Table Of ContentHydrothermal Analysis
in Engineering
Using Control
Volume Finite
Element Method
Hydrothermal Analysis
in Engineering
Using Control
Volume Finite
Element Method
Mohsen Sheikholeslami Kandelousi
Department of Mechanical Engineering,
Babol University of Technology
Davood Domairry Ganji
Department of Mechanical Engineering,
Babol University of Technology
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Nomenclature
A amplitude
Cp specificheatataconstantpressure
DB Browniandiffusioncoefficient
DT thermophoreticdiffusioncoefficient
Grf Grashofnumber (cid:3) pffiffiffiffiffiffiffiffiffiffiffi(cid:4)
Ha Hartmannnumber HBx sf=mf
Le Lewisnumber(a/D )
B
N numberofundulations (cid:3) (cid:4)
Nb Brownianmotionparameter ðrcÞ D ðf (cid:2)f Þ=ðrcÞa
(cid:3) p B h c(cid:5) f (cid:6)(cid:4)
Nt thermophoreticparameter ðrcÞ D ðT (cid:2)T Þ= ðrcÞaT
p T h c f c
Nu Nusseltnumber(cid:3)(cid:3) (cid:4) (cid:5) (cid:6)(cid:4)
Nr Buoyancyratio r (cid:2)r ðf (cid:2)f Þ= ð1(cid:2)f Þr bLðT (cid:2)T Þ
p 0 h c c f0 h c
Pr Prandtlnumber(u /a)
f f
J electriccurrent
T fluidtemperature
u,v velocitycomponentsinthex-directionandy-direction
U,V dimensionlessvelocitycomponentsintheX-directionandY-direction
x,y spacecoordinates
X,Y dimensionlessspacecoordinates
r nondimensionalradialdistance
k thermalconductivity
L gapbetweeninnerandouterboundaryoftheenclosureL¼r (cid:2)r
! out in
g gravitationalaccelerationvector
Ra Rayleighnumber
q00 heatflux
Rd radiationparameter
GREEK SYMBOLS
g anglemeasuredfromrightplane
z inclinationangle
g angleofturnofthesemiannulusenclosure
« eccentricity
a thermaldiffusivity
s electricalconductivity
f volumefraction
m dynamicviscosity
v vorticity
V dimensionlessvorticity
y kinematicviscosity
c streamfunction
vii
viii Nomenclature
C dimensionlessstreamfunction
Q dimensionlesstemperature
r fluiddensity
b thermalexpansioncoefficient
m0 magneticpermeabilityofvacuum
SUBSCRIPTS
c cold
h hot
loc local
ave average
nf nanofluid
f basefluid
p solidparticles
in inner
out outer
Preface
In this book we provide readers with the fundamentals of the control volume finite
elementmethod(CVFEM)forheatandfluidflowproblems.TheCVFEMcomprises
interestingcharacteristicsofboththefinitevolumeandfiniteelementmethods.Itcom-
binestheflexibilityofthefiniteelementmethodstodiscretizecomplexgeometrywith
theconservativeformulationofthefinitevolumemethods,inwhichvariablescanbe
easilyinterpretedphysicallyintermsoffluxes,forces,andsources.Mostotheravail-
abletextsconcentrateonsolidsproblems.Weappliedthismethodforflowandheat
transfer.ApplicationofCVFEMindifferentinterestingfieldssuchasnanofluidflow
andheattransfer,magnetohydrodynamics,ferrohydrodynamics,andporousmediais
considered.Severalexampleshavebeenpreparedinthesefieldsofscience.Thistext
is suitable for senior undergraduate students, postgraduate students, engineers, and
scientists.
ThefirstchapterofthebookdealswiththeessentialfundamentalsoftheCVFEM.
Thenecessaryingredientsinnumericalsolutionsarediscussed.Chapter2dealswith
solvingNavier-Stokesequations(invorticitystreamfunctionform)andenergyequa-
tions.Inthis chaptertwo basic importantproblemsare solvedviathe CVFEM.The
third chapter gives a complete account of the nanofluid hydrothermal behavior and
applicationofCVFEMtosolvesuchproblems.Alltherelevantdifferentialequations
arederivedfromfirstprinciples.Allthreetypesofconvectionmodes—forced,mixed,
andnaturalconvection—arediscussedindetail.Examplesandcomparisonsarepro-
vided to support the accuracy and flexibility of the CVEFM. Examples start with a
single-phasemodelandthenextendtoatwo-phasemodel.
