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Introduction and t h e equationso f fluid dynamics

1.1 General remarks and classification of f lu id mechanicsproblems discussed in this bookTh e problems of solid an d fluid behav iour are in m any respects similar. In bo th m ediastresses occur and in both the material is displaced. There is however one majordifference. The fluids cannot support any deviatoric stresses when the fluid is atrest. Then only a pressure or a mean compressive stress can be carried. A s weknow, in solids, other stresses ca n exist an d the solid material can generally s up po rtstruc tura l forces.In ad ditio n t o pressure, deviatoric stresses can how ever develop w hen the fluid is inmotion and such motion of the fluid will always be of primary interest in fluiddynamics. We shall therefore concentrate on problems in which displacement iscontinuously changing and in which velocity is the main characteristic of the flow.The deviatoric stresses which can now occur will be characterized by a quantitywhich has great resemblance to shear modulus and which is known as dynamicviscosity.U p t o this point the e quations governing fluid f low an d solid mechanics appe ar tobe similar with the velocity vector u replacing the displacement for which previouslywe have used the sam e symb ol. However, there is one fu rther difference, i.e. that evenwhen the flow has a constant velocity (steady state), convective ucceleration occurs.This convective acceleration provides terms which make the fluid mechanicsequations non-self-adjoint. N o w therefore in most cases unless the velocities arevery small, so that the convective acceleration is negligible, the treatment has to besomewhat different from that of solid mechanics. The reader will remember thatfor self-adjoint forms, the ap proxim ating equation s derived by the Ga lerkin processgive the minimum erro r in the energy n orm an d thu s are in a sense optimal. This is nolonger true in general in fluid mechanics, though for slow flows (creeping flows) thesituation is som ew hat similar .W ith a fluid w hich is in mot ion con tinua l preservation of mass is always necessaryand unless the fluid is highly compressible we require that the divergence of thevelocity vector be zero. We have dealt with similar problems in the context ofelasticity in Volume 1 and have shown that such an incompressibil i ty constraint

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2 Introdu ctio n and the equations of fluid dynamicsintroduce s very serious difficulties in the form ula tio n (C ha pte r 12, Volum e 1). In fluidmechanics the same difficulty again arises and all fluid mechanics approximationshave to be such that even if compressibility occurs the limit of incompressibilitycan be modelled. This precludes the use of many elements which are otherwiseacceptable.In this book we shall introduce the reader to a finite element treatment of theequations of motion for various problems of fluid mechanics. Much of the activityin fluid mechanics has however pursued a j inite difference formulat ion and morerecently a derivative of this known as the jinite volume technique. Competitionbetween the newcomer of finite elements and established techniques of finite differ-ences have appeared on the surface and led to a much slower adoption of the finiteelement process in fluid mechanics th an in structures. T he reasons f or this are perh apssimple. In solid mechanics or structural problems, the treatment of continua arisesonly on special occasions. The engineer often dealing with structures composed ofbar-like elements does not need to solve continuum problems. Thus his interest hasfocused on such continua only in more recent times. In fluid mechanics, practicallyall situations of flow require a two or three dimensional treatment and hereapproximation was frequently required. This accounts for the early use of finitedifferences in the 195 s before the finite element process was made available. How-ever, as we have pointed out in Volume l , there are many advantages of using thefinite element process. This not only allows a fully unstructured and arbitrarydomain subdivision to be used but also provides an approximation which in self-adjoint problems is always superior to or at least equal to that provided by finitedifferences.A methodology which appears to have gained an intermediate position is that offinite volumes, which were initially derived as a subclass of finite difference m ethods.W e have shown in Volume 1 tha t these a re simply anot her kind of finite element formin which sub dom ain collocation is used. W e do n ot see much adv anta ge in using tha tform of approximation. However, there is one point which seems to appeal to manyinvestigators. That is the fact that with the finite volume approximation the localconservation conditions are satisfied within one element. This does not carry overto the full finite element analysis where generally satisfaction of all conservation

