1

Reprinted from AIAA JOURNAL, Vol. 10, No. 12, December 1972. pp 1590-1599Copyright, 1972, by the American Institute of Aeronautics and Astronautics, and reprinted by permission of the copyright owner

Analysis of Flow of Viscous Fluids by the Finite-Element Method

J. TINSLEY ODEN* AND L. CARTER WELLFORD JR.tUniversity of Alabama, Huntsville, Ala.

General tinite-element models of compre<;sibleand incompressible fluid flow arc deri,·ed. These in>ol>e localapproximations of the 'elocit~· field, the density. and the tempemture for compressible fluids and the ,'e1ocity.temperature. and prl'S'i,re ror incompressihlc tluids. Thcuries or local solenoidal approximl.tions and mixed tinite-clement models ror coml,n'ssihle film lIn' lIerh·ed. A numhl'r or cumpulallonal s..ht'mes are lIe"elopt'll ror Iheuumerical solulion or hoth lransient and stl':IlI~'nOllllniformtlow prohlems illml\'illg illcol1lpressihlelIuids. Numericlllresulls ohtllinro rrom seH~rallest problel1lsllre gh'el1. It is Shllllil Ihat Ihe finile elel1lenlml,thod ha" grelll llolenlialfor U'ie ill tlo" prohll'I1I".and represenls II p,,"erful new 1001 for Ihe llllalysis of I'iwous flows.

..

Introduction

THE present paper is concerned with the application of theconcept of finite elements to the formulation and solution

of a wide range of problems in fluid dynamics. The method issufficiently general to treat a variety of unsteady and nonlinearflow phenomena in irregular domains. An intrinsic feature offinite-element approximations is that a mathematical model isgenerated by patching together a number of purely "Iocarapproximations of the phenomena under consideration. Thisaspect of the method effectively frees the analyst from traditionaldifficulties associated with irregular geometries, multi-connecteddomains, and mixed boundary conditions. Moreover. applicationsare firmly rooted in the physics of the problem at hand andpreliminary studies indicate that, for a given order of accuracy,the resulting equations are better conditioned than those obtainedby. say. finite difference approximations of the governing dif-ferential equations. I

Certain of the underlying ideas of the finite element methodwere discussed in 1943 by Courant.2 However. the formalpresentation of the method is generally attributed to the 1956paper of Tumer et a1.3 While the method has found wide applica-tion in solid and structural mechanics.4 its application to flowproblem~ has come only in rather recent times. Early uses of themethod were always associated with variational statements of theproblem under consideration. so that it is natural that steady.potential flow problems were the first to be solved using fi.1iteelements. We mention, in this regard. the works of Zienkiewicz,Mayer. and CheungS on seepage through porous media andMartin" on potential now problems. Finite element models ofunsteady compressible and incompressible flow problems wereobtained by Oden.7 -10 Applications of finite element methodsto a number of important problems in fluid mechanics have beenreported in recent years; among them, we mention the work ofThompson, Mack, and Linll on steady incompressible flow andTong.'2 Fujino,l.\ Argyris et a1..14- II> Reddi.17 BakeLls andHerting, Joseph, Kuusinen. and MacNeal19 on varioLl~ specialincompressible flow problems. The recent book of Zienkiewiczlcan be consulted for additional references.

In the present investigation. we extend the finite elementmethod to general three-dimensional problems of heat conduc-tion and flow of compressible and incompressible fluids. whereinno restriction is placed on the constitution or equation of state

Received December I. 1971; revision received May 24.1972. Supportof this work by the U.S. Air Force OffiL'e of Scientific Research underContract F44620·69-C·0124 is gratefully acknowledged.

Index categorics: Viscous Nonboundary-Layer Flows; BoundaryLayers and Convective Heat Transfer-Laminar.

• Professor and Chairman. Deparllllenl of Engineering Mechanics.Member AIAA.

t Graduate Studenl. and Engineer. Teledyne Brown Engineering.

of the fluid under consideration. Effectively. we develop finiteclement analogues of the equations of continuity. linear momen·tum, and energy of arbitrary fluids. The models are obtained fromlocal approximations of the density. velocity, and temperaturefields in each element and represent generalizations of thoseproposed earlier. 7.9

In addition. we treat the problem of fluids characterized byequations of state in which the thermodynamic pressure is notgiven explicitly as a function of the density. temperature. andvelocity gradients. There we develop mixed F1nite-element modelsby approximating locally the mean stress (or thermodynamicpressure) in each element We obtain a general model in whichthe equation of state is satisfied in an average sense over eachelemenL We then consider the special but important case ofviscoLl~ incompressible fluids, with emphasis on isotropic New-tonian fluids with constant viscosities. There we address ourselvesto certain problems connected with imposing the continuityequation (incompressibility condition) in the discrete model aooto special boundary conditions. A notion of solenoidal finiteelement fields is introduced We then describe computationalschemes for the solutions of the equations governing the modelfor uniform steady flow, nonunifonn steady flow. and unsteadyflow of viscous fluids. Numerical results obtained from applica-tions to a number of representative example problems arepresented.

