¨asovno odvisna simulacija, vizualizacija in meritve

Strojni�ki vestnik - Journal of Mechanical Engineering 51(2005)1, 13-27

13Èasovno odvisna simulacija, vizualizacija - Transient simulation, visualisation

Èasovno odvisna simulacija, vizualizacija in meritve kavitacije zmetodo PIV-LIF na razliènih osamljenih profilih

Transient simulation, visualization and PIV-LIF measurements of the cavitation ondifferent hydrofoil configurations

Matev� Dular - Rudolf Bachert - Brane �irok - Bernd Stoffel

Prispevek obravnava numerièno in preizkusno �tudijo kavitirajoèega toka okrog razliènih osamljenihprofilov. Za simulacijo neustaljenega toka je bil uporabljen programski paket Fluent. Dvofazni tok smoopisali z vpeljavo homogenega toka me�anice. Za popis nastanka in kolapsa kavitacijskega oblaka je biluporabljen kavitacijski model, utemeljen na poenostavljeni Rayleigh-Plessetovi enaèbi dinamike mehurèka.Narejene so bile trirazse�ne simulacije kavitirajoèega toka v razliènih razmerah za dva osamljena profila.

Za dva profila smo posneli slike kavitacijskih struktur v razliènih razmerah. Za doloèitev hitrostnegapolja zunaj in znotraj kavitacijskega oblaka smo uporabili metodo PIV-LIF. Izmerili smo frekvence trganjakavitacijskega oblaka.

Numerièno napovedane porazdelitve dele�a pare in hitrostnega polja smo primerjali s preizkusnimirezultati. Narejena je bila primerjava s preizkusi doloèenih in numerièno napovedanih povpreènih dol�inkavitacijske strukture vzdol� profila. Primerjali smo tudi numerièno napovedane in s preizkusi doloèenefrekvence trganja kavitacijskega oblaka. V vseh primerih smo dobili dobro ujemanje med rezultati preizkusovin simulacij. Poleg tega je simulacija pravilno napovedala nastanek znaèilne podkvaste kavitacijskestrukture.© 2005 Strojni�ki vestnik. Vse pravice pridr�ane.(Kljuène besede: kavitacija, dinamika tekoèin, vizualizacija, metoda PIV-LIF)

This paper concerns a numerical and experimental study of cavitating flow around different singlehydrofoils. The program package Fluent was used to calculate the unsteady flow, and the homogeneousflow principle was used to describe two-phase flow. The cavitation model based on a simplified Rayleigh-Plesset equation for bubble dynamics, was used to describe the appearance and collapse of a cavitationcloud. A 3D transient simulation of cavitating flow under different conditions for two hydrofoils was made.

Images of the vapour structures under different cavitation conditions for the two hydrofoils wereacquired. A PIV�LIF method was used to obtain the velocity field inside and outside the vapour cavity.Measurements of the cavitation cloud shedding frequencies were made.

Numerically predicted distributions of the water vapour and the velocity field were compared withexperimental results. The experimentally determined and numerically predicted mean cavity structurelengths along the hydrofoil were compared. Also, a comparison between the numerically predicted and theexperimental cavitation cloud shedding frequencies was made. In all cases a good correlation between theexperimental results and the numerical simulations was found. The simulation was also able to correctlypredict the formation of typical �horseshoe� cavitation structures.© 2005 Journal of Mechanical Engineering. All rights reserved.(Keywords: cavitation, compuatational fluid dynamics, visualization, particle image velocimetry (PIV))

Strojni�ki vestnik - Journal of Mechanical Engineering 51(2005)1, 13-27UDK - UDC 532.528:004.94Izvirni znanstveni èlanek - Original scientific paper (1.01)

0 INTRODUCTION

The occurrence of cavitation in hydraulicmachines leads to problems like vibration, an increaseof hydrodynamic drag, pressure pulsation, changes

0 UVOD

Pojav kavitacije v hidravliènih strojih vodik problemom, kakor so vibracije, poveèanjehidrodinamiènega upora, tlaèni utripi, spremembe v


14 Dular M. - Bachert R. - �irok B. - Stoffel B.

in flow kinematics, noise and erosion of solid sur-faces. Most of these problems are related to tran-sient behaviour of cavitation structures [1]. Hence,a study of unsteady cavitation behaviour is essen-tial for the correct prediction of these problems.

In the past decade a wide range of methodsfor the numerical simulation of cavitating flow weredeveloped. Most of the studies treat the two-phaseflow as a single vapour-liquid phase mixture flow.The evaporation and condensation can be modelledwith different source terms that are usually derivedfrom the Rayleigh-Plesset bubble dynamics equa-tion ([2] to [5]) or by the so-called barotropic statelaw, which links the density of the vapour-liquid mix-ture to the local static pressure ([6] to [8]).

The paper discusses an experimental andnumerical study of the unsteady phenomena of cavi-tating flow around two different hydrofoil configu-rations. The PIV (Particle Image Vecilometry) methodcombined with the LIF (Laser Induced Fluorescence)technique was used to determine the velocity fieldaround the hydrofoil (outside and inside the vapourcavity). Instantaneous images of the vapour struc-tures were recorded using a CCD camera. The LDA(Laser Doppler Anemometry) method was used todetermine the boundary conditions by measuringthe velocity in a plane in front of the hydrofoil. Thecommercial CFD (Computational Fluid Dynamics)code Fluent 6.1.18 was used for the 3D transientsimulation. A cavitation model, based on bubbledynamics equations, described in [4], was used todescribe the unsteady behaviour of the cavitation,including the shedding of the vapour structures.

1 EXPERIMENTAL SET-UP

The experiment was set up in a cavitationtunnel at the Laboratory for Turbomachinery andFluid Power, Darmstadt University of Technology.