conditions is achieved only in an assembly region of a few elements. This is nodisadvantage if the general approximation is superior.In the reminder of this book we shall be discussing various classes of problems,each of which h as a certain behaviour in the numerical solution. H ere we start withincompressible flows or flows where the only change of volume is elastic andassociated with transient changes of pressure (Chap ter 4). Fo r such flows full incom-pressible con strain ts have to be applied.Fu rth er, with very slow speeds, convective acc eleration effects are often negligibleand the solution can be reached using identical programs to those derived forelasticity. This indeed was the first venture of finite element developers into thefield of fluid mechanics thus transferring the direct knowledge from structures tofluids. In particu lar the so-called linear S tokes flow is the case where fully incompres-sible but elastic behaviour occurs and a particular varia nt of Stokes flow is that usedin metal forming w here the material c an n o longer be described by a cons tan t viscositybut possesses a viscosity which is non-newtonian and depends on the strain rates.

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General remarks and classification of f lui d m echanics problems discussed in this book 3Here the fluid (flow formulation) can be applied directly to problems such as theforming of metals or plastics and we shall discuss that extreme of the situation atthe end of Cha pter 4. Ho we ver, even in incompressible flows when the speed increasesconvective terms become im po rta nt. Here often steady-state solutions d o not exist orat least are extremely unstable. Th is leads us to such problems as eddy shedding whichis also discussed in this chapter.T he subject of turbulence itself is enorm ous, an d much research is devoted to i t . W eshall touch on i t very superficially in Chapter 5: suffice to say tha t in problems whereturbulence occurs, it is possible to use various models which result in a flow-dep end ent viscosity. The s am e ch ap ter also deals with incompressible flow in whichfree-surface and other gravity controlled effects occur. In particular we show themodifications necessary to the general formulation to achieve the solution of prob-lems such as the surface perturbation occurring near ships, submarines, etc.

Th e next area of fluid mechanics to which m uch practical interest is devoted is ofcourse that of flow of gases for which the compressibility effects are much larger.He re compressibility is problem -depen dent an d obeys the gas laws which relate thepressure to temperature and density. It is now necessary to add the energyconservation equation to the system governing the motion so tha t the temperaturecan be eva luate d. Such an energy equa tion c an of course be written fo r incompressibleflows but this shows only a weak or no coupling with the dynamics of the flow.Th is is not the case in com pressible flows where coupling between all equa tion s isvery stron g. In compressible flows the flow speed may exceed the speed of so und an dthis m ay lead to shock developm ent. This subject is of majo r imp ortan ce in the field ofaerodynamics and we shall devote a substantial part of Chapter 6 just to thisparticular problem.In a real fluid, viscosity is always prese nt but a t high sp eeds such viscous effects ar econfined to a narro w zone in the vicinity of solid bo und aries (houndury luyt.~).n suchcases, the rem aind er of the fluid can be considered to be inviscid. The re we can retur nto th e fiction of so-called ideal flow in which viscosity is no t present an d here va rioussimplifications are again possible.O ne such simplification is the int rodu ction of potential flow an d we shall mentionthis in Ch apte r 4. In Volume 1 we have already dealt with such potential flows und ersome circumstances and showed that they present very little difficulty. But unfortu-nately such solutions are n ot easily extendable to realistic problems.A third m ajor field of fluid mechanics of interest to us is tha t o f shallow water flowswhich occur in coastal waters or elsewhere in which the depth dimension of flow isvery much less than the horizontal ones. Chapter 7 will deal with such problems inwhich essentially the distribu tion of pressure in the vertical direction is almost hydro-static.In shallow-water problems a free surface also occurs and this dominates the flowcharacteristics.Whenever a free surface occurs i t is possible for transient phenomena to happen,generating waves such as for instance those that occur in oceans and other bodiesof water. We have introduced in this book a chapter (Chapter 8) dealing with thisparticular aspect of fluid mechanics. Such wave phenomena are also typical ofsome other physical problems. We have already referred to the problem ofacoustic waves in the context of the first volume of this book and here we show

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4 Introduction and the equations of fluid dynamicsthat the treatment is extremely similar to that of surface water waves. Other wavessuch as electromagnetic waves again come into this category and perhaps thetreatment suggested in Chapter 8 of this volume will be effective in helping thoseareas in turn .

