10−3 10−4

AC-CBS 180 930 290 700

SI-CBS 63,270 48,230 53,400 47,560

the right hand side of the momentum conservation equation in gravity directionthat must be evaluated at each time step. This term can be calculated only after theresolution of the energy equation (step 4 of the CBS algorithm), due to the fact thatthe nodal temperature values are unknown. This coupled system becomes stiff andcauses a restriction on the global time step value used in the SI procedure and thusthe solution to the simultaneous equations at the second step of the algorithm needsmore time.

Moreover, the SI scheme experiences difficulties in reaching the steady statewhen very refined meshes are used (Massarotti et al. [47]). On the other hand, theAC-CBS scheme does not show any problem in reaching the convergence whennatural convection flow occurs. The reason for this is the robust local time steppingprocedure used by the AC-CBS scheme.

The last example is the developing mixed convection in a region partially filledwith a fluid-saturated porous medium, confined between two vertical hot walls.In particular, because of the geometrical symmetry, half domain has been studied.Figure 4.37 shows the computational domain and the boundary conditions employedand the details of the computational grid near the entrance. The mesh employed has




u = v=0

(a) (b)

T = 1

v = 1, T = 0

L

yx

u = v=0T = 1

Figure 4.37. Mixed convection in a vertical channel partially filled with aporous medium. (a) Computational domain and boundary conditions;(b) detail of the structured computational grid near the entrance (8,591nodes and 16,800 elements).

x0 1

0

1

2

v/v

0

y = 3

y = 32 y = 0.5

Chang [48]

AC-CBS

1.5

0.5

0.5

Figure 4.38. Mixed convection in a vertical parallel plate channel partially filledwith a porous medium. Non-dimensional vertical velocity at differentheights of the channel and at Da = 10−5.

8,591 nodes and 16,800 elements and is refined near the wall, near the inlet regionand at the interface.

Figure 4.38 shows the dimensionless vertical velocity profile at different heightsof the channel (y) for a Darcy number equal to 10−5, obtained by using the AC-CBSscheme. The present results are compared with the numerical results of Chang and




Chang [48]. The parameters used for the present investigation are Ra = 0.72 × 103,Pr = 0.72, Re = 50, λ= 2.8 in the porous region and λ= 1 in the free fluid region,ε= 0.8. When y is small, there is a large velocity difference at the interface of thecomposite system (Figure 4.38). When y increases, the flow discharge in the porouslayer decreases, and the peak of the fluid velocity profile moves to the central axisof the vertical parallel-plate channel.

References

[1] Denham, M. K., and Patrik, M. A. Laminar flow over a downstream-facing in a two-dimensional flow channel. Transactions of the Institution of Chemical Engineers, 52,pp. 361–367, 1974.

[2] Aung, W. An experimental study of laminar heat transfer downstream of backsteps.Journal of Heat Transfer, 105, pp. 823–829, 1983.

[3] Ichinose, K., Tokunaga, H., and Satofuka, N. Numerical simulation of two-dimensionalbackward-facing step flows. Transactions of JSME B, 57(543), pp. 3715–3721, 1991.

[4] de Sampaio, P. A. B., Lyra, P. R. M., Morgan, K., and Weatherill, N. P. Petrov–Galerkinsolutions of the incompressible Navier–Stokes equations in primitive variables withadaptive remeshing. Computer Methods in Applied Mechanics and Engineering, 106,pp. 143–178, 1993.

[5] Massarotti, N., Nithiarasu, P., and Zienkiewicz, O. C. Characteristic-Based-Split (CBS)algorithm for incompressible flow problems with heat transfer. International Journalfor Numerical Methods in Heat and Fluid Flow, 8, pp. 969–990, 1998.

[6] Morgan, K., Weatherill, N. P., Hassan, O., Brookes, P. J., Said, R., and Jones, J.A parallelframework for multidisciplinary aerospace engineering simulations using unstructuredmeshes. International Journal for Numerical Methods in Fluids, 31, pp. 159–173, 1999.

[7] Spalart, P. R., and Allmaras, S. R. A one-equation turbulence model for aerodynamicflows. AIAA 30th Aerospace Sciences Meeting, AIAA paper 92-0439, 1992.

[8] Nithiarasu, P., and Liu, C.-B. An artificial compressibility based characteristic basedsplit (CBS) scheme for steady and unsteady turbulent incompressible flows. ComputerMethods in Applied Mechanics and Engineering, 195, pp. 2961–2982, 2006.