TheapplicationoftheCVEFMtoheatandfluidflowinthepresenceofamagnetic
fieldarediscussedindetailinChapter4.Twokindsofmagneticfieldsareconsidered:
a constant magnetic field and a spatially variable magnetic field. Several examples
includedinChapter4givethereaderafullaccountofthetheoryandpracticeassoci-
ated with the CVFEM. Chapter 5 gives the procedures for solving equations of
flowandheattransferinporousmedia.Someexamplesprovidereaderswithanoppor-
tunitytolearn about the effect ofactiveparameters. A sample FORTRAN code for
advection-diffusioninlid-drivencavitygeometryispresentedinAppendix.Readers
willbeabletoextendthiscodeandsolvealloftheexampleswithinthisbook.
Mohsen Sheikholeslami Kandelousi
Davood Domairry Ganji
ix
CHAPTER
1
Control volume finite
element method (CVFEM)
1.1 INTRODUCTION
Fluidflowhasseveralapplicationsinengineeringandnature.Mathematically,real
flows are governed by a set of nonlinear partial differential equations in complex
geometry.So,suitablesolutionscanbeobtainedthroughnumericaltechniquessuch
as the finite difference method, the finite volume method (FVM), and the finite
element method (FEM). In the past decade the FEM has been developed for use
intheareaofcomputationalfluiddynamics;thismethodhasnowbecomeapowerful
method to simulate complex geometry. However, the FVM is applied most in cal-
culatingfluidflows.Thecontrolvolumefiniteelementmethod(CVFEM)combines
interestingcharacteristicsfromboththeFVMandFEM.TheCVFEMwaspresented
byBaligaandPatankar[1,2]usinglineartriangularfiniteelementsandbyRawand
Schneider[3]usinglinearquadrilateralelements.Severalauthorshaveimprovedthe
CVFEMfromthentonow.Rawetal.[4]appliedanine-noddedelementtosolveheat
conductionproblems.Banaszek[5]comparedtheGalerkinandCVFEMmethodsin
diffusionproblemsusingsix-noddedandnine-noddedelements.CamposSilvaetal.
[6]developedacomputationalprogramusingnine-noddedfiniteelementsbasedona
controlvolumeformulationtosimulatetwo-dimensionaltransient,incompressible,
viscous fluid flows. Campos Silva and Moura [7] and Campos Silva [8] presented
results for fluid flow problems. The CVFEM combines the flexibility of FEMs to
discretize complex geometry with the conservative formulation of the FVMs, in
which the variables can be easily interpreted physically in terms of fluxes, forces,
and sources. Saabas and Baliga [9,10] referenced a list of several works related to
FVMs and CVFEMs. Voller [11] presented the application of CVFEM for fluids
and solids. Sheikholeslami et al. [12] studied the problem of natural convection
betweenacircularenclosureandasinusoidalcylinder.Theyconcludedthatstream-
lines, isotherms, and the number, size, and formation of cells inside the enclosure
strongly depend on the Rayleigh number, values of amplitude, and the number of
undulations of the enclosure.
1.2 DISCRETIZATION: GRID, MESH, AND CLOUD
In general there are three ways toplace node pointsinto a domain [11].
1
HydrothermalAnalysisinEngineeringUsingControlVolumeFiniteElementMethod
©2015ElsevierLtd.Allrightsreserved.
2 CHAPTER 1 Control volume finite element method
1.2.1 GRID
A basic approach assigns the location of nodes using a structured grid where, in a
two-dimensional domain, the location of a node is uniquely specified by a row
and a column index (Fig. 1.1a). Although such a structured approach can lead to
convenient and efficient discrete equations, it lacks flexibility in accommodating
complex geometries or allowing for the local concentration of nodes in solution
regionsof particularinterest.
1.2.2 MESH
Geometricflexibility,usuallyattheexpenseofsolutionefficiency,canbeaddedby
using an unstructured mesh. Figure 1.1b shows an unstructured mesh of triangular
elements. In two-dimensional domains triangular meshes are good selections
because they can tessellate any planar surface. Note, however, that other choices
ofelementscanbeusedinplaceoforinadditiontotriangularelements.Themesh
canbeusedtodeterminetheplacementofthenodes.Acommonchoiceistoplacethe
nodes at the vertices of the elements. In the case of triangles this allows for the
row1
row2
Grid
row3
row4
(a) col1 col2 col3 col4
Mesh
(b)
Cloud
(c)
FIGURE1.1
Differentformsofdiscretization[11],includingagrid(a),mesh(b),andcloud(c).
