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166 Computational Fluid Dynamics and Heat Transfer

Velocity residual

−6

−5

−4

−2

−3

−1

Velocity residual

−6

−5

−4

−2

−3

−1

Iterations

AC-CBS

SI-CBS

AC-CBS

SI-CBS

(a) (b)

Figure 4.36. ConvergencehistoriesforSIandAC.(a)Ra=7×10

andDa=10

−4

;

(b) Ra=10

and Da=10

−3

Table 4.2. CPU time (s) for natural convection in a porous cavity heated from the

bottom at different Rayleigh and Darcy numbers

Ra 10

7×10

Da 10

−3

−4

−3

−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 direction

that must be evaluated at each time step.This term can be calculated only after the

resolution ofthe energy equation (step4 ofthe CBS algorithm),due to thef act that

the nodal temperature values are unknown.This coupled system becomes stiff and

causes a restriction on the global time step value used in the SI procedure and thus

thesolution tothesimultaneous equationsatthe secondstepofthealgorithm needs

more time.

Moreover, the SI scheme experiences difﬁculties in reaching the steady state

when very reﬁned meshes are used (Massarotti et al. [47]). On the other hand, the

AC-CBS scheme does not show any problem in reaching the convergence when

natural convection ﬂow occurs.Thereason forthis is therobust localtime stepping

procedure used by theAC-CBS scheme.

The last example is the developing mixed convection in a region partially ﬁlled

with a ﬂuid-saturated porous medium, conﬁned between two vertical hot walls.

In particular, because of the geometrical symmetry, half domain has been studied.

Figure4.37showsthecomputationaldomainandtheboundaryconditionsemployed

andthe detailsofthe computationalgridnear theentrance.Themeshemployedhas

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Applications of ﬁnite element method to heat convection problems 167

u = v =0

(a) (b)

T =1

v = 1, T =0

u = v =0

T =1

Figure 4.37. Mixed convection in a vertical channel partially ﬁlled with a

porousmedium. (a)Computationaldomain andboundaryconditions;

(b)detailofthestructuredcomputationalgridneartheentrance(8,591

nodes and 16,800 elements).

v/v

y =3

y =32

y = 0.5

Chang [48]

AC-CBS

1.5

0.5

Figure 4.38. Mixed convection in a vertical parallel plate channel partially ﬁlled

witha porousmedium. Non-dimensionalverticalvelocityat different

heights of the channel and at Da=10

−5

8,591 nodes and 16,800 elements and is reﬁned near the wall, near the inlet region

and at the interface.

Figure4.38showsthedimensionlessverticalvelocityproﬁleatdifferentheights

ofthechannel(y)foraDarcynumberequalto10

−5

,obtainedbyusingtheAC-CBS

scheme.The present results are compared with the numerical results of Chang and

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168 Computational Fluid Dynamics and Heat Transfer

Chang[48].The parametersused forthe presentinvestigation areRa=0.72×10

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 the

compositesystem(Figure4.38).Wheny increases,theﬂowdischargeintheporous

layer decreases, and the peak of the ﬂuid velocity proﬁle moves to the central axis

of the vertical parallel-plate channel.

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5 Equal-order segregated ﬁnite-element

method for ﬂuid ﬂow and heat transfer

simulation

Jianhui Xie

MFG-Engineering Design & Simulation, Autodesk, Inc.,

San Rafael, CA, USA

Abstract

This chapter introduces ﬁnite element–based equal-order mixed-GLS (Galerkin

LeastSquares) segregatedfor mulation forﬂuidﬂowandheat transfer.This scheme

follows the idea used in ﬁnite-volume method, which decouples the ﬂuid pressure

calculation andvelocity calculationby takingthe divergence ofthe vectormomen-

tum equation and applying some clear insights regarding incompressible ﬂow.The

inherent velocity correction step in “segregated scheme” makes it attractive for

solution robustness and much faster compared to original ﬂuid velocity–pressure

coupling scheme because mixed-GLS ﬁnite-element formulation does not assure

themassconservationbetweentheinterfaceoftheelementsincomparisontoﬁnite-

volume method. With these advantages, this formulation has prevailed in recent

years in major ﬁnite element–based commercial codes used in the industry sim-

ulation models that usually import geometry from CAD and have large scale of

degree of freedoms to solve. Taking example of ﬂuid–thermal coupling problem,

thischapteralsodiscusses thestrategiesusedtosolvemultiphysicsprobleminnon-

linear iteration level and the block I/O technique that uses ﬁnite-element method

for handling massive data. Finally, this chapter illustrates some typical industry

conjugate heat transfer problems where both solid and ﬂuid parts are involved.

