Amano R.S., Sunden B. (Eds.) Computational Fluid Dynamics and Heat Transfer: Emerging Topics

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126 Computational Fluid Dynamics and Heat Transfer

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II. Finite Element Method

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4 The ﬁnite element method: discretization

and application to heat convection problems

Alessandro Mauro

1,2

, Perumal Nithiarasu

, Nicola Massarotti

and FaustoArpino

Civil and Computational Engineering Centre, School of Engineering,

Swansea University, Swansea, U.K.

Dipartimento per le Tecnologie, Università degli Studi di Napoli

“Parthenope”, Napoli, Italy.

Dipartimento di Meccanica, Structture, Ambiente e Territorio,

Università di Cassino, Cassino, Italy.

Abstract

Inthischapter,wegiveanoverviewoftheﬁniteelementmethodanditsapplications

to 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 the

incompressiblethermalﬂowequations.Thealgorithmisbasedonthetemporaldis-

cretization along the characteristics, and the high-order stabilization terms appear

naturallyfromthiskindofdiscretization.Theartiﬁcialcompressibility(AC)andthe

semi-implicit (SI) versions of the CBS scheme are presented and some examples

are 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. To

obtain a set of non-dimensional equations, let us consider ﬁve different cases of

convection heat transfer. We start with the mixed convection problem followed

by the natural and forced convection problems. The turbulent ﬂow and convective

ﬂow through porous media are also presented. For each case, we discuss one set of

non-dimensional scales.

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130 Computational Fluid Dynamics and Heat Transfer

Mixed convection

Mixed convection involves features from both forced and natural ﬂow conditions

that aretwolimiting cases ofmixed convection ﬂow.The buoyancyeffects become

comparable to the forced ﬂow effects at small and moderate Reynolds numbers.

Sincetheﬂowispartlyforced,areferencevelocityvalueisnormallyknown(exam-

ple: velocityat theinlet ofa channel). However,in mixed convectionproblems, the

buoyancy term needs to be added to the appropriate component of the momentum

equation.

In incompressible mixed convection ﬂow problems, the following non-

dimensional scales are normally employed:

∗

, u

∗

ref

, t

∗

ref

(1)

∗

ρu

ref

, T

∗

T − T

ref

− T

ref

where * indicates a non-dimensional quantity, L is a characteristic dimension, the

subscript ref indicates a constant reference value and T

is a constant reference

temperature, for example, wall temperature. The density ρ and viscosity µ of the

ﬂuid are assumed to be constant everywhere and equal to the inlet value.

Using the above scales, the non-dimensional form of the incompressible mixed

convection equations, in indicial notation, are written as:

Continuity equation

∂u

∗

∂x

∗

= 0 (2)

Momentum equation

∂u

∗

∂t

∗

+ u

∗

∂u

∗

∂x

∗

=−

∂p

∗

∂x

∗

∂

∂x



∂u

∗

∂x



∗

(3)

where γ

is a unit vector in the gravity direction.

Energy equation

∂T

∗

∂t

∗

+ u

∗

∂T

∗

∂x

∗

RePr

∂

∂x



∂T

∗

∂x



(4)

where Re is the Reynolds number deﬁned as:

Re =

ref

(5)

Pr is the Prandtl number given as:

Pr =

(6)

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Applications of ﬁnite element method to heat convection problems 131

Fluid circulation

Hot, vertical plate

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

and Gr is the Grashof number given as:

Gr =

gβTL

(7)

where ν =µ/ρ is the kinematic viscosity.Appropriate changes will be necessary if

an appreciable variation in these quantities occurs in a ﬂow ﬁeld.

Notethatsometimes,anon-dimensionalparameterreferredtoastheRichardson

number (Gr/Re

) is also used in the literature.

Natural convection

Natural convection is a limiting case of the more general case of mixed convec-

tion. Natural convection is generated onlyby the density differenceinducedby the

temperature differences within a ﬂuid system (Figure 4.1). Because of the small

density variations present in these types of ﬂows, a general incompressible ﬂow

approximation is adopted. In most buoyancy-driven convection problems, ﬂow is

generated either by a temperature variation or by a concentration variation in the

ﬂuid system, which leads to local density differences.

In practice, the following non-dimensional scales are adopted for natural

convection in the absence of a real reference velocity value:

∗

, u

∗

, t

∗

t α

(8)

∗

ρα2

, T

∗

T − T

ref

− T

ref

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132 Computational Fluid Dynamics and Heat Transfer

Upon introducing the above non-dimensional scales into the governing equa-

tions, and replacing Re with 1/Pr in the non-dimensional mixed convection

equations of the previous subsection, we obtain the non-dimensional equations

for natural convection ﬂows.

Often, another non-dimensional number called the Rayleigh number is used in

the calculations. This is given as:

Ra = Gr Pr =

gβTL

να

(9)

Forced convection

Forced convection is another limiting case of mixed convection. In forced convec-

tionﬂowproblems,thenon-dimensionalscalesarethesameasthemixedconvection

problem.The non-dimensional form of the forced convection governing equations

is obtained from that of mixed convection by neglecting the buoyancy term on the

right 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=

ref

L/α.

