Hydrostatic by Van Gent

8/13/2019 Hydrostatic by Van Gent

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Eyttrostaic abal ltydrotlynasic forcesw. van Gent..

ntrouction.In this lecture I restrict myself to th subjects rnentioned inthe title. In othr lecturs of this session of the congressvre will- Iearn rnuch nore about th practical applicatj.ons ofhydrodynanics in nature.t 1s useful to subdivid the topic in the two parts:- hydrostatic forces vhich are the forces in a fluid at restand- hydrodynanic forces lthich are th forces j"n a fluid inmot ion .The contnts of this 1ccur is based on naterial fron varl.ousbooks. They are nentiond in the list of rfernces in thnext ection. The first one (1) is advisd for thosi.ntrested in an inportant application relatd to lhiscongrss, Reference to the other books is nade by tneans of hnurflber in O.cneral Llquj,d Propertig.s watr is a liquid we have to consider the baslc proprtiesof a Iiquid first. some of th properties ar gneral, whileothers rnay be specific and rlated to th physical proprtisof the liquid, which itself is detrnined by its chnicalconposition.There are two gnral proprties, viz.- a liquid has a fixd volun and differs fror a gas and- a l iqu id has f lu id i ty and d i f fe rs f ron a so t id .By th first property a liqui.d is different frorn a gas, but inboth substances the particles can nove relatively to eachother. In a liquid th particles renain togther bcause ofthe hutual attraction forcs. Th actual shap at rest isprescribd by intrnal or external conditions. n externalcondition is th shape of th containing vessl with the sanevolune. In a vessl which has a larger volune than the liquidor vhen th vesst is open on th upper side, th xtrnalinftunc of gravity will cause that th I'j"quid seks thetowest possible lvl. Th most uppr free surface of theliguid is thn flat at rest and horizontal. You are faniliarwith this fact in daiLy lif itttn using cups and conmunicatingvessels .In the absence of gravity an internal condition controls theshape of the liquid volurne. It is the capiftary attraction,vrhich crats a surface tnsion. Through this effect the outerurfac of the liquid Cnds to assune th nininun possiblearea, l"ik th effct of a nembran. When there ar no otherextrnal conditi.ons th surfac is spherical, nhich is truefo r f re fa l I ng ra in drops .rn h t b rsr l t 1c t r r we have to dea l w i th th e f fec t o fg rav i ty , no t ld i rh the e f fec t o f cap i l la ry t t rac t ion .


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Th fluidity property of liquids is also the reason for thealternative nane; Iiquids are aLso narned fl"uids. Fluids differfron solids; solids hav rigidity or a pernanent reactionagainst a change in relative positions of particles. F1uids donot have pernanent reactions; a fluid in flowing rrotion hasresistance against changes but this rsistance disappears vhnthe fluid cones to rest. So in a state of rst the fluidityproperty is perfect, In a flowing fluid viscosity gives riseto internal friction. Thj,s effect depnds on the flou rale.n alternativ way to indicate the differnce btwen fluidsand solids is the differnce in elasticity properties. This isnentioned her for conpleteness sake. solids hav forrnelasticity and bulk elasticity. Fluids have no fornelastieityr which is the effect describd above. Fluids onlyhav bulk elasticity or conpressibil ity. This neans thatfluids have ven at rest a raction against change in volutn.PracticaIly, horver, fluids are said to be incornpressible,'hich is certainly true for vater in the context of thiscongress. t is worth noting that as a consequence of thesnalf cornpressibil ity of \,rater, sound travels relatively fastin water, i.e. hore than four ti\es fastr thn sound travelsin a i r .In general the voluhe of an amount of fluid depends on thetlperature and pressure. the latter quantity has to be givnnuch nore attention in the context of hvdrostatic anclhydrodynamic forcs,Hytlrogtatic Forces in a Fluitl at Rst,Top ics :Pressure :l-. Hydrostatic Plessure2 . Measures o f Pressure3. Hydroslatic Parailox and Pascalrs laitForc i4. Buoyancy and rchinedest Law5. EquiLibriurr and Stabil ity6. FLoat ng Bod iesHydroalynanic Forces in a Fluiil in Uotion.Topcs : IOT:7. Velocity and strean lins8. Stranline patterns9. Curvd streanlines and deformation10. ReIatj-on between velocity anil pressure. 11, Shear flow and flo\r stability12. tarninar and turbulent flouBod is in f lov :13. Boundary layers anal wafl friction14. F Iow spara t ion and ef fec t on d rag15, Lift forcs arld rneasures to create lift

1 r t r : d f ^ r r a < A h t - h h r r h h ' ^ v


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1. Hydrostatic prssur.In a ffuid at rest on earth the gravitational forc exerted bythis globe plays a doninant rol. The static pressure in afl"uid is built up by gravity together lrith the fluidrs ownponderosity.