[9] Nithiarasu, P., Liu, C.-B., and Massarotti, N. Laminar and turbulent flow calculationsthrough a model human upper airway using unstructured meshes. Communications inNumerical Methods in Engineering, 23, pp. 1057–1069, 2007.

[10] Denham, M. K., Briard, P., and Patrick, M. A. A directionally-sensitive laser anemome-ter for velocity measurements in highly turbulent flows. Journal of Physics E: ScientificInstruments, 8, pp. 681–683, 1975.

[11] Gemci, T., Corcoran, T. E., Yakut, K., Shortall, B., and Chigier, N. Spray dynamics anddeposition of inhaled medications in the throat, ILASS-Europe, Zurich, 2–6 September,2001.

[12] Li, W.-I., Perzl, M., Heyder, J., Langer, R., Brain, J. D., Englmeier, K.-H., Niven, R. W.,and Edwards, D. A. Aerodynamics and aerosol particle deaggregation phenomena inmodel oral-pharyngeal cavities. Journal of Aerosol Science, 27, pp. 1269–1286, 1996.

[13] Li, W.-I., and Edwards, D. A. Aerosol particle transport and deaggregation phenomenain the mouth and throat. Advanced Drug Delivery Reviews, 26, pp. 41–49, 1997.

[14] Martonen, T. B., Zhang, Z.,Yue, G., and Musante, C. J. 3-D particle transport within thehuman upper respiratory tract. Journal of Aerosol Science, 33, pp. 1095–1110, 2002.




[15] Whitaker, S. Diffusion and dispersion in porous media. AIChE Journal, 13,pp. 420–427, 1961.

[16] Vafai, K., and Tien, C. L. Boundary and inertia effects on flow and heat transferin porous media. International Journal of Heat Mass Transfer, 24, pp. 195–203,1981.

[17] Hsu, C.T., and Cheng, P.Thermal dispersion in a porous medium. International Journalof Heat Mass Transfer, 33, pp. 1587–1597, 1990.

[18] Chen, Y. C., Chung, J. N., Wu, C. S., and Lue, Y. F. Non-Darcy mixed convection ina vertical channel filled with a porous medium. International Journal of Heat MassTransfer, 43, pp. 2421–2429, 2000.

[19] Lewis, R. W., Nithiarasu, P., and Seetharamu, K. N. Fundamentals of the Finite ElementMethod for Heat and Fluid Flow, Wiley, Chichester, 2004.

[20] Zienkiewicz, O. C., Taylor, R. L., and Nithiarasu, P. The Finite Element Method for fluiddynamics, 6th Edition, Elsevier Butterworth-Heinemann, Oxford 2005.

[21] Brooks, A. N., and Hughes, T. J. R. Streamline upwind/Petrov-Galerkin formulation forconvection dominated flows with particular emphasis on incompressible Navier–Stokesequation. Computer Methods in Applied Mechanics and Engineering, 32, pp. 199–259,1982.

[22] Hirsch, C. Numerical Computation of Internal and External Flows, 1, Fundamentalsof Computational Fluid Dynamics, 2nd ed., John Wiley & Sons, Oxford, 2007.

[23] Zienkiewicz, O. C., and Codina, R. A general algorithm for compressible and incom-pressible flow, Part I, The split characteristic based scheme. International Journal forNumerical Methods in Fluids, 20, pp. 869–885, 1995.

[24] Babuska, I. The finite element method with Lagrange multipliers, NumerischeMathematik, 20, pp. 179–192, 1973.

[25] Brezzi, F., and Fortin, M. Mixed and Hybrid Finite Element Methods, Springer,Berlin, 1991.

[26] Chung, T. J. Computational Fluid Dynamics, Cambridge University Press, Cambridge,2001.

[27] Nithiarasu, P., An efficient artificial compressibility (AC) scheme based on the charac-teristic based split (CBS) method for incompressible flows, International Journal forNumerical Methods in Engineering, 56, pp. 1815–1845, 2003.

[28] Zienkiewicz, O. C., Nithiarasu, P., Codina, R., Vazquez, M., and Ortiz, P. An effi-cient and accurate algorithm for fluid mechanics problems, the characteristic basedsplit (CBS) algorithm. International Journal for Numerical Methods in Fluids, 31,pp. 359–392, 1999.