Finite Element Models of Fluid Flow

To fix ideas, consider the motion of a continuous mediumthrough some closed region R of three-dimensional euclideanspace. We establish in R a fixed inertial frame of reference definedby orthonormal basis vectors ij (i = 1. 2. 3). The spatial coordi·nates of a place P in R arc denoted XI and the components ofvelocity of the medium at P at time I arc denoted l'i(X\"\:2"X'.\. t) ==I'il.\'. I). The density and the absolute temperature at place P attime I arc dcnot\:d II(X. t) and O(.\'. I~ respectively. If To denotes aunifonn temperature at some reference time 10' we may usc as analtemat!: temperature measure the temperature change T(x, t) =O(x. t) - ·To·

To construct a finite element model of the fluid. we replace Rby a domain R consisting of a finite number E of subdomains r,..so that R ~ R = v~= I r,.. The subregions /'.. are the tlniteclements; they are generally of simple geometric shapes and aredesigned so as to represent a good approximation of R whenappropriately connected together. The geometry of each finiteelement r.. is characterized by a finite number N, of nodal places(the number may vary from elementta clement) and the nodes ofa typical element e arc defined by the spatial coordinates xf,'I;N = 1. 2..... N,.: i = I.2. 3: e = I.2.. '" E. The global finiteelement model R is obtained by connecting the E discrete elementsat appropriate nodal points by means of simple incidence

DECEMBER 1972 FLOW OF TIlE VISCOUS fLUIDS BY TilE FINITE ELEMENT METHOD 1591

(13)

mappings which merely identify the desired correspondencebetween local and global nodal labels. These mappings aredescribed e1sewhere20 and need not be discussed here. Theimportant feature of the model is its local character; that is. thebehavior of the medium can be idealized locally in a typicalelement independent of the behavior in other elements in themodel and independent of its ultimate location in the model.The final global model is then obtained by routinely connectingelements together through mappings which depend only uponthe topology of the model.

Following guidelines provided by the notion of determinism.22

we shall take as our fundamental dependent variables, the velo-city, mass density. and temperature change liS obvious mellsuresof these primitive characteristics. Therefore, consider 1I typicalfinite element r.. with N, nodes isolated from the globalmodel R. Let \·(x. I~ pix, I~ and T(x. II denote global approxima-tions of the velocity. density. and temperature change to bedctermined at x E R at time c, and let vee" PI"" and 1(., denote theirrestrictions to element re' Then finite-element approximations ofthese restrictions are constructed which arc of the form

\'kl = t/I~'(x)vr.,(t) (fa)

PkJ = cp~'(x)PI~,(11 (lb)

1(•• = X~'(x)Ti~,(C) (lc)

Here and henceforth the repeated nodal indices are summed fromI to N<; \j~.(t). pj~)(I~ and Tf~J(C) are the values of the velocity,density, and relative temperature at node N of element r,. attime c; i.e.

\.~(t) == v(e.(X~I' r) (2)

etc. The functions !/1\:,I(X~cp~I(X), and I.~'(x) are local interpolationfunctions defined so as to have the properties

"'.!/1~I(X) = O. xf.r,.; !/1~'(XM) = t5~; L !/1~)(x) = I (3)

!.= I

Notice that Eq. (I) implies that different forms of the interpolationfunctions may be used to approximate different local fields overthe same finite element. In certain cases (some of which arc to bediscussed later~ this may require that certain of the functionsvanish at certain nodal points or that indices N in each memberof Eq. (I) may have different ranges. Note also that "higher order-local representations can be obtained by also specifying valuesof derivative; of \', p, and T at the nodes.

We must also remark that, in the case of incompressible fluids.our formulation requires that, instead of the density Pte,(x, II weapproximate the pressure field P(X.I) over R. Thus, if PI"'(X, c) isthe restriction of pIx. I) to r,.. we assume

Pie) = µ~'(x)p~~J(r) (4)

The interpolation functions 11~I(X) also obey Eq. (3).

Mechanics of a Finite Element

I\inematic,

With the local velocity field given by Eq. (Ia~ all relcvantkinematical quantities associated with the motion of the clemcntarc dctcrmined by the nodal velocities \,""(t). For examplc. thecomponents of local acceleration are

OJ = DL'JDI = ikJol + l'i....I·., = .psi)' + t/ls,".p.\I1f!·~ (51where Vi" = d/'fll)/dl. commas denotc partial dili'erentiationwith respect to thc spatial coordinates (i.e. .p." m == iJ.pN(xl/iJx",~the repeated indices are, again. summed ovcr'therr lIdmissibleranges (i.m = 1.2.3: /lJ.N = 1.2 .... ,N •.~ and the elcmentidentification label (e) has been dropped for simplicity. Likewise.the modcls of the rate-of-deformation tensor Jij and the spintcnsor Il'ij are given by

dij = !(!/1N,v7 + !/1Nf~1 Wi) = !t1/JSil'~ - IjJNf~) (61and the local vorticity field (!)i is Wi = Ciji.p,S·ll';'. where l:.jIi is thepermutation symbol. Local approximations of various otherkinematical quantities can be calculated in a similar manncr.

.\lflmenlUm EqUlltiolls for II Fillile Hemell!

The equations governing the motion of a typical finite elementcan be obtained by constructing a Galerkin integral of Cauchy'sfirst law of motion over the element and by using the velocityinterpolation functions .pN(X) as weight functions in this integral.If this approach is taken. linear momentum is balanced in anaverage sense over the element. The arbitrariness of the choiceof IjJN(X) as weight functions, however. is removed if an alternatebut equivalent approach based on energy balances is employed.9

The behavior of the medium must be consistent with theprinciple of conservation of energy:

D/DI(K + U) = Q + Q (71where K is the kinetic encrgy, U is the internal energy, Q is themechanical power, and Q is thc heat:

'" = !J pl'i'il' U = f pl;(lv

Q = f pFh.d1' + L Sl'iA Q = f pJul1' +t ql1ilA 18)

In Eq. (8) /: is the internal energy density. FJ are the componentsof the body force vector per unit mass, Sj are components of the:iurface tractions. h is the heat per unit mass supplied frominternal sources, ([j arc the components of heat flux, and IIj is thenormal to the boundary surface, Noting that the material deriva-live of the kinetic energy for elcment e is

DK/Dr = f prpil''.

and introducing Eq. (51and (1al. we obtain

(D/Dt)K = [C"NIPMi'f + d;:}.\·LPpM,,~l'fJz{ (9)

where

e,\l\I. = 1..CP.\t.pN!/1Ldl'. dZ;,\'LP = 1.CPMIjJ".pI.IjJP....dl' (10)

and M. N. L. P = 1. ...• N,. The first quantity in parenthesisrepresents the local inertial force at node N in the kth direction.and the second term in the parenthesis represents the convectiveincrtial force at node N in the kth direction.