Two simple hydrofoils were used. The basicgeometry is a 50-mm-wide, 107.9-mm-long and 16-mm-thick symmetrical hydrofoil with a circular leading edgeand parallel walls (CLE � Circular Leading Edge hy-drofoil). In order to obtain three-dimensional cavita-tion effects the basic geometry was modified by cut-ting the leading edge at an angle of 25 degrees (ALE� Asymmetric Leading Edge hydrofoil) (Fig. 1).

The hydrofoil was inserted into the rectan-gular test section of a cavitation tunnel with a closedcircuit that made it possible to change the systempressure and consequently the cavitation number,

kinematiki toka, hrup in erozijo trdnih povr�in. Veèinateh problemov je povezana s prehodnim obna�anjemkavitacijskih struktur [1], zato je �tudija neustaljenegaobna�anja nujno potrebna za pravilno napoved prejomenjenih problemov.

V zadnjem desetletju je bilo razvitih mnogometod za numerièno simuliranje kavitirajoèega toka.Veèina teh metod obravnava dvofazni tok kotenofazni tok me�anice pare in vode. Uparjanje inkondenzacijo lahko modeliramo z razliènimi izvornimièleni, ki so navadno izpeljani iz Rayleigh-Plessetoveenaèbe dinamike mehurèka ([2] do [5])) oziroma stako imenovanim barotropiènim zakonom stanja, kipodaja odvisnost med gostoto me�anice pare in vodeod lokalnega statiènega tlaka ([6] do [8]).

Prispevek obravnava preizkusno innumerièno �tudijo neustaljenih pojavovkavitirajoèega toka okoli dveh razliènih profilov. Zadoloèitev hitrostnega polja (zunaj in znotrajkavitacijskega oblaka) je bila uporabljena metodavizualizacijske anemometrije (PIV) v kombinaciji stehniko lasersko inducirane fluorescence (LIF).Trenutne slike parnih struktur smo posneli s kameroCCD. Za doloèitev robnih pogojev smo uporabilimetodo laserske Dopplerjeve anemometrije (LDA),s katero smo doloèili hitrostni profil v ravnini predprofilom. Za trirazse�no nestacionarno simulacijosmo uporabili komercialni program za raèunalni�kodinamiko tekoèin Fluent 6.1.18. Za opisnestacionarnega obna�anja kavitacije, vkljuèno strganjem kavitacijskega oblaka, smo uporabilikavitacijski model, osnovan na enaèbi dinamikemehurèka, ki je podrobneje opisan v [4].

1 POSTAVITEV PREIZKUSA

Preizkus je bil opravljen v kavitacijskemkanalu v Laboratoriju za turbinske in hidravliènestroje na Tehni�ki Univerzi v Darmstadtu.

Za preizkus smo uporabili dve preprostigeometrijski obliki. Osnovna geometrijska oblika je50 mm �irok, 107,9 mm dolg in 16 mm debel simetrièniprofil s polkro�nim vpadnim robom (PPVR - CLE) tervzporednimi stenami. Z namenom, da bi dobilitrirazse�ne kavitacijske uèinke, smo osnovnogeometrijsko obliko spremenili tako, da smo vpadnirob profila odrezali pod kotom 25° in dobili profil znesimetriènim vpadnim robom (PNVR - ALE) (sl. 1).

Profil je bil vstavljen v pravokotni testniodsek kavitacijskega kanala z zaprtim obtokom, karomogoèa spreminjanje tlaka sistema in posledièno



which is defined as the difference between the sys-tem and the vapour pressure (at system tempera-ture) divided by the dynamic pressure:

(1).

Decreasing the cavitation number results in ahigher probability of cavitation occurrence or in an in-crease in the magnitude of the already-present cavitation.

The velocity in the reference plane in frontof the hydrofoil was held constant at 13 m/s (Re =208000 based on hydrofoil thickness). The devel-oped cavitating flow at different values of the cavi-tation number (2.5, 2.3, 2.0) at a 5° angle of attackwas observed.

2 EXPERIMENTAL EVALUATION OF THE CAVI-TATING FLOW

The shape and dynamics of the cavitationstructures and the velocity field around the hydro-foil were investigated. The images of the cavitatingflow were recorded with a CCD camera from the sideand from the top (Fig. 2).

The velocity field was determined with thePIV-LIF method. The experimental setup (Fig. 3) con-sisted of a laser, 2 CCD cameras, a PC and an acquisi-tion�control unit. A Nd-YAG laser vertical light sheetwith a wavelength of 532 nm (green spectrum) andapproximately 1 mm thick was used for the illumination.The position of the light sheet was 5 mm from the frontwall (where the hydrofoil is the shortest). The frequencyof the image capturing was approximately 2 Hz.

spreminjanje kavitacijskega �tevila, ki je definiranokot razlika med sistemskim tlakom in tlakom uparjanja(pri temperaturi sistema) deljena z dinamiènim tlakom:

Zmanj�anje kavitacijskega �tevila pomeniveèjo verjetnost pojava kavitacije oziroma poveèanje�e prisotne kavitacije.

Hitrost v ravnini pred profilom je bila medpreizkusom nespremenljiva v = 13 m/s (Re = 208000,glede na debelino profila). Obravnavali smo razvitikavitirajoèi tok pri razliènih vrednostihkavitacijskega �tevila (2,5, 2,3, 2,0) in 5° vpadnimkotom profila.

2 PREIZKUSNO VREDNOTENJEKAVITIRAJOÈEGA TOKA

Obravnavali smo obliko in dinamikokavitacijskih struktur ter hitrostno polje v okoliciprofila. S kamero CCD smo posneli slikekavitirajoèega toka z dveh pogledov (od zgoraj inod strani) (sl. 2).