Keywords: Finite elements, Fluid ﬂow, Heat transfer, Segregated formulation

5.1 Introduction

The quest for robust and efﬁcient strategies to solve large, sparse matrix sys-

tems has been an active area of research and development for decades. Galerkin

ﬁnite-element methods, with the advantages of handling irregular geometry and

multiphysicscoupling,haveplayedanimportantroleincomputationalﬂuiddynam-

ics area, but its computational difﬁculties and perceived shortcomings have led to

the development of alternative ﬁnite-element models.

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172 Computational Fluid Dynamics and Heat Transfer

Theoretically, the efforts have been to avoid the two major problems encoun-

tered. The ﬁrst problem is the “wiggles” velocity ﬁeld problem related to the

node-to-node oscillation of velocity components emanating from boundaries with

large velocity gradient. This problem is mostly pronounced under the condition of

high Reynolds numbers and coarse computational mesh, and therefore the inabil-

ity of the ﬁnite-element mesh to handle the steep gradient results in an imbalance

between advective and diffusive contribution. With getting best ﬁt to the prob-

lem, the weighted residual formulation produces a ﬁeld that oscillates about the

true solution (wiggle pattern). The second problem is the “saddle-point” prob-

lem originated by mixed Galerkin ﬁnite-element formulation. It is difﬁcult to

solve especially in three-dimensional large-scale industry model because the LBB

(Ladyzhenskaya–Babuska–Brezzi) stabilityconditionassociated withtheGalerkin

ﬁnite-element limits element choice with respect to the velocity and pressure

interpolation.

Inpracticalaspect,solutionschemes/solversandalgorithmshavebeenexplored

for betterCFD processor efﬁciency androbustness, which aremostly addressedby

commercial codes.

In this chapter, the equal-order mixed-GLS (Galerkin Least Squares) stabi-

lized formulation is presented ﬁrst. It is termed as a residual method because

the added least-squares terms are weighted residuals of the momentum equa-

tion, and this form of the least-squares terms implies the consistency of the

method since the momentum residual is employed. And this method is a gen-

eralization of SUPG (streamline-upwind/Petrov–Galerkin) and PSPG (pressure

stabilizing/Petrov–Galerkin) methods, which were developed to target the wig-

gle stabilization. Since this method plays a signiﬁcant stabilization role for the

coarse mesh and hybrid mesh, the equal-order velocity and pressure interpolation

is viable, and theLBB conditionisnot necessary any more, theconvenient bilinear

element interpolation becomes practical approximation although itwas unstable in

the Galerkin ﬁnite-element context.

The second emphasis in this chapter is on the practical view: The numer-

ical strategies for the solution of large systems of equations arising from the

ﬁnite-element discretization of the above formulations are discussed.To solve the

nonlinearﬂuidﬂowandheattransferproblem,particularemphasisisplacedonseg-

regatedscheme (which usesSIMPLEalgorithm from the ﬁnite-volume method) in

nonlinear level and iterative methods in linear level. The velocity–pressure cou-

plingformulationnot onlyresults ina large dimensionedsystem butalso generates

a stiffness matrix radically different from the narrowband type of matrix, which is

inefﬁcient to be solved. Segregated scheme was proposed with the idea of decou-

pling the pressure calculation and velocity calculation by taking the divergence

of the vector momentum equation and applying some clear insights regarding

incompressible ﬂow.The inherent velocity correction step in “segregated scheme”

makesitattractiveforsolutionrobustness andfastercomparedtovelocity–pressure

coupling scheme because mixed-GLS ﬁnite-element formulation does not assure

the mass conservation between the interface of the elements in comparison to

ﬁnite-volume method.