Di rec t i ono gravi tYatt ac on

F ig . 1 : co fuhn o f f l u i d g ivs p ressure , (6 )

p o d x d y


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At a point in a fl.uid we can think of a smatl horizontal-surface on which a force is exerted. Vertically above thatsurface ther is a coLurRn of lrater and air. Th total weightof that anount of vatr and air is the force which defines thestatic pressure. The total reight is proportional to thesurface area, so {e can spak of a force per unit of surfacearea and that is called pressure.The fluid weght is proportional to its mass density, i .e. i tsnass pr unt voluoe. Therefore also th hydrostatic pressurein a fluid depends on j.ts nass density, snall- differnqs innass density can have consequencs through its effect on thehydrostatic pressure. Faroiliar exanples are buoyant flows in awater kttle being heated ancl the dif,fernce in floatinq onsea water and fresh water. The aubiects of buovancv andfloating tdiIl be dealt r.rith nore extensvely.2. Measures of Pressur.

Rel t i vePressure

Absol epres ure B

Atmoshericpre5Sure( var ies wi th$eather andal i tude)

Re l 0

Abs 0

It lovabl0arum

Abso l tepressure A

F igure 2 : Var ious p rssure leve ls , (7 )The oficial neasure for pressure is one pascal (pa) , whichneans a force of one Nevton on a surface of one square rnetr,1Pa is a snall quantity if conpared with th atrnospheric

Vacuu;l

oaEul


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pressure in which we 1iv. In the atmosphere at th freesurfac lvel of the sea the pressure is about 100,000(one hundred thousand) Pa. This prssure l'vel is call 'ed onbar. The pressures which pl.ay a rol in swirnrning are lnuchsna l le r .Instrunents by which the pressure is nasured are usuallycalibrated fron the atnospheriq pressur level. hey indicatethe relative pressure anal ar set zero in the atnosphere. soundr vater thy measure the hydrostatic pressure on1y. Frornthe definition it follosls that the hydrostatj.c pressur rsproportional to the depth of submergenc. For havir fludslrnore wight) the incrase of prssure r.rith dpth is strongerthan for lightr fluids,3. Hydrostatic Paradox and Pascalrs aw.Th understanding of the hydrostatic pressure sems easy bythinklng of the coluhn of water above the point ofobservation. Howevr, ther are son special effects lthich donot follow directly fron this concept but hav to doessentially with the property of fluidity.

Figure 3a : Prssure on the bottom of a vessel, (6)First thre is the socalled hydrostatic paradox: If v havevessefs vrith different shapes but th sane bottoril area, thprssuf on these bottoms is t'h sane if th verticaL dstanceto the free vatr surface is the salne, The amoun of liquid inthe vessel- dos not nattr. Th force on the bottom can bnuch hj-gher than th weight of the liquid in the vssel. We.un "o.rl'.rde that the ide of pressure is nuch wider than justthe weight of a colunn of fluid/ which I usal as a neahs tointroduce the subj ect.


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dar

Figure 3b: Pressure on th hull of a body (6).second there is the law of Pascat, vhich states that thpressure at a cerain subnergence 1evel is the sane in alldirections. The force on a surface is the same for allorientations. This lat, follows froll the rquirenent that in afluid in a state of rest aII fluid elrnents nust be inequillbrlun. Then the forces on all sides of a snall cube rrustb th sane.The picture is that of a floating ship. lrhich is a body nottoo diffrnt fron the subJect on this congress. It has to benoted that ttre pressur on the hull is equal to the indicatedcolunn of vater irrspective whether the colunn of watr isthere in reality or only in our inagination,4. Buoyancy and rchiedest lJai.r,

Figure 4: fully submerged body (6) .