Hitrostno polje smo doloèili z uporabometode PIV-LIF. Za preizkus smo uporabili laser, dvekameri CCD, osebni raèunalnik in enoto za zbiranjein nadzor (sl. 3). Za osvetlitev smo uporabilinavpièno ravnino laserske svetlobe (laser Nd-YAG)z valovno dol�ino 532 nm (zeleni spekter) debelinepribli�no 1 mm. Polo�aj laserske ravnine je bil 5 mmod sprednje stene (kjer je profil najkraj�i). Frekvencazajemanja slik je bila pribli�no 2 Hz.

Sl. 1. Osnovni profil � PPVR (levo) in spremenjeni� PNVR (desno)Fig. 1. Basic � CLE (left) and modified � ALE hydrofoil (right)

2

( )

/ 2vp p T

vs

r

¥ ¥-=

×



The problem with using the PIV method incavitation flow is that the present vapour structuresreflect too much light so that the acquisition of theinformation about the velocity field inside the cavityis impossible. A relatively new technique of combin-ing the PIV method with the LIF method was used toobtain the information about the velocity field out-side and inside the vapour cavity [10]. For the seed-ing some special PMMA-Rhodamin B fluorescenceparticles that reflect the green light (532 nm) in theyellow spectrum (590 nm) were added to the water.

This way it was possible to use two CCD cam-eras (the first camera captured only light in yellow spec-

Problem uporabe metode PIV vkavitirajoèem toku je, da kavitacijske struktureodbijajo preveè svetlobe. Tako doloèitev hitrostnegapolja znotraj kavitacijskih struktur ni mogoèa. Da bidobili informacije o hitrostnem polju znotrajkavitacijskih struktur, smo uporabili razmeroma novotehniko, pri kateri zdru�imo metodo PIV z metodoLIF (glej tudi [10]). Za zrno smo vodi dodali posebneflourescentne delce PMMA-Rhodamin B, ki zelenosvetlobo (532 nm) odbijajo v rumenem spektru (590nm).

Tako je bila z uporabo dveh kamer CCDmogoèa hkratna doloèitev hitrostnega polja in

Sl. 2. Postavitev kamere za snemanje slik s strani (levo) in od zgoraj (desno)Fig. 2. Camera set up for recording the side-view (left) and top-view (right) images

Sl. 3. Postavitev preizkusa za meritve z metodo PIV-LIF

Fig. 3. Experimental setup for measurements with the PIV-LIF method

Osebni raèunalnik



trum and recorded only the images of the tracer particles(the vapour structure is not visible in the image), whilethe other captured the whole light spectrum and recordedthe image of the vapour cavity to determine the flowfield and the distribution of the vapour simultaneously.The two pictures where superimposed to get informa-tion about the velocity field and about the dimensionsof the vapour cavity in the picture (Fig. 5).

The velocity field was determined only froma side view, while the images of the vapour struc-tures were recorded from the top view also (Fig. 4).In the figure the flow is from left to right.

The sequence is made from characteristicsingle images. We can see that the cavitation behavesdynamically at the front wall (where the hydrofoil isthe shortest). In this region the separation of the cavi-tation cloud occurs. The separated cloud then travelsand implodes in a region with higher pressure. On theother side (where the hydrofoil is the longest) thecavitation is steady (without cloud separation).

Four typical situations in the cavitation cloudseparation process and two magnified details of theALE hydrofoil at s = 2.0 are shown in Fig. 5. After thecloud separation (1), just before the cloud separation(2), and two cases during the separation itself (3 and 4).The details show the re-entrant jet inside the attachedpart of the cavity (detail A) and a vortex behind theseparated cavitation cloud (detail B). The flow is fromleft to right. A significant vortex (re-entrant jet) can be

porazdelitev pare (prva je zajemala samo svetlobo vrumenem spektru in posnela le slike delcev(kavitacijskih struktur na slikah ni mogoèe videti),medtem ko je druga zajemala celoten svetlobnispekter in posnela slike kavitacijskih struktur). Da bidobili informacijo o hitrostnem polju in pripadajoèikavitacijski strukturi na eni sliki, sta bili obe slikizdru�eni (sl. 5).

Hitrostno polje je bilo doloèeno samo spogleda od strani, medtem ko so bile slikekavitacijskih struktur posnete tudi s pogleda odzgoraj (sl. 4) � tok teèe z leve proti desni.

Zaporedje na sliki 4 je sestavljeno izposameznih znaèilnih slik. Vidimo, da ima kavitacijaob sprednji steni (kjer je profil najkraj�i) znaèilnodinamièno obna�anje. Prihaja do trganjakavitacijskega oblaka, ki nato potuje s tokom in izginev obmoèju z vi�jim tlakom. Na drugi strani (kjer jeprofil najdalj�i) je kavitacija ustaljena in ne prihajado trganja oblaka.

Na sliki 5 so prikazana �tiri tipièna stanjav postopku trganja kavitacijskega oblaka za PNVRpri s = 2,0 ter dva poveèana detajla. Po odtrganju(1), tik pred odtrganjem (2) in dva primera medsamim trganjem (3 in 4). Detajla prikazujeta povratnicurek znotraj pritrjene kavitacije (detajl A) invrtinec za odtrganim kavitacijskim oblakom (detajlB). Tok teèe z leve proti desni. Na levih dveh slikah(1 in 2) je mogoèe znotraj pritrjene kavitacije opaziti

Sl. 4. Trganje kavitacijskega oblaka nad profilom PNVR pri kavitacijskem �tevilu s = 2,0

Fig. 4. Cavitation cloud separation on ALE hydrofoil at cavitation number s = 2.0



seen inside the attached part of the cavitation on theleft two pictures. The back flow is detectable from thecavity closure up to 75 % of its length. It can be seenfrom the right two pictures (3 and 4) that it causes theseparation of the cavitation cloud. The vortex remainspresent inside the separated cloud, which travels withthe flow and collapses in the region with higher pres-sure. The cavitation-cloud collapse forms a new re-entrant jet, which causes a new cloud separation andthe process is repeated.