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Equal-order segregated ﬁnite-element method 173

Accordingtohowtheprimitivevariablesofvelocity–pressurearetreated,ﬁnite-

elementmethodsforsolving Navier–Stokesequations canbecategorizedintothree

groups: (a) the velocity–pressure integrated method; (b) the penalty method; and

method in which the governing equations are treated simultaneously needs rela-

tively small number of iterations to achieve convergence, but a large memory and

computingtime.Thepenaltymethodrequireslessmemoryandlesscomputingtime

thantheﬁrstmethod,butitneedsanadditionalpostprocessortoobtainpressureﬁeld

and satisﬁes the continuity equation only approximately. Recently, much attention

has been paid to the segregated velocity–pressure method, where velocity and the

corresponding pressure ﬁeld are computed alternatively in an iterative sequence.

This method takes advantage of SIMPLE algorithm idea from ﬁnite-volume world

and uses it in the ﬁnite-element world. With the SIMPLE algorithm idea, it needs

much less memory and execution time, and with the velocity correction step using

continuity equation, it even satisﬁes the continuity equation better than the ﬁrst

method.

The second categorization of ﬁnite-element methods can be made according

to the orders of interpolation functions for the velocity and pressure. They are

mixed-order interpolation and equal-order interpolation. The mixed-order scheme

attempts to eliminate the tendency to produce checkerboard pressure distributions

by satisfying LBB condition. In this method the velocity is interpolated linearly

whereas the pressure is assumed to be constant within the element. Equal-order

interpolation was proposed by Rice and Schnipke [6]. They showed that equal-

orderschemeperformedbetterforthatpurposewithoutexhibitingspuriouspressure

modes.

A third categorization of the ﬁnite-element method for ﬂuid dynamics is

related to weight functions. Galerkin method has been widely used in discretiz-

ing the momentum equation. Brook and Hughes demonstrated the accuracy of

the streamline-upwind/Petrov–Galerkin method for the linear advection-diffusion

equation; Rice and Schnipke [5] adopted a monotone streamline-upwind ﬁnite-

element method, where they discretized the momentum equation by using con-

vectional Galerkin method with the exception of the advection items, which were

treated by the monotone streamline-upwind approach.

Followed by the success of the ﬁnite-volume methods, several ﬁnite-element

segregatedsolution schemes havebeen proposed. RiceandSchnipke [6]employed

equal-order interpolation for all variables and solved pressure directly; however,

their original formulation used a streamline unwinding scheme that may not be

straightforward enough to extend to three-dimensional ﬂows. Early equal-order

methods werein general oftransient kindand the equationshad tobe integrated on

timetoreachasteadystate.VanZijl[11]adaptedtheformulationbyapplyingSUPG

weighting functions to the convective terms; he later adapted the scheme further

by implementing SIMPLEST algorithm. Shaw [7] demonstrated the way to use

element matrix construction for equal-order interpolation in segregated scheme.

Haroutunian et al. [2] have shown that the implementation of iterative solvers

can result in a substantial reduction in the storage requirements and execution

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174 Computational Fluid Dynamics and Heat Transfer

time. Du Toit [10] tried conjugate gradient solver; Wang [13,14] demonstrated a

“consistent equal-orderdiscretization method” by introducing element-basednode

velocity to satisfy mass conservation, while keeping conventional node velocity

and temperature to satisfy momentum and energy equations. Wansophark [15]

evaluated segregatedschemes bycombining monotone streamline-upwind method

and adaptive meshing technique. However, relaxation factors, which need user’s

interpretation, have remained an issue that affects the users.