In what remains of this chapter we shall introduce the general equations of fluiddynamics valid for most compressible or incompressible flows showing how theparticular simplification occurs in each category of problem mentioned above.However, before proceeding with the recommended discretization procedures,which we present in Chapter 3, we must introduce the treatment of problems inwhich convection and diffusion occur simultaneously. This we shall do in Chapter2 with the typical convection-diffusion equ ation . Ch ap ter 3 will intro duc e a generalalgo rithm capa ble of solving most of the fluid mechanics problem s en countered in thisbook . As we have already m entioned, there are m any possible algorithms; very oftenspecialized ones are used in different areas of applications. However the generalalgo rithm of Ch apte r 3 produces results which are a t least as good a s others achievedby mo re specialized m eans. W e feel th at th is will give a certain unification to the wholetext an d thu s without ap ology we shall omit reference to m any o ther m etho ds or dis-cuss them only in passing.

1.2 The governing equations of fluid dynamics-81.2.1 Stresses in fluidsThe essential characteristic of a fluid is its inability to sustain shear stresses when atrest. Here only h ydrostatic stress o r pressure is possible. An y analysis mu st thereforecon cen trate on the m otion , and the essential independent variable is thus the velocityu or , if we ad op t the indicia1 nota tion (with the x , y , z axes referred to as x,, i = 1,2 ,3) ,

u l , i = 1 ,2 ,3 ( 1 .1 )This replaces the displacement variable which was of primary importance in solidmechanics.

Th e rates of strain ar e thus the primary cause of the general stresses, olJ , nd theseare defined in a manner analogous to that of infinitesimal strain as(1.2)au,pxJ+ au,px,2=

Th is is a well-know n tensorial definition of strain ra tes bu t fo r use later in variationalfor m s is written as a vector w hich is m ore co nvenien t in finite element analysis. De tailsof such matrix form s are given fully in V olume 1 but for completeness we mentionthem here. Thus, this strain rate is written as a vector 6). This vector is given bythe following form

ET = [E l l ,E22 ,2E121 = [i . l l ,E22, 21 (1.3)

iT= [ i , l , ~ 2 * , ~ 1 3 , 2 E l 2 , 2 E 2 ~ , 2 ~ ~ l l (1.4)in two dimensions with a similar form in three dimensions:

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The governing equations of f luid dynamics 5When such vector forms are used we can write the strain rates in the form

= s u (1.5)where S is kno w n as the stain op era tor and is the velocity given in Eq. ( I . 1).definition of two co nstants .Th e stress-strain relations for a linear (new tonia n) isotro pic fluid require theTh e first of these links the deviatoric stresses rlI o the deviatoric strain rutes:

In the above equa tion the qua nti ty in brackets is kno w n as the deviatoric strain , 6 sthe Kronecker del ta , and a repeated index mea ns summ ation; thusand

Th e coefficient p is kno w n a s the dy nam ic (shear) viscosity or simply viscosity an d isanalogous to the shear modulus G in linear elasticity.The second relation is that between the mean stress changes and the volumetricstrain rates. This defines the pressure as

or /= oI 22 033 i , ,= C l l + z z + i 33 (1.7)

where K is a volumetric viscosity coefficient analog ous to the bulk modulus K in linearelasticity a n d p o is the initial hydro static pressure independent of the strain rate (n otethat p and pa are invariably defined as positive when compressive).W e can immed iately write the constitutive relation for fluids from Eqs (1.6) and(1.8) as

11 6ijP ( 1 . s a )o r

p J = 2pC /. - 6 . . KJ p )e.. + 6. ./PO (1.9b)Tradit ion ally the Lam e notatio n is often used, putt ing

K - f p L f X (1 .10)but this has little to recommend it and the relation (1.9a) is basic. There is littleevidence ab ou t the existence of volumetric viscosity and we shall take

Ki..I 0 ( 1 .11 )in w hat follows, giving the essential constitutive relation as (no w dro pp ing the suffixo n P o)

( I . 12a)without necessarily implying incompressibility i t /= 0 .