3 NUMERICAL SIMULATION

The commercial code Fluent 6.1.18 was usedto calculate the cavitating flow. It is a 3D structuredmesh code that solves time-dependent Reynolds-averaged Navier-Stokes equations (URANS). Thenumerical model uses an implicit finite-volumescheme, based on a SIMPLE algorithm, associatedwith the multiphase and cavitation model.

3.1 Multiphase model

A single-fluid (mixture phase) approach wasused. The properties of the individual phases areincluded in the single-mixture phase properties. The

znaèilen vrtinec (povratni curek). Povratni tok jeprisoten od konca kavitacije do 75 % v njenoglobino. Z desnih dveh slik (3 in 4) opazimo, da jepovratni tok vzrok za trganje kavitacijskega oblaka.Vrtinec ostane znotraj odtrganega oblaka, ki potujes tokom in izgine v obmoèju z vi�jim tlakom. Kolapskavitacijskega oblaka oblikuje nov povratni curek,ki povzroèi trganje naslednjega kavitacijskegaoblaka.

3 NUMERIÈNA SIMULACIJA

Za simulacijo kavitirajoèega toka smo uporabilikomercialni program Fluent 6.1.18. Program uporabljastrukturirane trirazse�ne mre�e in re�uje sistem èasovnoodvisnih Reynoldsovo povpreèenih Navier-Stokesovihenaèb (RPNSE-URANS). Numerièni model uporabljaposredno metodo konènih prostornin, zasnovan naalgoritmu SIMPLE v povezavi z veèfaznim in kavitacijskimmodelom.

3.1 Veèfazni model

Uporabljen je bil postopek homogenegatoka me�anice. Lastnosti posameznih faz so vkljuèenev lastnosti enofazne me�anice. Gostota (en. 2) in

Sl. 5. Primeri s preizkusi dobljenega hitrostnega polja in kavitacijskih struktur (profil z NVR pri s = 2,0)

Fig. 5. Examples of experimentally obtained flow field and vapour distribution (ALE hydrofoil at s = 2.0)



density (Eq. 2) and viscosity (Eq. 3) of the mixtureare defined with the volume fraction a:

(2)

and

(3).

The continuity (Eq. 4) and momentum (Eq.5) equations for the mixture flow (index m) togetherwith the volume-fraction equation (Eq. 6) for the sec-ondary phase (index k) are solved:

(4)

(5)

(6).

3.2 Turbulence model

The RNG k-e turbulence model was appliedfor solving the transport equations of the turbulentkinetic energy and its dissipation rate.

The results acquired with the use of thestandard RNG k-e turbulence model did not agreewell with the experimental results. A significantbackflow and cavitation-cloud separation seen dur-ing the experiments could not be simulated. Also,the mean cavity-structure length along the hydro-foil was about 50% too small. To improve the simula-tion a modification of the turbulent viscosity wasapplied [9]. In regions with higher vapour-volumefractions (lower mixture densities) a modification ofthe RNG k-e turbulence model was made by reduc-ing the turbulent viscosity of the mixture (Fig. 6):

(7)

(8).

This modification limits the kinetic energyand consequently allows the formation of a re-en-trant jet and the cavitation-cloud separation.

3.3 Cavitation model

The vapour mass fraction is given by thetransport equation:

viskoznost (en. 3) me�anice sta definirani sprostorninskim dele�em pare v me�anici a :

in

Program re�uje kontinuitetno enaèbo (4) ingibalno enaèbo (5) za tok me�anice (indeks m) skupajz enaèbo prostorninskega dele�a (en. 6) za drugofazo (indeks k):

3.2 Turbulentni model

Za re�evanje prenosnih enaèb turbulentnekinetiène energije in njene trosilne hitrosti je biluporabljen turbulentni model RNG k-e.

Rezultati simulacije, pri kateri smo uporabiliobièajni turbulentni model RNG k-e se niso ujemali zeksperimentalnimi rezultati. Pomembnega povratnegatoka, ki smo ga opazovali pri preizkusu ni bilo moèsimulirati. Poleg tega je bila povpreèna dol�inakavitacijske strukture vzdol� profila glede na preizkusza pribli�no 50 odstotkov premajhna. Z namenom,da bi izbolj�ali simulacijo, smo spremenili èlenturbulentne viskoznosti [9]. V obmoèju z velikimdele�em pare (majhne gostote me�anice) smoturbulentni model RNG k-e spremenili z zmanj�anjemturbulentne viskoznosti me�anice (sl. 6):

Sprememba omeji kinetièno energijo inzaradi tega omogoèi nastanek povratnega curka tertrganje kavitacijskega oblaka.

3.3 Kavitacijski model

Masni dele� pare je podan s prenosnoenaèbo:

(1 )m v lr ar a r= + -

(1 )m v lm am a m= + -

( ) ( ) 0m m mvtr r

¶+Ñ × =

¶

r

( ) ( ) ( )T

m m m m m m m m mv v v p v v g Ftr r m r

¶é ù+Ñ × = -Ñ +Ñ × Ñ +Ñ + +ë û¶

vr r r r r r

( ) ( )k k k k mv mta r a r

¶+Ñ =

¶

r&

( )2

t

kf C

mm r

e= × ×

( )( )

( )1 kjer je / where 1

n

m v

v n

l v

f nr r

r r

r r-

-= + >>

-



(9).

Source terms Re and R

c define the vapour

generation (liquid evaporation) and the vapour con-densation, respectively. The source terms are functionsof the local flow conditions (static pressure, velocity)and the fluid properties (liquid and vapour phase den-sities, vapour pressure and surface tension). The deri-vation of the source terms can be found in [3].They are given by:

(10)

- and by:

(11).

where Ce and C

c are empirical constants, k is the

local kinetic energy, g is the surface tension, fv is the

vapour mass fraction, fl is the liquid mass fraction

and fg is the mass fraction of the gases in the water.

