chapter 9: vortext cavitation

Chapter 9

Vortex cavitation

Objective: Description of the structure ofa cavitating vortex and its appearance on shippropellers

Vortices are important for ship propellersbecause the pressure in the core is low andthus cavitation may occur. Vortex cavitationis not erosive when it is not excessive, but it isa source of noise. And noise is used to detectships and is thus important for navy ships.Radiated noise was and is used to find thelocation of ships. Since low frequency noisereaches longer distances these frequencies weredetected with towed arrays. This detectionmethod has been replaced by detection bysatellites. However, for close range detectionnoise is still very important, because the maintarget of torpedoes is a noise source. Tipvortex inception often determines the speed atwhich the propeller starts to radiate cavitationnoise, which remains important for navy ships.

There is also increased interest in cavitationinception of commercial ships because ofthe suspected interference of the radiatednoise with marine life. Although very littleis known about the relation between noiseand biological damage, there is a tendency toreduce noise radiation. Especially cruise shipswant are keen on having a green label, so theywant to operate without cavitation.

As a whole, tip vortex cavitation is there-

fore important for navy ships and for researchvessels which have to operate without self-noise. Developed tip vortex cavitation isnot very critical in general, except when itis excessive. The dynamics of cavitating tipvortices are still poorly understood, but itis likely that these dynamics play a rle inbroadband noise radiated in some cases andin cavitation induced pressure fluctuations.These induced pressures may be influenced bythe interaction between sheet cavitation andthe cavitating tip vortex.

Inception of vortices is an important issuefor model testing, because cavitation incep-tion of a vortex is strongly dependent on theReynolds number. As a result, tip vortexcavitation occurs much later at model scalethan at full scale, and this can lead to largeerrors in the prediction of cavitation.

To understand vortex cavitation it is nec-essary to understand the structure and rep-resentation of the non-cavitating vortex first.A vortical flow structure can be described interms of flow velocities by the shape of thestreamlines. A vortical structure is a circulat-ing flow in a certain plane in a fluid, whichmeans that the flow lines are closed. Theclosed streamlines have to contain only fluid,because when there is a body in the closedstreamlines there is not a vortex but we re-gard it as circulation around the body.

65

66 G.Kuiper, Cavitation in Ship Propulsion, June 22, 2012

9.1 Vortex, vorticity and

vortex induced veloci-

ties

It is, however, more convenient to describe theflow in terms of vorticity instead of in termsof velocities. Then a vortex is defined as aconcentration of vorticity.

9.1.1 Vorticity

Vorticity is defined as rotation of fluid ele-ments. Vorticity is generated by friction forcesin the fluid, as follows from the considerationof forces on an infinitesimal small fluid ele-ment. In an inviscid fluid there are only pres-sure forces acting on a fluid element (Fig.9.1).The fluid element will deform due to theseforces, but will not rotate. Without rotationthe equations of motion are described by theLaplace equation. (Note: When there is rota-tion at time t in an inviscid fluid, the rotationwill not change due to the pressure forces, butit can be changed by fluid accelerations in thez-direction. The governing equation of motionis the Euler equation). Rotation is only gener-ated by frictional forces, as shown in Fig.9.2.The fluid element at time t+dt is not only de-formed, but also rotated, which is measuredby the average angle of the bisectrices of theelement.

Vorticity is defined as the curl of the equa-tion of motions, and in two dimensions it iswritten as

ω =∂u

∂y− ∂v

∂x(9.1)

which is twice the angular velocity of thefluid element in Fig.9.2. Vorticity is definedin a plane and it is thus a vector,which meansthat it has not only a strength but also adirection.

When a vortex is defined in terms ofvorticity, a vortex is a concentration of

Figure 9.1: Deformation of a fluid element dueto pressure forces

Figure 9.2: Defprmation and rotation of a fluidelement due to pressure and friction forces

vorticity in one direction in a body of fluid.In Fig. 9.3 a cross section of the fluid per-pendicular to the direction of the vorticityis given. The strength of the vorticity hasbeen indicated by the intensity of the color.In the center there is a strong vorticity,around the center the vorticity is lower andin the region between streamlines C and Dthere is no vorticity, as in the region outside D.

A measure of the strength of a vortical struc-ture is the circulation Γ. This is the integral ofthe velocity over a closed streamline (Stokes’

June 22, 2012, Vortex cavitation67

Figure 9.3: The structure of a vortex

law).

Γ =

∫V ds (9.2)

where s is the closed streamline. The circu-lation also is the integral of the vorticity overthe area of the closed streamline, so

Γ =

∫ωda (9.3)

in which a is the area inside the streamline.As long as there is vorticity between variousstreamlines the circulation will change withthe streamline.