Since the local form of the energy balance is

p(D/:/Dt) = pt. + pr/.f.,i = 7;kVk,i + qi .• + ph (II)

wherein 7;. is the Cauchy stress tensor. the material derivativeof the internal energy for the elcment can be written as

DU..IDI = r~ f 7;l!/1N,idl' + Q,. (12)'.

where Q, = f.. (qi.l + ph)dv is the heat of the element. For anelcment of Iluid of volume v, and surface area A •. the powcr ofexternal forces is

!l = [L. 7;illi.p,ydA + C RSkpR ] I{

where

CRNk = J. CPRFk!/1NdF

The first and second qualllities in the parenthesis represent theforce at node N in the kth direction duc to the surface stressdistribution and the force at node N in the kth direction due tothe body force. respectively, If we define thc generalized forcePSi according to

P.'<k = L. Si.p"dA + CRNkpR 114)

thcn the generalized forces develop the same amount of mechan-ical power as the external forces in the continuum element. and

fl" = p,virfFinally. introducing Eqs. (n(12). and n,. into (7~ we obtain an

encrgy equation for a typical finite element e. Then. making the

1592 J. T. ODEN AND L. C. WELLFORD JR. AIAA JOURNAL

Energy Equalion for an Elemenl

The Conlinuily Equaliun fur an Element

argument that this result must hold for arbitrary values of thenodal velocity I{. we obtain a~ lhe general equations of motion(momentum) for a compressible fluid elcment:

C,\II'ILpMi'f + d';"",.pp'\/l'~;l'r + J..7;kr/J"',idv - P,"k = 0 (IS)

The constitutive equation for stress in terms of the approximatcvelocity and pressure cxpressions must be introduced to completethis equation.

Constituth'e Equations, Equations of State,and Mixed Models

F(n. dij. p. n = 0 (26)

Assuming that we cannot (or do not choose tol eliminate nfrom Eq. (25) by use of Eq. (26) we propose a mixed finite-elementmodel in which the restriction ne(x. I) of n to element,." is assumedto be of the form

Equations (15),(18). and (22) [or Eq.(23)] describe the generalequations of motion. continuity. and energy (or heat conduction)of a typical element in a finite-element model of an arbitraryfluid. To apply these equations to a specific fluid. it is nece~sary toeliminate c. 1;j. ilk. and possibly '1 by introducing appropriateconstitutive equations which uniquely define these functions interms of "",. Pie, and 7;",. This is the customary procedure infinite-element formulations.

In the case of compressible fluids. however. the mean stress orthe thermodynamic pressure often appears implicitly in theequation of state of the tluid. and it may be impossible or im-practical to obtain 7;j explicitly as a function of p. ('if and T.In ~uch cases. we propose that a "mixed" finite-elcment formula-tion be used. the basis of which is now to be described.

Consider a c1as.~of fluids described by constitutive equationsfor the stress tensor of the form

T,j = Jijn(p. d". n+ i;hJ. dn' T) 125lHere n is the so called ther~odynamic pressure and i;j is thedissipative stress. Generally 7;j is given explicitly as a functionof p. dn• and T (e.g .. for a class of Stokesian fluids, i;j = 2Jldij.JI being the viscosity): however. n is defmed implicitly by anequation of state:

(19)

The local form of the continuity equation is

P + (P('dk = 0 (16)

whcrc p == Cp/CI. Introducing Eqs. (tal and (I b) into (16) yieldsat point x the residual

r.(x) = If) ....(x)P'''' + (If)Ii(X)r/JR (X)).kpN/{ (17)

We can guarantee that thc rcsidual vanishes in an average senseover the element by requiring that it be orthogonal (with respectto the inner product <f. g) = J,..fgdv) to the subspace spannedby the functions !P.\I(x).Then < r., If).\/) = .f '. r.lf),\ldl' = O. andwc obtain the finite element model of the continuity equation:

1l,IINpN + b~I.'IRPSI'~ = 0 (18)

Ilere M. N. R = I. 2..... N. and all'" and b~/NR denote the localarrays

By introducing Eq. (Ia) into the local fonn of thc first law.Eq. (t I~ wc obtain thc residual

r.. = If)RpRt; + If)RpRr/JNI'fr.,i - 7;kr/JI'I,;l-t' - qk,k - If)RpRIl (20)

I-Iere it is understood that f'~ 7;k' and 'lk are functions of the localfields dcfined in Eqs, (I). As before. we rcquire that the residualbe orthogonal to thc subspacc spanned by the I.M(X) functions:

< f." 1..\1) = J..f.l.,l/dl' = 0 (21)

Introduction of Eq. (20) into Eq. (21) gives the general finitec1emcnt analogue of the energy equation:

J.. If)RCX,\ldl'pR + J. If)Rr/JN£/XMdl'pRl'i'' = J. 7;kr/JN,il.\Idv/~·

- J..lM'kilkdl' + L. l..IIqkllkdA + J..If)RII1..lldl'pR (22)

In order to obtain a useable exprcssion for the energy equationconstitutive equations must be introduced for stres.~ 1;k' internalenergy density r., heat tlux qk' and heat produced by internalsources II.