4 SIMULATION AND RESULTS

Different types and resolutions of mesheswere tested. A structured C-type mesh with about360000 nodes was eventually chosen. Standard wallfunctions were applied, hence the y+ value liesbetween 30 and 80. The time-step solution wasconsidered converged when the residuals fell below5*10-4. Approximately 40 iterations per time step wereneeded to obtain a converged solution. Theconditions applied for the simulation are thefollowing:

Izvirna èlena Re in R

c podajata nastajanje

(uparjanje kapljevine) in kondenzacijo pare. Èlenasta funkciji lokalnih razmer v toku (statiènega tlaka,hitrosti) in lastnosti tekoèin (gostot kapljevine inpare, uparjalnega tlaka, povr�inske napetosti).Izpeljava èlenov je podana v [3].

Izvorna èlena sta podana z:

- in z:

kjer sta Ce in C

c empirièni stalnici, k lokalna kinetièna

energija, g povr�inska napetost, fv masni dele� pare,

fl masni dele� kapljevine in f

g masni dele� plinov v

vodi.

4 SIMULACIJA IN REZULTATI

Preizku�eni so bili razlièni tipi in gostotemre�. Nazadnje je bila uporabljena strukturirana mre�atipa C s pribli�no 360000 vozli�èi. Ker so bileuporabljene obièajne stenske funkcije, je vrednosty+ le�ala med 30 in 80. Za konvergirano re�itevposameznega koraka smo vzeli stanje, ko so ostankizna�ali manj ko 5*10-4. Za konvergirano re�itevposameznega èasovnega koraka je bilo potrebnihpribli�no 40 iteracij. Pogoji, pri katerih smo izvajalisimulacije, so naslednji:

Sl. 6. Sprememba turbulentne viskoznosti

Fig. 6. Modification of the turbulent viscosity

( ) ( )m m m e cf v f R Rtr r

¶+Ñ × = -

¶

r

( )( )

21 ; pri / when

3v

e e l v v g v

l

p pkR C f f p pr r

g r

-= × × × × × × - - <

( )2; pri / whe

3v

c c l l v v

l

p pkR C f n p pr r

g r

-= × × × × × × >



- Boundary condition: Imposed velocity at inlet andstatic pressure at outlet. Initial transient treatment: Alow velocity is initially applied to the flow field, forwhich no vapour appears. The velocity is then in-creased until the considered operating point is reached.

- Because the experimental cavitation number isbased on upstream pressure, the losses gener-ated in the test section have to be taken into ac-count in the calculation of the numerical cavita-tion number. The desired operating point (cavita-tion number) is reached by comparing the experi-mental and numerical upstream pressures.

- Different time-step values were tested; eventu-ally a time step of 2*10-5 s was used.

- Although the vapour-fluid mixture flow can reachthe speed of sound at relatively low velocities,compressibility effects were neglected in order tosimplify the calculation and decrease the compu-tational time.

4.1 CLE hydrofoil

The instantaneous vapour volume fraction dis-tributions together with the velocity field are shown inFig. 7. The flow is from left to right. Four cases and twodetails are shown (like in Fig. 5 � after the cloud separa-

- Robni pogoji: hitrost na vstopu in statièni tlak naizstopu. Na zaèetku simulacije smo predpostavilimajhno hitrost toka, pri kateri kavitacije �e ni.Hitrost toka smo nato veèali, dokler nismo dosegli�elenih razmer.

- Ker je bilo kavitacijsko �tevilo pri preizkusudoloèeno na mestu pred profilom, smo morali zadoloèitev numeriènega kavitacijskega �tevilaupo�tevati tlaène izgube, ki nastanejo v testnemodseku. �elene razmere (kavitacijsko �tevilo) smodosegli iterativno s primerjavo eksperimentalnegain numeriènega tlaka na mestu pred profilom.

- Preizkusili smo veè dol�in èasovnega koraka; nakoncu smo uporabili korak dolg 2*10-5 s.

- Èeprav lahko me�anica pare in kapljevine dose�ezvoèno hitrost �e pri razmeroma majhnih hitrostih,so bili uèinki stisljivosti, zaradi poenostvitvesimulacije in zmanj�anja raèunskega èasa,zanemarjeni.

4.1 Profil s PVR

Na sliki 7 so prikazane trenutne porazdelitveprostorninskega dele�a pare skupaj s pripadajoèimhitrostnim poljem. Tok teèe z leve proti desni. Kakorna sliki 5 so prikazani �tirje primeri (po odtrganju

Sl. 7. Primeri numeriène napovedi hitrostnega polja in porazdelitve pare (profil s PVR pri s = 2,0)

Fig. 7. Examples of numerically predicted flow and vapour field (CLE hydrofoil at s = 2.0)



tion (1), just before cavitation-cloud separation (2) andtwo pictures showing the situation during the separa-tion itself (3 and 4)). A single vortex (re-entrant jet) can beseen in the case before cloud separation (bottom-leftpicture). The back flow is present only inside the vapourcavity. It penetrates the cavity up to 75% of its length. Inthe case after the cloud separation, two vortices can bedetermined. One inside the attached vapour cavity (re-entrant jet) and the other one downstream of the sepa-rated cavitation cloud (forming the re-entrant jet).

It is obvious that in all cases the numeri-cally predicted velocity field shows a good correla-tion with the experimental results (Fig. 5 and 7).

A sequence of instantaneous vapour volumefraction distributions for the CLE hydrofoil at cavitationnumber 2.5 is shown in Fig 8. Flow is from left to right.The time delay between two successive images is 0.4ms. It is clear that the attached cavitation first grows,until the separation of a cavitation cloud occurs. Theattached cavitation part then decreases while the cavita-tion cloud travels with the flow and collapses in a higherpressure region. Meanwhile, the attached cavitationstarts to grow again and the process is repeated.

A typical �horseshoe� vapour structurecould be observed during the experiment. Fig. 9shows two experimentally recorded (left) and nu-merically predicted (right) cavitation structures. Theflow is from left to right.