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Th forenentioned properties of hydrostatic pressur are thse for understandin buoyanc forces. . Most faniliar is theforce on a full,y subrlerged body which is an upth-rust of thewater on he body equal to th weight of the displacd volumeof water. This is linovn as rchimedes' Lanq. More generally.Drssures on the hull of a ship or on th skin of a body, lthenthay a"" put together become forces on the ship or on thbody, The iorce ay have any direction depnding on th shapet tft" uoay and on the prssure distribution. This will bconeclear in the sections on hYdrodynamic forces 'thn there isalso flow around the bodY.5. Equilibriurn and stabil itY.

clv

oFtl

\ /oa

Figure 5: Partly subrnrgd ody in two positions (6) 'when a body is not fully subderged th. situation ofhtdrostatic io."" on the body is rnore con|plicatd' But the'niioarr"tiotr of forces instea of p ssures i helpful' Thbody vhich is ftoating on water has its own weightr a force Piing i" the centr of gravity G. Fo-r quilibriuh it s"".""ty that the upthrust or buoyance force is equal to theveiqht, -but the centre of this forc is at a differenti;;ii;" i. ro- . sltnmetric bodv like t.I -sh-iP in the pictureon the ] f t F and G ar on the sahe vr t ica r i rne 'ihe situaion becones nore corplicated whn th body is notymrnetric or not in a synhetric position like the ship in thepictur on lhe ri-ght 'ftn.tt trr" snip hai rotated from its original upright position'i i i"--""t.. t gravity G rmains the iame'

(Note that it isp . " " i l r . - inJ t - t te roa ins ide th sh ip noves , e 'g ' o i l in a

c1I PtlruII


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tanker, then also c rnay change its position relative to theship.) The cntre of buoyancy witl chang in genral, say toFd. The originat line through F and c and the vertical l inethrough Fd intrsect at M. This point is call-ed thentacentr. Nol,r we can s that the gravity force pull-s downthe ship and the buoyance pusttes up th ship in suctr a vaythat th original position of th ship is obtained again. Thisstabilizing effect is related to the relative position of Mand G. When M is bIow c the shiD is not stabl.5 . F l oa t i ng Bod is .

Figur 6 r F loa t ing bod ies , (7 ) .In this Figur you see three types of floating bodies, abal,loon in the air, a subrlarine in the r.rater anil a surfaceship at sealvl. 11 pictules show stable conditons, but itis irnportaht to note the differnt distances between l( 'ight) and Fb (buoyance). lso not that the balloon and thesubtnarine both have fixed positions of the buoyance forc.The balloon is most stable through the lower position ofbasket vith wight, The surfac ship has not such a 1ovposition of the weight and is only stable thanks to theflexibl position of th buoyance force. Th subharine has notth stabili.zing features of the balloon or the surface ship;now stabi.lity stronqty depends on the position of Fb binghiqher than the Dosition of w. The smalf distanqe bet 'en both

. E _ - _ F' 8 B+ lm F -rr\ /I\\ T , / t a lv \l-lI iw


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indicates that stability issubmerged body.? . V loc i ty and St reaml ns .

St reaml inesstrearntube

weak in suctr a case of a fuLly

and velocity vectors (top) ; elernentary( b o t t o n ) , ( 5 ) .


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8. strealrline Pattern. pattern of stram lines reveals atand relative rnagnitude of the velocityf lov .

a glance the direclionthroughout th field of

Figure I : strearnline pattrns of flow around body, (5) .The flow around a body vhich moves in stil l watr is anunsteady f1ow. f v observe such a flow fron a fixdnosition, we se particles noving away from the body in the?ront region and particles noving toward,s the body in the aftregion a illustrated in th top picture. rf the body has acostant speed it is possibl to transforrr the streamlinesinto that f a steady flov by us of the principle of rlativenotion. That occurs if ve bserve the flow frorn a positonwhich noves with the body. What we se then is shown in thebotton picture. This procdure is the basic principle .ofwatertun-nel and ltind.tu;neI experirnents. Herein bodis which


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nornally nove at constant speed through a fluid at rst, e.g.Iike shlps, airplanes and svinners are held statonary in astrean of fluid rroving at the sarne relativ speed, I-n thissituations al.] kinds of measurnents and visuali-zation of flovbecon nuch easier.9. Curved Streanlines and Deforhation.Nov we want to have a closer look at vhat happens noreprec is ty in th f low ing f lu id .