We remark that an alternate equation of heat conduction canbe obtained for the clement by rewriting Eq. (II) in terms of theentropy density 'I(x. t) and the internal dissipation a(x. t) =pO'i - qu - ph. For these choices of variables, a proceduresimilar to that used to obtain Eq. (15) yields the general equationof heat conduction for an element.

pM J,. If).\I1.N(To + l.p'fP) e~+ r/JRV~'I.".) dv + J,. qiXN,;<fV

= qN + aN (23)

Here q....and aN are the generalized normal heat flux and thegeneralized internal dissipation at node N

qN = f. h1.Ndv + L.qillibdA a,,' = 1.. al.Ndl' (24)

Again, the procedure used to derive Eq. (23) is equivalent toGalerkin's method and specific forms can be obtained whenconstitutive equations for 'I and qi are furnished.

n" = {J",(x)n·'·UI 127)The interpolation functions {("~Ix) have properties similar toEq. (3). Introducing Eq. (II and (27) into (26). we obtain a residualr,."'. Then. the condition <,.:.{I",) = 0 leads to the N" localequations

1.(J",F({J",n"'. r/JX,i/'~' If),,,p'''. I.", TN) dl' = 0 (28)

Equation (281 represents the /inite-element analogue of theequation of state, Eq. (26). and insure that it is satisfied in aweighted average sense over each finite element. IntroductionofEq. (27) into (25) and incorporation of the result into Eqs. (I5~(18~ and (22) [or Eq. (23)] (along with constitutive equations forelk. e. and" as functions of p. T. and I'i,j) yields a complete systemof equations in the nodal values of velocity. density. temperature,and thermodynamic pres~ure.

Incompressible Fluids

We shall now consider purely mechanical behavior of incom-pressible fluids. Here two principal considerations are involved:I) all motions are volume-preserving and 2) the stress tensor isnot completely determined by the motion. The first conditionreveals that p is now a known constant and the continuity equa-tion reduces to the incompressibility condition

divv = dii = ('i,i = 0 (29)

The second consideration suggests that rr = - p. P being thehydrodynamic pressure. and that the local pressure of Eq. (4)should be selected as an unknown in place of p.

Considering now p to be known and following essentially thesame procedure used to obtain Eq. (15), we obtain for the generalequation of motion of incompressible elements

ml/sl': + n:~I",pl':.('r + hMI,kpl'I + J. i;kr/J,\1.idI1 = PMk (30)

where III.1(N and nZ;NP are the mass and convected mass "matrices-respectively. h\l,\'k is an array of pressure coefficients. and P.llk

are the components of the generalized force defined by Eq. (I4l

DECEMBER 1972 FLOW OF VISCOUS FLUIDS BY THE FINITE ELEMENT \IETHOD 1593

(32)

Solenoidal Approximalions

LOin.LOin.

Fig. 1 Finite-element model for the calculation of COlleUe fluw.

.Eo

so

The quantity fiNk is the generalized nodal force due to prescribedpressures p on the boundary.

Similar procedures can be used for various other specialboundary conditions. For example. it is not uncommon to use asa condition at the impermeable boundary with zero velocity thespecification of the pressure gradient as a function of the bodyforce and the acceleration (i.e.. p" = pF" - pa" where the sub-script II refers to the normal direction to the wall). This boundarycondition can either be applied by retaining the momentumequation at the nodes in contact with the fixed wall or by applyinga discrete version at the nodes in contact with the wall. The firstmethod. which is followed herein. statisfies the boundary con-dition in an average sense ovcr the boundary elements. Thesccond method satisfies thc boundary condition cxactly at thewall nodcs.

The first method is obtained directly from the discrete momen-tum equation. Eq. (33), by setting l'~ = 0 (for boundary nodcs N)and t;k = 0, sincc t:k depends only on l'~ there. Then

"'A/NV~ + IJMNkpN = fllk - f Pllkl/1MdAA.

where fMk = s.. pFkl/l.lfdl' is the generalized forcc at node M ducto the body forces Fk' Transforming the surfacc intcgral via theGreen-Gauss theorem and collecting terms, we arrive at thediscrete analogue of the pressure-gradient boundary condition,

y.IfNkpN = J\(k - 1Il,\l,,;i:: (36)

wherein Y.\(Nk = J,.,.l/1.1(11N.k d\'.In the second method, a discrete version of this type of boun-

dary condition can be obtained for element e by merely intm-ducing Eq. (4) into the local statement of the boundary conditionand evaluating the result at the coordinates of each of the wallnodes. While this leads to a cruder approximation, it is neverthe-less much easicr to apply in actual calculations.

The pressure tcrm h.IINkP''' can be eliminated from the discretemomentum equation Eq. (33) by constructing solenoidal finite-element approximations of the local velocity. We accomplish thisby introducing so called "bubble functions~ !Xi(x) which vanishon the boundaries of each element and which satisfy (at leastapproximately) the condition

IXi; = -l/1...,;lx)vf (37)Any particular integral of Eq. (37) vanishing on oR,. is of the form

IXi(X) = IX/N(xlvj (38)

The incompressibility condition. Eq. (29) is now satisfied locallyby local velocity approximations of the form

r/(x. r) = i/i/i(X)!·j: i/i/ = (jNN(X) + !X/N(x) (39)

The remaining terms in Eq. (33) are altered accordingly.The use of local solenoidal vclocity fields makes it possible to

eliminate the hydrodynamic pressure term from the discrete

n~ ...p = J, PI/IMI/I,\,I/IP' ",til'

hMNk = - J.,LNI/1M.k dl' (31)

The derivation of the incompressibility constrainl can best becast in physical terms: Observe that thc work done by the pressureP due to a change in volume dl' is clearly P..V' vdl' = Ped~!dl'.But p" can perform no work since for incompressible fiow V . I'iszero. It follows that for a finite element of an incompressible fluid

J,. P..v}~/dl' = J. ILMI/IN,i dVP'\(ll( = 0

This relation must hold for arbitrary pM_; thus thc finitc elementmodel for the incompressibility condition is.

ri'NI'/'i = 0: dIN = J /l.III/IN.i dv".

We note that the energy equation. Eq. (22~ is also modifiedslightly for incompressible elements due to the fact that CPNpN isnow replaced by its presumedly known value and Trk dependson pN. These alterations arc straightforward and need not bedisplayed here.

or

The special case of an incompressible Newtonian flow withconstant viscosity is of special interest because a number ofclassical solutions to various flow problems are available forcomparisons. Of course, since the viscosity is constant. the energyequation is uncoupled from the momentum and continuityequations and they may be solved separately. For the Newtonianfluid the dissipalive stress tensor is simply Trk = 2l1dik,IL being theviscosily. Thus, for the finite element, t:k = IL(I/IN kl'iN + I/IN i!l/land Eq, (30) reduces to .,

lilA/Nil;''' + 1l.\;NPll~ !-k + hMNkpN + Z.lfNl'f + II'.:,'!. l·f = PMk(33)

Boundary Conditions

where

Z.IIN = t. JLI/IM,il/lN.i dv and II'~\~N = J. /ll/l.II.il/lN,k dv (34)

To these equations we add the incompressibility conditions,Eqs. (32), to form a determinate set.