4.3 ALE hydrofoil

A 3-dimensional simulation of cavitatingflow around a hydrofoil with an asymmetric leading

oblaka (1), tik pred odtrganjem oblaka (2) ter dvestanji med trganjem oblaka (3 in 4)) ter dva detajla.Pred odtrganjem oblaka (leva spodnja slièica) lahkovidimo vrtinec (povratni curek). Povratni tok, ki jeznotraj pritrjene kavitacije, zasledimo do 75 % v njeniglobini. V stanju po odtrganju kavitacijskega oblakalahko razloèimo dva vrtinca. Prvega znotraj pritrjenekavitacije (povratni curek), drugega pa nekoliko zaodtrganim kavitacijskim oblakom (nastajajoèipovratni curek).

Oèitno je, da se v vseh primerih numeriènonapovedano hitrostno polje dobro ujema spreizkusnimi rezultati (sl. 5, 7).

Na sliki 8 vidimo zaporedje trenutnihporazdelitev prostorninskega dele�a pare za profils PVR pri kavitacijskem �tevilu 2,5. Tok teèe z leveproti desni. Èasovni korak med slièicami je 0,4 ms.Vidimo, da se pritrjena kavitacija zveèuje, dokler sene odtrga kavitacijski oblak. Pritrjena kavitacija senato sprva manj�a, odtrgani oblak pa potuje s tokomin izgine v obmoèju z vi�jim tlakom. Med tem sezaène pritrjena kavitacija zopet veèati in postopekse ponovi.

Med preizkusom smo lahko opazili znaèilnepodkvaste kavitacijske strukture. Na sliki 9 vidimodve s preizkusi posneti (levo) in numeriènonapovedani (desno) kavitacijski strukturi. Tok teèe zleve proti desni.

4.3 Profil z NVR

Narejena je bila trirazse�na simulacijakavitacije na profilu s po�evnim vpadnim robom. Na

Sl. 8. Zaporedje trenutnih porazdelitev prostorninskega dele�a pare pri s = 2,5 (pogled od strani)

Fig. 8. Sequence of instantaneous vapour volume-fraction distributions at s = 2.5 (side view)



edge was performed. Instantaneous views of the 0.1vapour volume-fraction isosurface are shown (Fig.10). The flow is from left to right. The time delaybetween successive images is 0.6 ms. Similar to theexperiment [10] a significant dynamic cavitation be-haviour can be seen near the front wall (pulsationsof the cavitation region with the separation of a cavi-tation cloud), while the cavitation at the rear wallremains steady (with no cloud separation).

4.4 The mean cavity length

The experimentally measured mean cavitystructure length along the hydrofoil was compared to the

sliki 10 vidimo trenutne poglede na izopovr�ino zvrednostjo 0,1 prostorninskega dele�a pare. Tok teèez leve proti desni. Èasovni korak med slièicami je 0,6ms. Podobno kakor pri preizkusu [10] vidimo vobmoèju blizu prednje stene oèitno dinamiènoobna�anje kavitacije (utripi kavitacije s trganjemkavitacijskega oblaka). Ob zadnji steni ostajakavitacija ustaljena (brez trganja kavitacijskegaoblaka).

4.4 Povpreèna dol�ina kavitacije

Primerjali smo eksperimentalnoizmerjeno in numerièno napovedano povpreèno

Sl. 9. Eksperimentalni posnetek (levo) in numerièna napoved (desno) podkvaste kavitacijske struktureFig. 9. Experimental image (left) and numerical prediction (right) of the �horseshoe� cavitation structure

Sl. 10. Numerièno napovedan èasovni razvoj kavitacijske strukture pri s = 2,0

Fig. 10. Numerically predicted time evolution of the cavitation structure at s = 2.0



numerically predicted one (Fig. 11). The mean cavitationstructure was determined by averaging 50 instantaneousimages of the cavitation. For the CLE hydrofoil the lengthwas determined in the middle of the hydrofoil, while forthe ALE hydrofoil the length corresponds to the positionof the laser light plane (5 mm from the front wall � wherethe hydrofoil is the shortest).

The results show that the mean cavitylength grows when the cavitation number isdecreased. The trend is correctly predicted in thenumerical simulation. Since the determination of theboundary of the mean cavitation structure(experimental or simulated) is relatively inaccurate,the results of numerical simulation are acceptable.

4.5 The dynamic behaviour of cavitation

Experimental and numerical values for vapour-cloud shedding frequencies for different operating condi-tions are presented in Fig. 12. The experimental sheddingfrequencies for the CLE hydrofoil were obtained from simi-lar past measurements of the dynamic pressures on thesurface of the hydrofoils ([11] and [12]). The sheddingfrequencies for the ALE hydrofoil were deduced from theimages of a high-speed movie (approximately 2800 fps).Frequencies for different cavitation numbers from 1.8 to2.7 were measured. The numerical frequencies were de-duced from a simulation of about 10 cycles.

It can be seen (Fig. 12) that the numericallypredicted and experimental shedding frequencies showthe same trend. The shedding frequency decreases whenthe cavitation number is decreased. In the case of theCLE hydrofoil the predicted frequencies are a little higher,

dol�ino kavitacijske strukture vzdol� profila (sl.11). Povpreèno kavitacijsko strukturo smodoloèili s povpreèenjem 50 trenutnih slikkavitacije. Za profil s PVR smo dol�ino doloèilina sredini profila, za profil z NVR pa na mestulaserske ravnine (5 mm od sprednje stene � kjerje profil najkraj�i).

Rezultati ka�ejo, da se povpreèna dol�inakavitacije poveèuje, ko manj�amo kavitacijsko�tevilo. Odvisnost je pri numerièni simulaciji pravilnonapovedana. Glede na to, da je doloèanje povpreènevelikosti kavitacijske strukture (eksperimentalne alinumeriène) razmeroma nenatanèno, so rezultatinumeriène simulacije sprejemljivi.