The classical mixed velocity–pressure interpolation Galerkin ﬁnite-element

method, often referred to as “mixed v–p form,” was the workhorse of incom-

pressible ﬂow solvers in the 1970s to 1990s and has proved highly successful for

two-dimensional and small three-dimensional problems, offering high order of

accuracy and strong convergence rates [5]. The high mark of this approach is the

discretization of the continuity equation in a manner adherent to certain mathe-

matical constraints, and the ability to simultaneously solve the equation with the

discretized momentum equations in either a Picardora Newton–Raphson iterative

scheme.Thediscretized continuityequation is theknown causeof “indeﬁniteness”

inthe resultingJacobian matrix, leadingto unboundedlargeand negativeeigenval-

ues. Despite this poor matrix conditioning, most two-dimensional problems with

small bandwidth are amenable to efﬁcient solution with optimized variations of

classical Gaussian elimination. For example, skyline method allows for memory-

efﬁcientsolution and in fact is still used today. But even tremendous technological

advances in computer processor speed and random access memory capacity will

never make direct methods a viable alternative for large three-dimensional prob-

lems. This is one reason why many researchers have abandoned the traditional

mixed-interpolationapproachinfavorofalternativesthatallowthereadyuseofiter-

ative solvers.These alternative formulations, including penalty methods, pressure

stabilization methods, and pressure projection methods[2], bring“deﬁniteness” to

the matrix system.

Transientanalysisisatraditionalwaytoovercomepoorperformanceofiterative

solvers [9]; the idea is to take advantage of iterative solvers that thrive on a good

initialguess–whichtransientanalysiswithsmall-enoughtimeincrementdeliversat

eachtimestep.Furthermore,thesmallerthetimestepthemorediagonallydominant

the matrix system, because the time derivatives occur on or near the diagonals. It

is noticed that the transient analysis is inefﬁcient for a broad class of problems

that want steady solution; hence, the result from integrating in time is not really

practical as an end-all cure, instead transient term like inertial relaxation is more

effective in steady analysis.

In general, ﬁnite-element method based simulation has following basic steps:

Preprocessor phase: The preprocessor phase prepares all the required data for the

core processor computation; it includes CAD input, mesh generation, and geom-

etry decoder. After three-dimensional modeling from CAD packages, the model

geometry and part/assembly relations are established in parameter control.

The CAD input module reads and transfers the CAD data into the mesh ready

geometry data with the ordered hierarchy of assembly/part/surface relations.

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Equal-order segregated ﬁnite-element method 175

Mesh generation, which is responsible for generating valid ﬁnite-element

meshesandpossiblespecial meshes(e.g., boundar y mesh forﬂuid ﬂow), andmesh

reﬁnement.

Decoder does the job of producing ﬁnal working ﬁnite-element nodes, element

connectivity, node mergeamongparts, mesh clean up, contact, and setting uppart,

surface, node-based loadings with priority rules.

Processorphase:Processorphaseisthekernelpartoftheﬁnite-elementapplication;

it targets on solving real physical problems which are in discretized partial differ-

ential equation (PDE) forms and nonlinear formulations.The processor takes care

of several levels of iterations, which includes time iteration for transientproblems;

nonlinear iteration in a single time step handles the coupling among multiphysics

variables like velocity, pressure, temperature, turbulence, and multiphase; the

iterative solver for linear equations requires iteration for the converged solution.

Intheinnerlevel,elementmatricesarebuiltupviashapefunctionofthespeciﬁc

element type and system matrix is assembled from the element-level matrix and

then eventually global linearalgebraic equations aresolved to producethe primary

result.

The element-based and node-based results are the output from the processor;

typical variables are velocity, pressure, temperature, turbulence kinetic energy and

dissipation rate, phase fraction, etc.

The processor phase is usually the biggest user of system memory and CPU

time for ﬁnite-element-based simulation.

Postprocessor phase: Postprocessor also refers to result environment, which

provides interactive graphics environment to visually inter pret the node- and

element-based simulation results from processor phase. Some typical operation

includessliceplane,streamline,isolineandisosurface,makeanimation,annotation,

derived date calculation, inquire data, curve plot and report production.

5.2 Finite-Element Description

A ﬁnite-element simulation program should select particular elements to form the

base element library. Elements for ﬂuid ﬂow and heat transfer analyses are usually

categorized by the combination of velocity and pressure approximations used in

the element.

In ﬁnite element–based ﬂuid ﬂow and thermal problems, which is basically

advection–diffusionanalysis,followingelementsareusedfortwo-dimensionaland

three-dimensional models.

5.2.1 Two-dimensional elements

Quadrilateral element

The four-node quadrilateral element can be used to model either two-dimensional

Cartesian or axisymmetric geometries. Each node has four deg rees of freedom for