Incomnrl'ssible Newtonian Fluids

If either the total stress or the velocity is specified at the boun-dary of a finite element modc~ no particular difficulties areencountered: the prescribed stress is introduced directly intoEq. (14) to obtain consistent generalized nodal forces, and thenodal velocities are prescribed to satisfy the "nonslip" boundarycondition at a fixed or moving wall or the specified velocitydistribution on other surfaces, In this respect, the procedurediffers very little from that employed in the finite element analysisof solids. However. in fluids, specification of the stress at aboundary may not uniquely determine the pressure; moreover.the boundary conditions may represent constraints on thepressure or its gradient rather than the total stress. In thesesituations, it may be necessary to develop special analogues ofthe boundary conditions.

Consider. for example, the generalized force of Eq. (14) in thecase of an incompressible fluid, Ignoring the body force temltemporarily. we observc that

PNk = f 1Ikllil/1NdA = f (- pbik + Trk) lIil/l",dAA. A,

AIAA JOURNAL

Fig. 2 Velocity profile at I:uioustillll' poillts ror transient Couette

flow.

- EXACT SCLUTtON

<>----<> FINITE ELEMENTSOLUTION

J. T. ODEN AND L. C. WELLFORD JR.1594

VElOCITY (in/sec.)0 .05 ./0I ,I I

.20

,18c:UJ .16

5 .14a..

~ .12l-I-0 .1!IIUJ:L .08I-:l!0 .06a:IJ..UJ .04~~ .02(f)

0

t =.00010 sec. t= .00000l1l1C. t=.OOI67sec. t=.00376sec. t =.ClO669sec. t=- sec.

momentum equations, Eqs. (33~ Thus the momentum equationscan be solved directly for the local velocity fields. Then thepressure calculation can be performed independently based onthe computed velocity field.

The pressure must satisfy Poisson's equation.

P.ii = 1;Ui(X. c~) + (lFi.i - plIi,; (40)

where 'l;/{x. I'~) is the dissipative part of thc stress tensor. Thiscan be verified by taking thc divcrgcnce of thc local momentumequation.22 Thc solution of this cquation by mcans of finiteeleme\lts is very well documentcd."

of this type. With minor changes, these techniques could beapplied to three-dimensional incompressible flow problems or.in genera~ to any isothermal incomprcssible viscous fluid.

Consider the solution of the system of equations consisting ofthe momentum equation. Eq, (33). and the continuity equation,Eq. (32). Three cast..'Selm be analyzed based on the properties ofthe now; stcady unifoml flow: stcady nonuniform flow: andtransient or unsteady now. Thc models dcvcloped earlier lcadto systems of linear algehraic equations in the first case, non-linear algcbraic equations in the sccond. and nonlinear dif·fercntial cquations in thc third,

Numerical Techniques

In order to obtain a prcliminary estimate of the applicabilityof the forcgoing theory, wc considcr a series of classical problcmsin two-dimensional incompressiblc Newtonian flow. Severalnumerical techniques are developed herein to handlc problems

Stendy, Uniform Lillear Flow

For the stcady fiow problem, thc term "'M""vZ and the con-vective tcrm IIM""l'll~Vr in Eq. (33) vanish and the linearizedmomentum cquation takes the form

hMN1P'''1 + ZM""vZ + 1I'~~....v;"1 = PMl (41)

.08

NCOE9

NODES

07

.06

~.05c,;~.E;:04I-

§~ 03

.02

.01

.002 .003 .004 .005 .006

TIME (SEC).007 .008 .009 .010 .012

Fig. 3 Time hislory of the x velocitycomponl'lIts at nodes 7, 8, and 9 for

transient Couelle 1101'1.

01

DECEMBER 1972 FLOW OF TIlE VISCOUS FLUIDS BY THE FINITE ELEMENT METHOD J595

-I

-10

""'---.---- -----------

,-

.012.011.010.009.008.005 .006 .007TIME (SEC)

.004.003.002.001

f\/ \

/ \~NODE 7

I '" "\ I\ I~I

o

14

12

10

8

-;' 6

Q)( 4.;'" 2"CC- O

~ -2

g..J -4W> -6

-8

fig. 4 Time history of y velocit)' compollfnt at nodes 7 and 9 for transiellt Couelle flow,

where v is the veclOr of unknown velocity components andpressures and p is the density parameter. Let v be a solutionveClor correspondil~g to a particular value of density p. and letI' + 151' be a solution vector corresponding to the value of density

rZ;",1' = f..l/I,I,l/INl/Il'.m d\'. Thus, wc obtain the system of norrlinear equations

w;;,wvz'd: + Ir,\fHlP'" + z,'I,..vf + \I,,~~,..l!~= PMk (43)

A natural choicc of a lechnique for solving Eqs. (43) is Iheincremental loading melhod used in nonlinear structuralmechanics.20 This conclusion is based on Ihe observation Ihal ifdensily p is assumed to be Ihe loading parameter. when lhedcnsity equals zero thc sct of equalions reduces to Ihc stationarylinear systcm. Eq. (42), which can be solved using methodsdcscribcc\ above.