[29] Chorin, A. J. A numerical method for solving incompressible viscous flow problems.Journal of Computational Physics, 2, pp. 12–26, 1967.

[30] Malan, A. G., Lewis, R. W., and Nithiarasu, P. An improved unsteady, unstruc-tured, artificial compressibility, finite volume scheme for viscous incompressible flows:Part II. Application. International Journal of Numerical Methods in Engineering, 54,pp. 715–729, 2002.

[31] Turkel, E. Review of preconditioning methods for fluid dynamics. Applied NumericalMathematics, 12, pp. 257–284, 1993.

[32] Weiss, J. W., and Smith, W. A. Preconditioning applied to variable and constant densityflows. AIAA Journal, 33, pp. 2050–2057, 1995.

[33] Gaitonde, A. L. A dual time method for two dimensional incompressible flow calcula-tions. International Journal of Numerical Methods in Engineering, 41, pp. 1153–1166,1998.




[34] Choi, Y. H., and Merkle, C. L. The application of preconditioning in viscous flows.Journal of Computational Physics, 105, pp. 207–223, 1993.

[35] Turkel, E. Preconditioning methods for solving the incompressible and low speedcompressible equations. Journal of Computational Physics, 72, pp. 227–298, 1987.

[36] Cabak, H., Sung, C. H., and Modi, V. Explicit Runge-Kutta method for three-dimensional internal incompressible flows. AIAA Journal, 30, pp. 2024–2031, 1992.

[37] Tamamidis, P., Zhang, G., and Assanis, D. N. Comparison of pressure-based and arti-ficial compressibility methods for solving 3D steady incompressible viscous flows.Journal of Computational Physics, 124, pp. 1–13, 1996.

[38] Lin, F. B., and Sotiropoulos, F. Assessment of artificial dissipation models for three-dimensional incompressible flow solutions. Journal of Fluids Engineering (ASME),119, pp. 331–340, 1997.

[39] Zhao, Y., and Zhang, B. L. A high-order characteristics upwind FV method for incom-pressible flow and heat transfer simulation on unstructured grids. Computer Methodsin Applied Mechanics and Engineering, 190, pp. 733–756, 2000.

[40] Manzari, M. T. An explicit finite element algorithm for convection heat transfer prob-lems. International Journal of Numerical Methods Heat Fluid Flow, 9, pp. 860–877,1999.

[41] Kim, E. A mixed Galerkin method for computing the flow between eccentric rotatingcylinders. International Journal of Numerical Methods in Fluids, 26, pp. 877–885,1998.

[42] Wasfy, T., West, A. C., and Modi, V. Parallel finite element computation of unsteadyincompressible flows. International Journal of Numerical Methods in Fluids, 26,pp. 17–37, 1998.

[43] Massarotti, N., Nithirasu, P., and Zienkiewicz, O. C. Natural convection in porousmedium-fluid interface problems – A finite element analysis by using the CBS pro-cedure. International Journal of Numerical Methods for Heat and Fluid Flow, 11(5),pp. 473–490, 2001.

[44] Nield, D. A., Kuznetsov, A. V., and Xiong, M. Effect of local thermal non-equilibriumon thermally developing forced convection in a porous medium. International Journalof Heat Mass Transfer, 45, pp. 4949–4955, 2002.

[45] Lauriat, G., and Vafai, K. Forced convection and heat transfer through a porous mediumexposed to a flat plate or a channel. Convective Heat and MassTransfer in Porous Media,Kluwer Academic, Dordrecht, pp. 289–327, 1991.

[46] Basak, T., Roy, S., Paul, T., and Pop, I. Natural convection in a square cavity filledwith a porous medium: Effects of various thermal boundary conditions. InternationalJournal of Heat Mass Transfer, 49, pp. 1430–1441, 2006.

[47] Massarotti, N., Arpino, F., Lewis, R. W., and Nithiarasu, P. Explicit and semi-implicitCBS procedures for incompressible viscous flows. International Journal of NumericalMethods in Engineering, 66(10), pp. 1618–1640, 2006.

[48] Chang, W.-J., and Chang, W.-L. Mixed convection in a vertical parallel-plate channelpartially filled with porous media of high permeability. International Journal of Heatand Mass Transfer, 39(7), pp. 1331–1342, 1996.


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4 The ﬁnite element method:discretization and application ... · vertical direction, caused by the temperature differences induced in the ﬂuid. Both structured and unstructured