1.3 Element and interpolation shape functions 3
approximation of a dependent variable over the element, by linear interpolation
betweenthevertexnodes.Higher-orderapproximationscanbearrivedbyusingmore
nodes(e.g.,placingnodesatmidpoints)oralternativeelements(e.g.,quadrilaterals).
When consideringan unstructuredmeshrecognizing the followingis important:
1. The quality ofthe numerical solutionobtained iscritically dependent onthe
mesh.Forexample,avoidinghighlyacuteanglesisakeyqualityrequirementfor
ameshoftriangularelements.Thegenerationofappropriatemeshesforagiven
domainisacomplextopicworthyofamonographinitsownright.Fortunately,
for two-dimensional problems inparticular,there isasignificant range of
commercial andfreesoftware that can be used to generate qualitymeshes.
2. Thetermunstructuredisusedtoindicatealackofaglobalstructurethatrelates
the position of allthe nodes inthe domain.Inpractice, however, a local
structure—theregionofsupport—listingthenodesconnectedtoagivennodeiis
required. Establishing, storing,and using this local data structure isone ofthe
criticalingredientsin using an unstructuredmesh.
1.2.3 CLOUD
Themostflexiblediscretizationistosimplypopulatethedomainwithnodepoints
that have no formal background grid or mesh connecting the nodes. Solution
approaches based on this “meshless” form of discretization create local and struc-
tures,usuallybasedona“cloud”ofneighboringnodesthatfallwithinagivenlength
scale ofa given node i[13](Fig.1.1c).
1.3 ELEMENT AND INTERPOLATION SHAPE FUNCTIONS
Abuildingblockofdiscretizationisthetriangularelement(Fig.1.2).Forlineartri-
angular elements the node points are placed at the vertices. In Fig. 1.2, the nodes,
movinginacounterclockwisedirection,arelabeled1,2,and3.Valuesofthedepen-
dent variablef are calculatedand stored at these nodepoints.
Inthisway,valuesatanarbitrarypoint(x,y)withintheelementcanbeapprox-
imatedwith linear interpolation
f(cid:1)ax+by+c; (1.1)
wherethe constant coefficients a,b, and csatisfythe nodal relationships
fi¼axi+byi+c, i¼1,2,3: (1.2)
Equation (1.1) can be more conveniently written in terms of the shape function
N , N ,and N , where
1 2 3
(
1 Atnodei
Niðx,yÞ¼ (1.3)
0 Atallpointsonsideoppositenodei
4 CHAPTER 1 Control volume finite element method
3
A123
1
2
3
3
3
Ap23 Ap31
p 1 p p
1
1
Ap12
2 2
2
FIGURE1.2
Anelementindicatingtheareasusedinshapefunctiondefinitions[11].
X3
Niðx,yÞ¼1 Ateverypointintheelement (1.4)
i¼1
such that, over the element the continuous unknown field can be expressed as the
linear combinationof the values at nodes i¼1,2,3:
X3
fðx,yÞ(cid:1) Niðx,yÞfi: (1.5)
i¼1
Withlineartriangularelementsastraightforwardgeometricderivationfortheshape
functions can be obtained. With reference to Fig. 1.2, observe that the area of the
elementisgiven by
(cid:2) (cid:2)
A123¼12(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)111 xxx11 yyy11(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)¼12½ðx2y3(cid:3)x3y2Þ(cid:3)x1ðy3(cid:3)y2Þ+y1ðx3(cid:3)x2Þ(cid:4) (1.6)
1 1
andtheareaofthesubelementswithverticesatpoints(p,2,3), (p,3,1),and(p,1,2),
wherep isan arbitrary and variable point inthe element, are given by
(cid:3) (cid:4)
Ap23¼ ðx2y3(cid:3)x3y2Þ(cid:3)xpðy3(cid:3)y2Þ+ypðx3(cid:3)x2Þ
(cid:3) (cid:4)
Ap31¼ ðx3y1(cid:3)x1y3Þ(cid:3)xpðy1(cid:3)y3Þ+ypðx1(cid:3)x3Þ : (1.7)
(cid:3) (cid:4)
Ap12¼ ðx1y2(cid:3)x2y1Þ(cid:3)xpðy2(cid:3)y1Þ+ypðx2(cid:3)x1Þ
Basedon these definitions,it follows that the shape functionsare given by
N ¼AP23=A123, N ¼AP31=A123, N ¼AP12=A123: (1.8)
1 2 3
Notethat,whenpointpcoincideswithnodei (1,2,or3),theshapefunctionNi¼1,
and when point p is anywhere on the element side opposite node i, the associated
subelement area is zero, and, through Eq. (1.8), the shape function Ni¼0. Hence