This solution proccss can be described concisely in vectornotation. Suppose Ihat the collection of momentum equationsand continuity equations are expressed in vector form as

Equations (41) and (32) represent, for the two-dimensional case.a set of 3N. algebraic equations in 3N. unknowns. Taking noleof the fact Ihal r~'N = -Ir,y,III: we can exprcss these equationsin matrix form as

[Z.H,.. + W,~iH 1I'1,~ 1r,IINIJ [[I'~J] [[PMIJ]II',~,~ Z.II,.. ;- W,~,~ IrMN2 [1'~J = [Pm] (42)

-rkM -r~M 0 [pH] [0]The coefficielll matrix is square and symmctric. In add ilion,

upon asscmbling thc elcments, the global form is sparse andbandcd. Boundary conditions must be applied in accordance withthe previous discussion and the resulting set of equations can besolved for the velocity and pressure variables using standardGauss-elimination codes.

Steady, Nonuniform Flow-The Method of Incremfntal Oellsities

The momcntum equation for stcady nonlinear flow can beoblained from Eq. (33) by setting the local inertial term m~fNi·/Jequal to zero and using the fact that Il;;NP = pr;;NP' where

f(I'. p) = 0 (44)

y

I

fig. 5 finite - element model ofsteady lIow of a lubricant through

a plane slider bearing.

~ .3000 19

i.23331:

Eo.......'"uc:2'"o

o 06667

Slider Beoring

1.333 2000."" -x(inches)

Moving Surfoce •

'.

1596

.3

~.2

<!>z;;;o:e':I!....~.III-

wU

~o

J. T. ODEN AND L C. WELLFORD JR.

Cl----<) FINITE ELEMENTSOLUTION

0.05 :, ~.05 :, .6666.05VELOCTTY(in/MC.X 10~

DISTANCE FROM LEADING EDGEOF THE BEARING (In)

AIAA JOURNAL

Fig. 6 x "elocity component for the lubricatiunproblem.

p + bp. Then f(\' + b\·. P + bp) also vanishes. A Taylor's seriesexpansion can be introduced in the neighborhood of \' if flv. p)is continuously differentiable at \'. Thus

flv + by. p + bp) = f(\·. p) + 8.)\' + Cbp + higher order terms

where B is an N x N matrix and C is an N x I vector: B;i\·. pI =vh(v, p)/VVj: Cj(I" p) = Cf(I', p)/('P i, j = I. ... , N. Thus. withinterm~ of first order. we have the linear system &5v = - Cbp.Given the initial value of the solution vector Vo and the initialvalue of the density Po = O. the scries expansion forms the basisof .1Il iterative scheme for which the 11 + \Til iterate is given by

11+ 1

v"+ I = v. - B - 1(\,•• p")e(\',,. p"),5p,,. I: P.,. I = L bpi (45)i-I

This procedure amounts to a piecewise linearization and thenumber of iterations should be determined by accuracy rcquire-ments.

Unstelldy Flow

We shall present results in the next section in which the solutionof the system of nonlinear differential equations. Eqs. (32) and(33~ were integrated numerically using a self-correcting 4th orderRunge-Kutta technique. To outline the essential features, con-sidcr first Eq. (33) rewritten in the form

v~ = //lI..'J - IF.llk (46)

where

F"k = -IlM ....pV~V~ - hMJ'iJlN - :AfJ'iuZ - \I',~~....V~ + P"H (471

and //lUI - I is thc inverse of the mass matrix IIIUI' The firstderivative of the prcssure variable can be formulated in similarfashion. We differentiate Eq, (46) with respect to time to obtain

i'~ = -//lL.II-1h.lf ....J1'" + //lUI -IH,If4

where

ll.lfk = -IlMJ'iPI!~ - 1l.Z;NPV~I{ - :,\INV: - \I',~~NI'i\'+ PmBy d:fTerentiating Eq. (32) twice we observe that r~Li:t = 0:consequently

XVNP"" = Yu: xu~' = r~,.//l,_~lh\,,'·k: ,I'v = rt'L//l/~I/J.\fl (48)

If hlV is the inverse of xu .....we obtain the explicit expression for thederivalive of the prcssure variable

p" = h"u}'u (49)

Equations (46) and (49) are now of the form x = F(x. (~ whichcan be directly intcgrated by standard Runge-KUlla schemes.

Some Numerical Results

We shall now cite representative numerical results obtainooby applying the theory and methods presented earlier to specificproblcms in two-dimensional flow of incomprcssible Newtonianfluids. For demonstration purposes, we shall employ six-nodetriangular elements of the type shown in Fig. I. for which the localvelocity and pressure fields are given by the quadratic polyno-mials

()--Q FINITE ELEMENT SOLUTION

w....j.CLIIIZ

~~:Io~.1Wvz;!VIQ

.3333VELOCITY (h/ .. c.• 10' I

.6666

2 3

1.0000

2 3 2 3

1.6666

2 3

2.0

Fig. 7 y velocit~' cumponent forthe lubrication prohlem.

DISTANCE FROM LEADING EDGE OF BEARING (in)

DECEMBER 1972 FLOW OF VISCOUS FLUIDS BY THE FINITE ELEMENT METHOD 1597

.004

Fig.8 Variation of IlTl'Ssore on tbe fixed slider bearing wall.

component at nodes 7, 8, and 9 arc included. As can be secn, inthe initial starting period negative tangential velocity com-ponents occur at several nodes. This was apparently due to eitherthe applied stress boundary condition or lhe coarseness of themodel used. Because of this constraint small transverse velocitycomponents were computed. Thcy were symmetric with respectto the lines connecting nodes 11 and 15 and 3 and 23. The timchistories of the transverse velocity component at nodes 7 and 9are given in Fig. 4.

r

2.01.0

DISTANCE FROM LEADING EDGE OF BEARING (IN,)

o

Ol;

"u:IIIdo .00l&l0::;lCIlCIll&l

g: .001

.003

0--0 FINITE ELEMENTSOlUTION

Couelte Flow

['k = (a ....+ b....iXi + CNilXiXl)V/

p = (aN + bNiXi + CNij(IXi)pN (50)

Here the six independent constants aN' bNi• CNii (CNil = CNj;:;,j. k = 1,2.3: N = 1.2, .... 6) depend only on the local coor-dinates of the six nodes of the element. These local approximationsdetermine all of the relevant arrays and matrices in the localmomentum, energy, and continuity equations described earlierfor each element.