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6 Introd uct ion and the equations of f luid dynamicsIn the above,

(1.12b)

All of the above relationships are analogous to those of elasticity, as we shall noteagain later for incompressible flow. We have also mentioned this in Chapter 12 ofVolume 1 where various stabilization procedures are considered for incompressibleproblems.Non-linearity of some fluid flows is observed with a coefficient p depending onstrain rates. We shall term such flows non-ne wto nian .

au- d u . 2 duTi/ = 2p .j j , = p [ ( G + )6

~ ~ ~~ . ~ > ~ ~ ~ -- -I .- I_~. II XIXI __I_x. - x^i__ _._1.2.2 Mass conservationIf p is the fluid density then the balance of mass flow pu; entering and leaving aninfinitesimal con trol volume (Fig. 1 .1 ) is equal to the rate of change in density

( 1.13a)ap T3 - pi) 5 v ( p u ) = 03 p u ) + p.) + p w )= 0

d t 8.u; d tor in trad ition al Cartesian coord inate s

( 1 I 3b)d ddt a x aY a z

Fig. 1.1 Coordinate direction and the infinitesimal control volume1.2.3 Mom entu m conservat ion or dynamic equi l ibr iumI X X I I Y X I ~ -I- _L - - _x_x ~ m ---- -------- - ~ ~ _ - -~---No w the balance of mom entum in the lth direction, this is p u l ) u , eaving an d enteringa control volume, has to be in equilibrium with the stresses 0, and body forces pf/

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The governing equations of fluid dynamics 7giving a typical com pon ent eq uat ion

o r using (1.12a) ,(1.14)

(1.15a)with (1.12b ) implied.Cartesian form:Once again the above can, of course, be wri t ten as three sets of equat ions in

ar,, dr , , aryz ap3 2 8 3at u i?? a- 3.u 3y 32 ax- - p f , = op . ) + - p u ) + ~ p u v ) + ~ p z l w )

( 1 . 1 )etc.

_ _.2.4 Energy co nservation and equation of statex I _ x x I _ _ ^ X X I I I I - I - C _ I I ; _ ~ x - ~ ICWe note tha t in the equa t ions of Secs 1.2.2 an d 1.2.3 the indepen dent variables are 1.1(th e velocity), p ( the pressure) and p (the densi ty). Th e deviato ric stresses, of co urse,were defined by Eq. (1.12b) in term s of velocit ies an d hence are n ot ind epen den t .

Obviously, there is one variable too m an y fo r this equa tion system t o be capable ofsolut ion . Howev er, if the den si ty is assumed con stan t (as in incom pressible fluids) or ifa single relat ion ship l inking pressure a nd densi ty can be establ ished ( as in isotherm alflow with small com pressibi l ity) the system becom es com plete a n d is solvable.More general ly , the pressure (y), densi ty p ) and abso lu te t empera ture T ) arerelated by an eq uat ion of s ta te of the form

P = P b > -1 (1.16)Fo r an ideal gas this takes, for instance, the form

Pp = - RT (1.17)where R is the universal gas con stant .In such a general case, it is necessary t o su pplem ent the govern ing equ ation systemby the equ at ion of energy conservation. Th is equ ati on is indeed of interest even if i t isnot coupled, as i t provides addi tional informat ion abo ut the behav iour of the system.Before proceeding wi th the derivat ion of the energy conservat ion equat ion we mustdefine som e further quan ti t ies. Th us we introduce e , the intrinsic energ]' per unit mass.This i s dependent o n the s tate of the f luid, i .e . i ts pressure a nd temperature o r

e = e ( T , p ) (1.18)Th e total energy per uni t mass, E , includes of course the kinetic energy per unit ma ssand thus(1.19)

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8 Introd uct ion and the equations of flu id dynamicsFinally, we can define the enthulpy as

(1.20)and these variables are found to be convenient.ally being confined to boundaries). T he conductive h eat flux qi is defined asEnergy transfer ca n take place by convection and by conduction (rad iation gener-

dd X jq ; = - k - T (1.21)

where k is an isotropic thermal conductivity.T o complete the relationship it is necessary to determine heat source terms. Thesecan be specified per unit volume as qH due to chemical reaction (if any) and mustinclude the energy dissipation due to internal stresses, i.e. using Eq. (1.12),

(1.22)The balance of energy in a unit volume can now thus be written as

d d da pE)+ pujE) p i ) T ~ U , ) pf,u, H = 0 (1.23a)at axj d X ior mo re simply

(1.23b)d+ p u , H ) 7 , / U , ) pLu, H = 0at ax,Here, the penultimate term represents the work done by body forces.