- -r'- F --1 - -T - - - r - r - 1l l l- I . I -J

l r ro to t iono l Roto t iono l

Ill--+-

Fgure 9 : Typs o f no t ion o f f lu id e lemnts , (5 ) .Tr^to basic notions of an elenentarv fluid Iemnt can bedistinghuishd, viz. hotion atong curvd streanline anddefornation. In th upper picturs the effec of dforrnationin a fl-o , r,rith straight streanlines is shorarn. This effctdepends on the difference in vetocity along adjacentstranl-ins. n the top right picture w se that a fluid


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etement is deforrned and rotated. In th lol1'er picturs thesatne effect is shovrn for a flow with curvd streanfines. onthe left the differnce in velocity between streanl,ines issuch that he fluid element is strongly deforned, but as awhole not rotatd. Thrfore this is stil l called irrotationalf1ow. on the botton right picture the difference in velocitybetven streanlines is such the fluid efernt is not defomedbut roated as a vrhole, Fron this il lustrations we see thatfluid etements can unilergo different tratmnts. This can haveirnportant consquencs for the description of th flow byusing nathenatics. but' that will not be done hre.10. Relation betwn velocity and Pressure.Having discussed no\^' he kinqrat ical aspects of a fluid innotion, vj-z. its vlocity distributior and its stramlines. wehave to rturn to the pressure. The pressure distribution ha6to be understood in ordr o learn about hydrodynanic forcessinilarly as hydrostatie forcs.

Figur 10VelocityandPressuredistribu-ti-on on a


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If w could follow th flov along a streanlin rr r'ouldexprience that at places of lon velocity th pressure is highand vice versa. In the picture this is denonstrated vithgraphs for pressure (P) and for velocity (V). Renenber thatEtreanlines are wider away in a lov, veloeity region and closetogther at higher velocity.This velocity-presEur relationship, I'hich is calledBernoullj.rs 1aw, is very irnportant and reprsents rnajoreffects 1n the fLolr. However, it is not accurate forprediction of a force on a body. For that purpose we also takeinto account the friction on the bodv surfac. s lonq as wedisregard fric-Lion ve encounter the ;ocalled paradox fdrlenber , which says the force on a body in a 6teady Etranis zero. That is contrary to intuition.11. Shear Flow and Flov stability.In a flo\,t ve have to deal also with instabilitis. This neansthat above a certain speed the stramlines loose their rgularpattrn. Thy becone vry cornplicatd and can no longer bev isua l i zed .

"/,///,/,/,///////,////////////////////t

'///////////////////////////////m

--

Figure 11 : crowth of a disturbance in a florr, (5) .


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The underlying rnechanisrn s by no neans simple, but acualitativ analysis nay b helpful in understanding' Assurnhat a laninar flow taks plac parallel to a plan boundarypicture a) . such a flow hs zer speed at the ltal' l due to theifect or viscosity which prevents slip atong the wall' Thflow appears as layered wj.th increasing sped avJay from thewaff. ir l sone distnce from the wall a disturbance is producedwhich yields at a given instant local undulations of theiii."riin.= (pictues b). The acconpanyl-ng variation in thevlocity wili- give ris to a pressure pattern athich tnds to.no^nt'tn" unulations (pictures c an o1 . on th ottler sidthre are effects fron viscosity and of th presence of thwa11 \irhich may darnpn th undulatj 'ons. In practical .situationsii "penat on- the itovr configuration whther the original flol'i s s tab l or no t .

12. Laninar and Turbul-nt FIow.Laninar flot' iE defined as that in lthich th streamlinesrenain distinct fron each other over thir entire length'laninar flov nay be ither steady of unstady and ethruniforrn or nonu-niforn' Turbulent flow is the opposite oflaninar flolri once the hterogeneous mixing process started byiiw instalii it i,es xists, evn the instananous- streamlinesbecone lhoroughly confused. strictly speaking turbulent flowis intrerentty-uniteady and nonuniform, but it is convenint todistinghuish- betvreen Lhe secondary motion of the turbulenceand th prirnary motion of the fluid. Th latter can bclassifia as ither stady or unsteady, unifonn or nonuniformand rolational or irrotationat vrithout regard to theturbulence itself.

LominorLaminar flow in a duct, (5) .

i

Figur 12a :


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Turb len tFigure 12b : Turbulent flor^' in a duct, (5).Th ffect of turbulence is a lateral mixinq process rrhichresults in an egualization of the rnean veloity vctor acrossthe fIoJ section.13. Boundary layer and wall friction.s real fluid flows ovr the surface of a body, th effct ofviscosity is fj.rst that the ftuid sticks to ti-Ie surface bvadhesive forces. But away fron the surface the fluid trie; tofolLo' ttle rnain strean. The Laninar flow is in a layer ofvarying hickness; it begins frorn no thickness at the front ofthe body whre the fluid first contacts it and ir;rases alondthe surface in th dirction of the rnotio . such a larninarboundary layer nay change into a turbulent boundary layer.