Concerning convergence and accuracy of this particularapproximation, we mention that Ziamal24 has obtained the errorestimate

II' - vi ~ (Kfsin (l)h2 (51 )where v is a given continuously differentiable field. v is a finite-element interpolant (i.e~ v coincides with I' at the nodal points~h is the maximum diameter of all finite elements in a given mesh,o is the smallest angle between any two sides of a triangle. andK is a constant independent of () and h. Clearly. long nat elementslead to poorer conditioned systems than networks of isoscelestriangles. In the case of elliptic and parabolic problcms, estimatesof the type in Eq. (51) lead directly to convergence and errorestimates also involving Ifsin 0 and, for energy convergencc. Kh4•

While the study of thc stability and convergence of finite-elementapproximations of hyperbolic problems is scarcely beginning.preliminary rcsults seem to indicate that the local character ofthe approximation lead to inherently bctter conditioned systemsthan conventional difference schemcs of equal accuracy.

The problem of unsteady COllette .flow through the domainindicated in Fig. I is considered. The following boundaryconditions were applied: I) The x vclocity component wasassumed to be cqualto 0.1 in.fsec. and lhe)' velocity componenlwas assumed 10 be equal to zero at y = 0.2 in, 2) The x and }'velocity components were prescribed as zero at )' = O. 3) Thestress on the boundaries x = a and x = 2.0 in. was set equal tozero. 4) The gradient of the pressure in the direction perpendicularto the wall was zero at )' = O. As initial conditions. we set thepressure and the velocity vector equal to zero at all interior nodesall = O.The value of the mass density used was 0.00242 Ibf.-sec.2fin,4 and the viscosity here and in all subscquent results wasassumed to be 0.00362 Ibf.-sec.fin.2

In Fig. 2 the tangential velocity profilc at x = 0.5 in. is presentedat various times and shows good agreement with the exactsolution.21 In Fig. 3, time histories of the tangential velocity

An Incompressible Lubrication Problem

Thc two-dimensional flow of lubricant bctwcen a slide blockor a slider bcaring and a moving surface was determined. A fifteenelemcnt model, as illustrated in Fig. 5 was constructed. Thcrollowing boundary conditions were applied: I) The x-vclocitycomponent was 0.01 in.fsec and the y velocity component waszero along y = O. 2) The x and y velocity components werecquated to zero along the slidcr bearing wall. 3) The stress at allunconstrained boundary nodes was prcscribed as a hydrostaticpressure of 0.00 I Ibf.fin.2 4) Thc gradient of the pressure pcrpen-dicular to the bearing wall was set equal to zero.

In the initial calculations, the convective inertial tcrms wereignored. The finite element solution for thc velocity profile in thex direction is prcscnted in Fig. 6. The transverse velocity profileis prescnted in Fig. 7. and the pressure along the inclined bearing

NOT TO SCALE

DISTANCE FROM LEADING EDGE lin.>1.0 2.0

I III

A

! i~..J .3-Q.

~ .2-0:...~ .1-Z

~o

R=O

.002o

~R=240

I

9\\

.3

o-.002

.!'i~w~..Jll.

~l-SIn

:::E0e: .1wuzt!CIl

5

Fil:. 9 x velocity component atsection A as computed by Ihe in-

cremental densities technique.

X VELOCITY COMPONENT AT SECTION A Cin./llec.)

1598

10

9

Ii

.... ~:S~~UJ>

8

~\\\\\

"- '--

J. T. ODEN AND L. C. WELLFORD JR.

DSTANCES FRCt.4LEAONG EDGE (inlN::lTIN SCALE

YI ~ "'88~ ~ _, ~C?OO 0 \'" -

NODE AL_--------NODEB

AIAA JOURNAL

Fig. 10 Finite-element model of flow O\'er aflat plate aDd time histories of the x velocit)'

components at nodes A and B.

7.1 .2 .3 .4

TIME (SEC,).5 .6 .7 .8

is indicated in Fig. 8. The transverse velocity profile changesdirection between x = 0.3333 and x = 0.6666 in. resulting in ahack flow in the bearing. The nonlinear convective terms were thenincluded, and the bearing problem was solved by the incrementaldcnsities mcthod described earlier. The resulting velocity profilein the x direction at a Reynolds number of 240 is included inFig. 9, For Reynolds numbers larger than 240 more significantchanges were obscrved in the velocity profile: these rcsults arenow being evaluated with a more dctailed model.

U()undar~-I.ayer Flow

The problcm of transient boundary-laycr formation ovcr al1at platc was allltlyzed using the finite elemcnt grid of Fig. Ill.This model is primarily useful for study of the velocity profileaway from the leading edge of the platc. Thus rcsults arc presentedfor spatial points at least I in. from the leading edgc where thcmodel converges to the exact solution. In ordcr to predict nowpatterns at the leading edge. where high velocity gradients areencountercd.a more detailed model of the now would be required.

The rollowing boundary conditions were specified: II The xvelocity component was set cqual to 10. in.jsec. and the r velocitycomponent was equated to zero at x = -0.05 in. 2) The x and J'velocity components were zero at J' = O. 3) The gradient.of thepressurc in the direction normal to the wall was zero at y = O.4) The pressure was zero at all boundary nodes at which thevelocity was not specified.

The initial conditions specified th:ll the x velocity components

were 10. in.fsec. and the .v velocity components were zero at allunconstrained nodcs at t = O. The pressure was also zero at allnodes at time zero. A mass dcnsity of 1. Ibf.-sec.2/in.4 was assumed.

Computed time histories of the x velocity components atnodes A and B arc presented as representative variations inFig. 10. In Fig. 11. the steady state finite-element solution iscompared to the Blasius boundary-layer solution at specificpoints on the plate. Again, excellent agreement is obtained witha rather coarse mesh.