Some examples

The following results are obtained solving the above equations for forced, natural

and mixed convective ﬂows by using an artiﬁcial compressibility (AC) version and

a semi-implicit (SI) version of the characteristic-based split (CBS) algorithm and

ﬁnite element discretization technique that will be explained later.

Theﬁrstproblemconsideredisthebackward-facingstep,thatisatestcasecom-

monly used to validate numerical solutions of the incompressible Navier–Stokes

equations. In fact, several data are available in literature for this problem (Denham

and 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 work

are 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 in

theexperiments, asin thepresent calculations, was placed1.3 timesthe stepheight

upstream of the step.All the horizontal walls and the step have been imposed with

no-slip velocity boundary conditions.

For non-isothermal cases, in addition to the above boundary conditions (Aung

[2]), anon-dimensionaltemperatureequaltooneisimposedalong thebottomwall,

whilethetemperatureoftheinletﬂowisassumedtobeequaltozero.Allotherwalls

are assumed to be adiabatic. Figure 4.2 shows some details of the computational

grids used for this problem.

Figure 4.3 shows the horizontal velocity distribution along different vertical

sections of the channel, obtained by using the AC and the SI versions of the CBS

algorithm.

The temperature proﬁles at two different vertical cross sections downstream of

thesteparepresentedinFigure4.4forthecaseofafullydevelopedinletproﬁlewith

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Applications of ﬁnite element method to heat convection problems 133

p = 0

q = 0

T = 1v = 0,u = 0,

v = 0,

v = 0, q = 0,

u = 0,

10h8h6h4h2h0.8h0

−1.3h

h 2h

u = u(y)

v = 0

T = 0

(a)

(b)

(c)

Figure 4.2. Backward-facing step:(a)computational domainand boundary condi-

tions; (b)unstructured mesh(955nodesand 1,704elements); (c)detail

of the structured mesh near the step (4,183 nodes, 8,092 elements).

02468

SI-CBS

−1.3

AC-CBS Denham and Patrik [26]

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

Re=229.

y/δt

0 0.25 0.5 0.75 1

0.5

1.5

2.5

AC-CBS

SI-CBS

Aung [2]

AC-CBS

SI-CBS

Aung [2]

x/Xr = 0.482 x/Xr = 0.946

y/δt

0.5

1.5

2.5

Figure 4.4. Backward-facing step. Temperature proﬁles at two different sections

downstream of the step (Re=233).

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134 Computational Fluid Dynamics and Heat Transfer

= u

= 0

= u

= 0

T = 1 T = 0

Insulated

= u

= 0

= u

= 0

Figure 4.5. Natural convection in a square cavity. Computational domain and

boundary conditions.

Re=233.The proﬁlesare practically thesamefor boththe schemesand theyshow

a very good ag reement with the experimental data available from the literature for

this problem (Aung [2]). It should be observed that the position along the vertical

section is related to the thickness of the thermal boundary layer in the channel.

Anotherbenchmarkproblemthatiscommonlyemployedforcodevalidationisa

well known natural convection problem in whichthe ﬂuid-ﬁlled square cavity with

the vertical walls subjected to different temperatures is used. The two horizontal

wallsareassumedtobeadiabaticasshowninFigure4.5.No-slipvelocityconditions

are applied on all walls. The ﬂow is driven by the buoyancy forces acting in the

vertical direction, caused by the temperature differences induced in the ﬂuid. Both

structured and unstructured meshes are used (Figure 4.6).

Figures 4.7–4.9 show the temperature contours and streamlines obtained with

theAC-CBS algorithm at different Rayleigh numbers.

The transient ﬂow over a circular cylinder is a popular test case for validating

temporal discretization of numerical schemes. The inlet ﬂow is uniform and the

cylinder is placed at a centreline between two slip walls. The distance from the

inlet 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 cylinder

surface. The problem has been solved both on two and three dimensions. Figure

4.10 shows the mesh used for two-dimension case. As seen, the mesh close to

the 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 components

at a real non-dimensional time of 98.02. The time history of the vertical velocity

component at an exit central point is plotted with respect to time and is shown in

Figure 4.12.

In order to show that the method works also on three dimensions, a coarse

mesh with 17,382 nodes and 69,948 tetrahedral elements was used to recompute

thesolution.ThismeshwasgeneratedusingthePSUEsoftwareavailablewithinthe

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Applications of ﬁnite element method to heat convection problems 135

(a) (b)

(d)(c)

Figure 4.6. Naturalconvectioninasquarecavity.(a)Mesh1:1,251nodesand2,888

elements; (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 10,596

elements.

Figure 4.7. Natural convection in a square cavity. Streamlines and isotherms at

Ra=10

School of Engineering, Swansea University (Morgan et al. [6]).The ﬁnite element

mesh and thecontours of velocitycomponents are shown in Figures 4.13 and 4.14,

respectively, atarealnon-dimensionaltimeof83.33.Asseen,thesolutionobtained

is very similar to the one shown in Figure 4.11 for a two-dimensional case.