1.2.5 Navier-Stokes and Euler equationsThe governing equations derived in the preceding sections can be written in thegeneral conservative form

(1.24a)d*-+ VF V G Q = 0ato rd 9 dF; dG;- + - + - + Q = Oat dx ; axi (1.24b)

in which Eqs (1.13), (1.15) or (1.23) provide the particular entries to the vectors.notation,Th us, the vector of independent unk now ns is, using b oth indicia1 an d Cartesian

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The governing equations of f luid dynamics 9

Gi = or71

-72;-73

dT- ~ j j ~ ~ )k-ax

(1.25b)

( 1 . 2 5 ~ )

(1 .25d)with

Th e com plete set of (1.24) is know n as the Naviev-Stokes equation. A particularcase when viscosity is assumed to be zero and no heat conduction exists is knownas the Euler equation T ~ ,= k = 0).Th e abov e equations ar e the basis fro m w hich all fluid mechanics studies start a ndi t is not surprising that many alternative forms are given in the literature obtainedby combinat ions of the various equations.* The above set is, however, convenientand physically meaningful, defining the conservation of important quantities. I tshould be noted th at only eq ua tio ns written in conservation for m will yield thecorre ct, physically m eaningful, results in problems where sh ock discontinuities arepresent. In Appendix A, we show a particular set of non-conservative equationswhich are frequently used. There we shall indicate by an example the possibilityof obtaining incorrect solutions when a shock exists. The reader is therefore

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10 Introduction and th e equations of fluid dynamicscautioned not to extend the use of non-conservative equations to the problems ofhigh-speed flows.In man y actual si tuations o ne or an other feature of the f low is predom inant. Fo rinstance, frequently the viscosity is only of im po rta nc e close to the bound aries a twhich velocities are specified, i.e.rL where u, = Uo r on which tractions are prescribed:

I?, where n p i i = iIn the above ni are the direction cosines of the o utward norm al .In such cases the problem can be considered separately in two parts: one as theboundury luyer near such bo undar ies and ano ther as inviscidjoM. outside the bou nd-ary layer.Further, in many cases a steady-state solution is not available with the fluidexhibiting turbulence, i.e. a random fluctuation of velocity. Here it is still possibleto use the general Navier-Stokes equ ation s now w ritten in terms of the mean flowbut with a Reyn0ld.y viscosity replacing the molecular one. The subject is dealt withelsewhere in detail a nd in this volume we shall limit ourselves to very brief remarks.Th e turbulent instability is inherent in the simple Navier-Stokes equ ation s an d it is inprinciple always possible to obta in the transient, turb ulen t, solution modelling of theflow, providing the mesh size is cap able of reproducing the ran dom eddies. Such com -puta tions , thou gh possible, a re extremely costly a nd hence the Reynolds averaging isof practical importance.Tw o impo rtant points have to be m ade concerning inviscidflow (ideal fluid flow as itis sometimes known).Firstly, the Euler equ ation s are of a purely convective form:

dF- 0 F; = F; U)at ax; (1.26)and hence very special methods for their solutions will be necessary. These methodsar e applicab le and useful mainly in conzpressib/e,floiz.,as we shall discuss in Ch ap ter 6.Secondly, for incompressible (o r nearly incompressible) flows it is of interest to in tro -duce a potential that converts the Euler equa tions to a simple self-adjoint form. Weshall mention this potential approximation in Chapter 4. Although potential formsare applicable also to compressible flows we shall not discuss them later as they failin high-speed supersonic cases.