References

I Fix. (j, and Larsen. K .. "On lhe Convergence of SOR Iteralionsfor Finite Element Approximalions 10 Elliptic Iloulldary ValueProblems:' SIAM Journal of NUlllerical Alw~l'sis. Vol. 8. No.3. 1970.pp. 536-547.

2 Courant. R .. "Variational Methods for the Solulion of Problemsof Equilibrium and Vibrations." Bullelin. American MalhematicalSociely. Vol. 49, Jan. 1943. pp. 1-23.

J Turner. M. J .• Clough. R. W .. Martin. H. C .. and Topp. L. P .."Sliffness and Deflection Analysis of Complex Slructures." Journal oJthe Aeronautical ScienCt's. Vol. 23. No.9. 1956. pp. 805-823.

4 Zienkiewicz. O. C .. The Finile Element ,'!,fetlrod in EngineeringScience. McGraw-HilI. London. 1972.

S Zienkiewicz, O. C .. Mayer. P., and Cheung. Y. K .. "Solulion ofAnisotropic Seepage Problems by Finite Elements." Proceedings oj IheAmcrican Society of Cillil Engineer.\'. EM 1, Vol. 92. 1966, pp. 111-120.

6 Martin. H. c.. "Finile Elemellt Allalysis of Fluid Flows. '" AFFDL-

1.0- BLASIUS SOLUTION

~ FINITE ELEMENTSOLUTION

Fig. I t Comparison of finite-clemenl andRlasiu....solutions for the stcady-state bound-ary layer on the front portion of the plale.

2 3 4 5 6 7 8 9 10 II

DISTANCE FROM THE LEADING EDGE OF THE PLATE (in)

DECEMBER 1972 FLOW OF VISCOUS FLUIDS BY THE FINITE ELEMENT METHOD 1599

TR-68-150, Wright-Pallerson Air For~'CBase. Ohio, 1969.pp. 517-535.7 Oden, J. T.. "A General Theory of Finite Elements; " Applica-

tions." Infernational Journal for Numerical Methodv in Engineering.Vol. I. No.3, 1969. pp. 247-259.

80den, J. T. and Somogyi, D .. "Finite-Element Applications inFluid Dynamics." Journal of the Engineering Mechanics Division.ASCE. Vol. 95, No. EM4, 1969. pp. 821-826.

• Oden. J. T., "Finite Element Formulation of Problems of FiniteDeformation and Irreversible Thermodynamics of Nonlinear Con-tinua," Recent Advances in Matrix Methods of Structural Analysis andDesign, Universily of Alabama Press, Tuscaloosa, Ala.. 1970.

10 Oden, J. T., "A Finite-Element Analogue of the Navier StokesEquations," Journal of the Engineering Mechanics Division, ASCE,Vol. 96. No. EM4, 1970, pp. 529-534.

II Thompson, E. G .. Mack, L. T., and Lin. F. S., "Finite-ElementMethod for Incompressible Slow Viscous Flow with a Free Surface."Developments in Mechanics, Proceedings of the 11th MidwesternMecllanics Conference, Iowa State Univcrsity Prcss. Ames Iowa.Vol. 5, 1969. pp. 93-111.

12 Tong, P.• "The Finite Element Method in fluid Flow Analysis."Recenf Advances in Matrix Methods of Structural Analysis and Design.edited by R. II. Gallagher. Y. Yamada, and J. T. Chien. Universily ofAlabama Press. Tuscaloosa. Ala .• 1971. pp. 787-808.

13 Fujino. T., "Analysis of Hydrodynamic and Plale StructureProblems by the Finite Element Method," Recenf Adrances in MatrixMethods of Structural Analysis and Design, Ediled by R. H. Gallagher.Y. Yamada, and 1. T. Oden, University or Alabama Press, Tuscaloosa,Ala., 1971. pp. 725-786.

14 Argyris, 1. H .• "Two and Three-Dimensional Potential Flow by

the Method of Singularities," The Aeronautical Journal of the RoyalAeronautical Society, Vol. 73. Nov. 1969, pp. 959-961.

1 S Argyris. J. H.. Scharpf, D. W.. "The Incompressible LubricationProblem." The Aeronautical Journal of the Royal Aeronautical Society.Vol. 73. Dec. 1969. pp. 1044-1046.

'6 Argyris. J. H.. Mareczek. G .• and Scharpf, D. W.. "Two andThree-Dimensional flow Using finite Elements," The Aeronau/icalJournal of /111'Royal AerO/Jau/ical Sode/}'. Vol. 73, Nov. 1969.'pp.961··964.

17 Reddi, M. M.• "Finile-Element Solution of the IncompressibleLubrication Problem." Transactions of the ASM E, Journal of Lubrica-tion. Vol. 91,1uly 1969. pp. 524--533.

'8 Baker, A. J .• "A Numerical Solution Technique for Two-Dimen-sional Problems in Fluid Dynamics Formulated with the Use ofDiscrete Elements," TN-TCTNIOO5, Hell Aerosystems Co .. NiagraFalls. N.Y., 1970.

,. Herting. D. N.. 10seph. J. A., Kuusinen. L. R.• and MacNeal.R. H., "Acoustic Analysis of Solid Rocket Motor Cavities by a FinitcElement Method," NASTRAN: User's Experience. Vol. I, NASATMX-2378, 1971.

20 Oden, 1. T., Finile Elemen/sof Nonlinear Continua. McGraw-Hili.New York, 1971.

21 Schlichting, H .. Boundary-Layer Theory, McGraw-Hili. NewYork, 1968.

22 Eringen, A. c.. Mechanics ofCon/inua, Wiley, New York. 1967.23 Ladyzhenskaya, O. A., The Mathematical Theory of Viscous

tncompressible Flo....., Gordon and Breach. New York. 1969.24 Zlamal, M,. "On the Finite Element Method," Numerische

Mathematik. Vol. 12, 1968. pp. 394--409.

,-,.