F igur 13 : Boundary layers a tong f la t p la te , (7 ) .In he picture these phenornnaare il lustrated for a sinplbody, a srnooth flat plate. The boundaty layers occupy anxtrnely thin and usuatly invisible part of th fI; (ththicknss is exaggerated j-n the picture). Nevertheless it isof great inporlance since it is the essential reason for thexj.stenc of a flictional drag force exerted by the fluid on

Lmi boundry Turbu l n t boundal a y e r I ayer


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the body' The dragsurface, snooth or force also dependsrough. on the quality of th

) spothboundaY

14 . F low spa ra t i on and e f fec t on d rag 'we have looked at a number of delai ls of th fl"ow, but now we

Figure 14 : snooth and rough surfaces, (7) 'laninar flo\^t occurring over smooth or rough surfaces possesssil;--J;"-;;;p;ii i". i both cases' as depicted n th picture(a) and (b) on the left. Thus in faninar frow, surlacerouqhness has no effct on the flow.i"-"iu"i"nt flow, ho$rver, the fluid notion is influenced'When turbulent flow occurs over snooth solid boundaries it isalways separated fron the boundary by a fitn of laninar flow'hi;';";-'b; xplained as foLrows. Th presence of a boundaryi"-" i"il"r""t^ flov will reduce the freedom of the turbulentii*itrq pto"a". and in the region very crose to the boundary;;;]"; irnains laminar. Now a rough boundary of a turburntii" .pp"u.. as smooth as long as the roughness particls orl.i""i* are conpretel"y subnrged in the laninar filrn';;;; ; . ; , vhen th height of the-roughnss lements. exceeds theirri"ttt"it. of the larninar fil-n the turbulence vill' b augnented""-iii iu.i""t fi lrn is no longer ffective' A diffrentlrio"ity profile of the flow rsults as depictd in thpictures la) and (b) on th right.

a) Sooth soundry

b) Rough oundarY b) Rouqh oundrY


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l r i l l pay attention to a broaderflow. Much of the hydrodynaniceffect of separation.viv of bodies in rnotion in aforces can b understood by the

Lt^.r1.uFb.d ou.r . l tow

...: . ( ':\P o - l l v . p r . . . u . . 9 F . d t . n c

Flgure 15 r Separa t ion o f f low a t a f t end o f body , (3 ) .

V A F | E X


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The fluid elernents in the boundary layer loose energy andmomentum in conparlson with fluid elenents in the outer flow.Near the af end of a body the pressure tends to j.ncrease. sa result the fluid partLcles fron th boundary Layer can nottnove furthrt they cone to rest, accunulate and are given arotary notion by the lain stream. This rotation can grow andan eddy of increasing size is devel-oped. he eddy cannot beretained in the prsnc of the body but breaks away. Thenanother is all,owed to form and the procss repeats itself.


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2 0

(r) ?lO ?llERl| O CIRCII{R cylINDE.R t{ NON.VISCOSS IO.a: :O D&.|,C,

{B) ct l r D l . t aEri{orJs NIw :RS :{ 1la c D R o lo; co.= r.z,

(C) cr l I :aDA E1'/ i t r { 8d - ro4 aao ro5i yoRlEI saREt. i I tn cD.. l ,Z.

(D) rr: yr: 's?LrrtER" DEvIa :r { l r i ; c- . 1,s,

(E) CYLILDEI tsOvE lMc^], RErNO ,DSrWlER t tY C- . O,t .

Figure 16( ) STRrrg, l l E sEctron: Separation and

r l t $ c ) . I N 1 g : cR D E O P 0 . C 6 .tu rbu len t wakes , (3 ) .