4.5 Dinamièno obna�anje kavitacije

Na sliki 12 so prikazani eksperimentalni innumerièni rezultati vrednosti frekvenc trganjakavitacijskega oblaka v razliènih razmerah.Eksperimentalne vrednosti frekvenc trganja za profils PVR smo dobili s prej�njih meritev dinamiènih tlakovna povr�ini profila ([11] in [12]). Frekvence trganja zaprofil z NVR smo dobili s filmov, ki smo jih posneli shitro kamero (pribli�no 2800 slièic na sekundo). Merilismo frekvence za razlièna kavitacijska �tevila med 1,8in 2,7. Numeriène frekvence trganja smo izraèunali izsimulacije v trajanju pribli�no 10 ponovitev.

Vidimo (sl. 12), da imajo numeriènonapovedane in s preizkusi doloèene frekvencetrganja enak znaèaj. Frekvenca trganja se manj�a,ko manj�amo kavitacijsko �tevilo. V primeru profilas PVR so napovedi frekvence nekoliko previsoke,

Sl. 11. Numerièno napovedana in s preizkusi doloèena povpreèna dol�ina kavitacijeFig. 11. Numerically predicted and experimentally determined mean cavity length



v primeru profila z NVR pa nekoliko manj�e.Frekvence trganja niso popolnoma nespremenljive.Nihanja v vrednosti lahko ocenimo na +/- 10 % odsrednje vrednosti za profil s PVR in +/- 18 % zaprofil z NVR. Ker je dinamika kavitacije postopek,ki je za simulacijo zelo zahteven, imamo lahkosimulacijo za sprejemljivo, èe so simulirane ineksperimantalne frekvence trganja enakegavelikostnega reda.

5 SKLEPI

Narejeni sta bili numerièna ineksperimantalna �tudija kavitirajoèih tokov vrazliènih razmerah na dveh razliènih profilih.

Posneli smo slike kavitacijskih strukturin doloèili povpreène dol�ine kavitacije. Zuporabo razmeroma nove metode PIV-LIF nam jeuspelo doloèiti hitrostno polje zunaj in znotrajkavitacije. S filmov, ki smo jih posneli s hitrokamero, smo doloèili frekvence trganjakavitacijskih oblakov.

Za simulacijo kavitacije smo uporabili tr�niprogram za raèunalni�ko dinamiko tekoèin Fluent6.1.18. Rezultati numeriène simulacije ka�ejo dobroujemanje z eksperimentalnimi meritvami. Oblikekavitacijskih struktur (na primer podkvasta struktura)so bile pravilno napovedane. Napovedanepovpreène dol�ine kavitacije se razmeroma dobroujemajo z eksperimentalno doloèenimi. Hitrostnopolje znotraj kavitacije in zunaj nje je zelo dobrosimulirano. Pravilno je napovedan povratni curek, kipovzroèi trganje kavitacijskega oblaka. Prav tako

while for the ALE case the predicted frequencies arelower. The shedding frequencies are not perfectlyconstant. Fluctuations can be estimated in a range of+/- 10 % from the mean value for the CLE hydrofoil and+/- 18 % for the ALE hydrofoil. Cavitation dynamics is aparticularly complicated process to simulate. The simu-lation is considered acceptable if numerically simulatedshedding frequencies are of the same order of magni-tude as the experimentally obtained ones.

5 CONCLUSIONS

Numerical and experimental investigationsof different cavitating flow conditions on two differ-ent hydrofoils were performed.

Images of the cavitation structures wererecorded and the mean cavitation lengths were de-termined. With the use of a relatively new PIV-LIFmethod we were able to determine the velocity fieldsoutside and inside cavitation. From films that wererecorded by high-speed camera the cavitation-cloudshedding frequencies were determined.

A commercial CFD program, Fluent 6.1.18, wasused for the simulation of the cavitation. The results ofthe numerical simulation show a good correlation withthe experimental measurements. The shapes of the cavi-tation structures (for example, the �horseshoe� shape)were correctly predicted. The predicted mean lengthsof the cavities agree relatively well with the experiment.The velocity fields inside and outside the vapour cav-ity are particularly well simulated. The backflow phe-nomenon that causes cavitation-cloud separation wascorrectly predicted. The simulation also shows good

Sl. 12. Eksperimentalne in numeriène frekvence trganja kavitacijskega oblakaFig. 12. Experimental and numerical shedding frequencies

PPVR pres.CLE-exp.PPVR num.CLE-num.PNVR pres.ALE-exp.PNVR num.ALE-num.



simulacija dobro napove dinamièno obna�anjekavitacijskih oblakov. Simulirane frekvence trganjakavitacijskih oblakov so za primer profila s PVRnekoliko vi�je za primer profila z NVR pa nekolikoni�je od eksperimetalno doloèenih.

Naslednji korak je izbolj�ava simulacije zupo�tevanjem uèinkov stisljivosti. Izziv je tudizmanj�anje raèunskega èasa simulacije, ki je presegal300 ur na osebnem raèunalniku s procesorjemPentium IV � 2,4 GHz (glej [13]).

Predstavljeni rezultati obetajo mo�nostnapovedi dinamiènih uèinkov kavitacije, na primerkavitacijske erozije, z izkljuèno numeriènimi orodji.

empirièna konstantaempirièna konstantasilamasni dele�masni dele� paremasni dele� plinovte�nostni pospe�ekturbulentna kinetièna energijamasni toktlaktlak uparjanjahitrost me�anicehitrost k-te fazeprostorninski dele� k-te fazedisipacijska hitrost turbulencepovr�inska napetostviskoznost me�anicegostota kapljevinegostota k-te fazegostota me�anicegostota parekavitacijsko �tevilo

prediction of the dynamic behaviour of cloud cavi-tation. The simulated frequencies of vapour-cloudshedding are generally a bit higher than the experi-mentally obtained ones for the CLE hydrofoil andlower for the ALE hydrofoil.