1.3 Incompressible or nearly incompressible) f lowsWe observed earlier th at the Navier-Stokes equ ation s are completed by the existenceof a sta te relationship giving [E q. (1.16)]

P = P P , TIIn (nearly) incompressible relations we shall frequently assume that:

1 . Th e problem is isothermal.

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Incompressib le or nearly incom pressible) f lows 112. The variation of p with p is very small , i .e. such tha t in pro duct term s of velocity

and densi ty the la t ter can be assumed cons tant .Th e first assum ption will be relaxed , as we shal l see later, al lowing som e therm alcoupling via the dependence of the fluid propert ies on temperature. In such caseswe shal l introduce the coupling i terat ively. Here the problem of densi ty-inducedcurrents or temperature-depend ent viscosi ty (C hap ter 5 ) will be typical.If the assum ptions introdu ced abov e are used we can sti ll allow for small com pres-sibi l i ty, noting that densi ty changes are, as a consequence of elast ic deformabil i ty,related to p ressure chang es. Th us we can write

d p = dp (1.27a)Kwhere K is the elastic bulk mo dulus . This can be wri t ten as

(1.27b)(2d p = po r

( 1 . 2 7 ~ )with c =

(and con densing the general form) as

being the acoustic wave velocity.Equ at ions (1.24) an d (1.25) can now be rewri tten om it ting the energy t ran sport

au a l a p 3-+ - Z l , U , ) +- - 7 - 6= 0at as p as, p ax(1 .28a)(1.28b)

Wi th j = 1 . 2 , 3 h is represents a system of four equa t ions in which the variables areii a n d p .W ritten in term s of Cartesian coo rdin ates we have, in place of Eq. (1.28a),

1 ap 3 au all.(2 at a s 8.v 3:p-+p-+ p- = 0

where the first term is dro pp ed for com plete incompressibi l i ty ( c =x nd3u a a i 1- + - I / - ) +-(Z/Y) + - u l \ > ) +- -at as a * a: P a\

with s imi lar forms for and 3 In bo th forms

(1.2921)

( 1 2 9 b )

where 7 = p / p is the kin em atic viscosity

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1 2 Introduction and the equations o f f luid dynamicsTh e reade r will no te tha t the abo ve equation s, with th e exception of the convectiveacceleration terms, are identical to those governing the problem of incompressible (orslightly compressible) elasticity, which we have discussed in Chapter 12 of V olume 1.

1.4 Concluding remarksWe have observed in this chapter tha t a full set of Navier-Stokes eq uatio ns can bewrit ten incorporat ing both compressible and incompressible behaviour. At thisstage it is worth remarking that1. M or e specialized sets of equa tions such as those which govern sh allow-water flowor surface wave behaviour (C hapte rs 5, 7 a n d 8) will be of similar form s and neednot be repeated here.2 . The essential difference from solid mechanics equations involves the non-self-adjoint convective terms.

Before proceeding with discretization an d indeed the finite element solu tion of thefull fluid equations, i t is important to discuss in more detail the finite elementproce dure s which a re necessary to d eal with such convective tran spo rt terms.W e shall d o this in the next cha pter w here a sta nd ar d scalar convective-diffusive-reactive equation is discussed.

References1. C .K. Batchelor. A n Introduction to Fluid Dyna mics, Cambridge Univ. Press, 1967.2. H. Lamb. Hydrodynamics, 6th ed., Cambridge Univ. Press, 1932.3. C. Hirsch. Numerical Computation of Internal and Exte rna l Flows, Vol. 1, Wiley, Chichester,4. P.J. Roach. Computational Fluid Mechanics, Hermosa Press, Albuquerque, New Mexico,5. H . Schlichting. Boundary Layer Theory, Pergamon Press, London, 1955.6. L .D. Landau and E. M. Lifshitz. Fluid Mechanics, Pergamon Press, London, 1959.7. R. Temam. The Navier-Stokes Equation, North-Holland, 1977.8 . I.G. Currie. Fundamental Mechanics of Fluids, McGraw-Hill, 1993.

1988.1972.