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The result of separation and ddy formation is the fonnationof a turbulent rrak. Th turbulence in th wake is of othernature than in a boundary layer. any situations can be seenin the pictur.To return now to the probl-en af drag forces on objects in afluid flov, the effects of bounclary layrsr separation ands'akes nay b absrved. Fundarnentally drag is caused by theconponents of the nornal and tangential forces transrnittedfron the fluid to the surface lemenls of the body' The nornalf,orces are those of, pressure hich in gneral nay becalculated by applying Bernoutli 's La to the stleantubadjacent to the body' The tangntial forces ar those of sheara the urface of te object rising frorn viscous effects inthe boundary layer.15. fi.ft Forces and Deasures to Create Lift.Based on considerations of the effect of visclsity, lanl"narflow and turbulent flovJ, we have discussd the resistance ofbodies rnoving in water. This force acts in longtudinaldirection i,. in the direction of the flov. Thre iE also aforce perpndicular to this direction. t is called the liftforce and is of great irnportance in th hydrodynanlcbroDulsion of bodis.in"- t i f t force also is a rsult of the pressures around abody. To obtain tif it is necessary that in the. flow acirulation is cratd. To undersland this effect it is usefulto hink of long bodes like airplan wings' The lorlgestdirnension is not in the direction of the fLow butDrDendicular to that. The other dinnsions are relatively..it. rnis facilitates that we can discuss the properties ofthe flow in a plane parallel to th flolr. The flow in adjacentDlans does not dLffer froru each other.

Figure 1?a : circulatory flow creating lift, (5) .


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Figure 17b , 1?c : c i rcu la to ry f lov ' c ra l inq f i f t , (5 )In the pcture strearnlines around a cylinder are coniderd.The effect of boundary layers is disregarded. Althoughboundary layers pLay a specific role in the cralion ofcirculatory flow its effect on the nagnitude of the Lift inrelation to circulation is srnaLl. (Hence lre can useBernou l l l ' s Law) . In p ic tu re (a ) there is no c i rcu la t ionr on lyan approaching flow fron the left. In this synmetric flowpattern there is no reason to expect a vertical forc on thecylinder as prssures are equal on both sids.The si,tuation becones different ra,hn he flov is conbind lrl"tha pure circulation (pictur (b) ), Ieading to the flow patternin picture (c) . Far alray fron the cylinder th approachingfloi,r is stil l uniform as th vetocitles of the circulatoryfLolr are snalL there, Cl-ose to the cylindr, hot'ever, the flowpattern is 6trongly distorted in conparison nith picture (a),The flow over the top has higher velocity than the flov alongthe botton. s a consequenc of the lalr of Brnoulli thepressure at the top is lorrer than at th botton and there isan upvard force called lift.


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Th lift force can be very strong in conparison with-the dragforce and it is vry useful. In nature flying and swl'mnrngdeDnd on this phenrnenon, while it is util izd by nen in; ; i i ;g , sh ip p lopu ts ion , puhp ing , w indn i l l s and aga in f ly ing 'It lr i l be cter tfrat by sone means we have to conlrol thecirculation around the body.

F igure 18 | Pro f i le vh ich cra tes c i rcu la t ion , (5 ) 'The difference betven the body in this picture and theorevi.ous one is clar, It is not a cytindr but a profil' The

(d)

(c/


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2 4most inportant featur of a profile is that the strearnlines atthe trailing edge follow the body contour as snooth aspossible. This is seen in pictur 1c1 vtricn is aiiierent trorn(a ) . The d i f fe rence is due to boundary layr e f fec ts , l rh ichcreate the requj-red circul-ation, picture lly, to nake the flowsrnooth. s a consquenc there is lift.l-6. Drag Forces on the Hunan Body.To conclude this lcture \,re stlol^rsorn practical infornaEronabout hydrodynanic forces on a human bdy \rhen it is placd ina strean in various positions. The resulls are obtaind frorlneasuretents in a vrindtunnel-, so strictly ve hav to speak ofaerodynanc forces. But th results are fresented in such avay that they also apply to circurnstance in water,

D/qI2 0 3 f l i 12 f+'Figure 19 : Hunan body in a constant strean, (3).The human body is of the shape sinilar to a cylinder of aIenght to diantr ratio bet\a'en 4 and 7. cylj-nders ar ofgreat technical interest and therefor a usfuL reference. Thedrag of the hunan body is expressed as an effectiv crosssecfion which is confronted with the f1ow. This cross sectionis calculated fron th tneasured drag D and fror the referencepressure-q .o f t lg . fJ9" , q is propor t iona l to the mass dens i tyof the fluid nultiplied by the vetocity ssuared.The cross sectional areas in the pictuie ire given in squarefeet. The given nunerical values re effectiv in regard ofdrag; thy nay differ from the ral cross sctional reas inregard of th geometry. s a refernc it can be rnentionedthat the total suifac area of this hunan body is 20 squarefee t .

g f t t

References:1 . C o u n s i l n a n , J . 8 . ,

1 9 6 1 . .trTh science of swinning", prentic Ha1I,rrTechnische Strnungslehre", Springr verfag,


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Hydrostatic by Van Gent

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