The next step is to upgrade the simulationby considering the compressibility effects. Anotherchallenge is to reduce the computational time, whichexceeded 300 hours for a simulation using a PentiumIV � 2.4 GHz processor [13].

The presented results promise the possi-bility of the prediction of dynamic cavitation effects,like cavitation erosion, using only CFD.

6 SIMBOLI6 NOMENCLATURE

Ce

- empirical constantC

c- empirical constant

F N forcef - mass fractionf

v- vapour mass fraction

fg

- gas mass fractiong m/s2 gravitational accelerationk m2/s2 turbulence kinetic energym kg/s mass flowp Pa pressurep

vPa vapour pressure

vm

m/s mixture velocityv

km/s k-th phase velocity

ak

- k-th phase volume fractione m2/s3 turbulence dissipation rateg N/m surface tensionm

mPa s mixture viscosity

rl

kg/m3 liquid densityr

kkg/m3 k-th phase density

rm

kg/m3 mixture densityr

vkg/m3 vapour density

s - cavitation number

7 LITERATURA7 REFERENCES

[1] �irok, B., M. Dular, M. Novak, M. Hocevar, B. Stoffel, G. Ludwig, B. Bachert (2002) The influence ofcavitation structures on the erosion of a symmetrical hydrofoil in a cavitation tunnel; Journal of Me-chanical Engineering, vol. 48, no. 7, Ljubljana, Slovenia.

[2] Kubota, A., H. Kato, H. Yamaguchi (1992) A new modelling of cavitating flows: A numerical study ofunsteady cavitation on a hydrofoil section; Journal of Fluid Mechanics, vol. 240, 59-96.

[3] Sauer, J. (2000) Instationär kavitierende Strömungen - Ein neues Modell, basierend auf Front Capturing(VoF) und Blasendynamik � PhD thesis; Universität Karlsruhe (TH), Karlsruhe.



[4] Singhal, A.K., Li H., M.M. Atahavale, Y. Jiang Y. (2002) Mathematical basis and validation of the fullcavitation model; Journal of Fluids Engineering, 124, 617-624.

[5] Dular, M., B. �irok, B. Stoffel, B. Bachert, R. Bachert (2003) Numerical simulation of cavitation on a singlehydrofoil in a cavitation tunnel; Slovensko dru�tvo za mehaniko, Kuhljevi dnevi, Zreèe.

[6] Coutier-Delgosha, O., R. Fortes-Patella, J.L. Reboud (2001) Evaluation of turbulence model influence onthe numerical simulations on unsteady cavitation; Proceedings of ASME FEDSM 01, 2001 ASME FluidsEngineering Division Summer Meeting, New Orleans, Louisiana, May 29-June 1.

[7] Hofmann, M., H. Lohrberg, G. Ludwig, B. Stoffel, J.-L. Reboud, R. Fortes-Patella (1999) Numerical andexperimental investigations on the self - oscillating behaviour of cloud cavitation - Part 1: Visualisation;Proceedings of the 3rd ASME / JSME Joint Fluids Engineering Conference, San Francisco CA.

[8] Lohrberg, H., B. Stoffel, R. Fortes-Patella, J.L. Reboud (2001) Numerical and experimental investigationson the cavitation flow in cascade of hydrofoils; Procedings of the Fourth International Symposium onCavitation, California Institute of Technology, Pasadena, California, USA, 20-23 June,

[9] Reboud, J.L., B. Stutz, O. Coutier (1998) Two-phase flow structure of cavitation: experiment and modellingof unsteady effects: Third International Symposium on Cavitation, Grenoble, France.

[10] Bachert, R., B. Stoffel, R. Schilling, M. Frobenius (2003) Three-dimensional, unsteady cavitation effectson a single hydrofoil and in a radial pump � Measurements and numerical simulations; Part one: Experiments;Procedings of the Fifth International Symposium on Cavitation, Osaka, Japan, November 1-4.

[11] Hofmann, M. (2001) Ein Beitrag zur Verminderung des erosiven Potentials kavitierender Stömungen �PhD thesis; Technische Universität Darmstadt, Darmstadt.

[12] Boehm, R. (1998) Erfassung und hydrodynamische Beeinflussung fortgeschrittener Kavitationszuständeund ihrer erosiven Aggressivität � PhD Thesis; Technische Universität Darmstadt, Darmstadt.

[13] Frobenius, M., R. Schilling, R. Bachert, B. Stoffel (2003) Three-dimensional, unsteady cavitation effectson a single hydrofoil and in a radial pump � Measurements and numerical simulations; Part two: Numericalsimulation; Proceedings of the Fifth International Symposium on Cavitation, Osaka, Japan, November1-4.

Naslova avtorjev:Matev� Dularprof.dr. Brane �irokUniverza v LjubljaniFakulteta za strojni�tvoA�kerèeva 61000 [email protected]@fs.uni-lj.si

Rudolf Bachertprof. dr. Bernd StoffelTehnièna univerza DarmstadtMagdalenenstrasse 4D-64289 Darmstadt, Nemèija

[email protected]@tfa.maschinenbau.tu-darmstadt.de

Authors� Addresses: Matev� DularProf.Dr. Brane �irokUniversity of LjubljanaFaculty of Mechanical Eng.A�kerèeva 61000 Ljubljana, [email protected]@fs.uni-lj.si

Rudolf BachertProf.Dr. Bernd StoffelDarmstadt University of Tech.Magdalenenstrasse 4D-64289 Darmstadt, Germany

[email protected]@tfa.maschinenbau.tu-darmstadt.de

Prejeto: 29.6.2004

Sprejeto: 2.12.2004

Odprto za diskusijo: 1 letoReceived: Accepted: Open for discussion: 1 year

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