When the vorticity between two contoursis zero (as between streamlines C and D inFig. 9.3) the circulation of both contours isthe same. The circulation around stream-lines A, B and C will be increasing. Thestrength of a vortex Γ is now defined byits maximum circulation. The location of avortex is generally taken somewhere in thecenter of the region with vorticity. Sincevorticity has the tendency to roll-up into acylindrical shape this is often not very difficult.

Note that the description above assumesthat there is one vortex under consideration.

The situation is much more complex in e.g.a boundary layer. There the vorticity occursin many directions and scales and it canbe difficult to distinguish coherent vorticalstructures (e.g.[26]). This can also occur inthe collapse region of a cavity, where multiplevortices are interacting with each other. Inour discussions we will focus, however, onsingle vortices, eventually interacting withdistinct other vortices.

9.1.2 Vortex induced velocities

A vortex can now be divided into two regions.The first region is the vortex core, which is theregion with vorticity. This vortex core is of-ten indicated as the vortex itself. The secondregion is the region outside the vortex core,where there is no vorticity. These velocitiesare therefore called vortex induced velocities,although it would be better to talk about vor-tex core induced velocities. The term inducedis used because the relation between the vor-tex core and the velocities outside the core issimilar as in a magnetic field. The mecha-nism which establishes this induction in a flowfield is viscosity. It is important to note thata vortex consists both of a vortex core andof induced velocities outside the core. Bothare components of one vortex and are directlylinked to each other. There is also no strictlimit to the extension of a vortex. Its exten-sion is infinite, although at larger distancesfrom the core the induced velocities becomenegligibly small.//

In an incompressible flow without viscositythe circulation in a body of fluid remainsconstant (Thomson’s law). The body of fluidcan, however be transformed and translated.This means that the vorticity can be changedover time. An example is the stretchingof a vortex when the vortex exists in anaccelerating flow. The strength of the vortexremains the same, but the radius decreases,resulting in an increase of the local vorticity


Figure 9.4: Vorticity downstream of a liftingwing

in the core. The location of the vortex isdefined by the translation of the fluid element,which means that a vortex moves with the(local)flow.

9.1.3 The source of vorticity

As has been mentioned vorticity is generatedin a boundary layer, where large velocity gra-dients occur. So viscosity is a necessity for thegeneration of vorticity. A vortex is defined as aconcentration of vorticity in free flow. So therehas to be separation to generate a vortex. Ex-amples are the wake of a wing with a certainspanwise loading distribution (Fig. 9.4). Nearthe trailing edge of the wing there is separation(the Kutta condition in potential flow) and aplane of vorticity occurs in the separated flowat the trailing edge. Separation can also oc-cur in flow direction, as is often the case withbilge vortices (Fig. 9.5). Here also vorticity isreleased in the free flow. In both examples thevorticity is in a plane, but will soon roll-up, aswill be discussed later.

Figure 9.5: Vorticity generated by longitudinalseparation in a boundary layer

Figure 9.6: Vortex roll-up of two separate vor-tices

9.1.4 Vortices behind a wing

Co-rotating vortices have the tendency to rollup into a more or less cylindrical shape. Thiscan be explained by considering two schematicvortices A and B as in Fig. 9.6.

Vortex A will follow the streamline of vor-tex B. Vortex B will follow the streamline ofvortex A. As a result they will rotate aroundeach other and eventually form a single vortex.

A well known example is the roll-up ofthe vortex plane behind a wing, as given inFig. 9.4. This plane develops as in Fig 9.7.The ends roll up and develop into a tip vortex,in which the core has a spiral structure. Theresult is a tip vortex with a vorticity distribu-


Figure 9.7: Vortex roll-up behind a wing

tion and strength depending on the distancefrom the wing.

To get an idea about the amount of roll-upbehind a wing we will consider an wing with anelliptical contour, because its loading can betreated analytically. The spanwise circulationdistribution is

Γ(x) = Γ0(1− x2)0.5 (9.4)

where x is the spanwise position, made non-dimensional with the semi-span and Γ0 is themaximum circulation at the center of the wing.

At the trailing edge of the wing a free vor-tex sheet is formed with a strength of ω(x) =δΓ(x)δ(x)

The strength of the trailing vortex sheetover the span is then

ω(x) = −Γ0x(1− x2)−0.5 (9.5)

A problem in calculations is that thestrength at the tip is infinite, but the inte-gral over a finite distance from the tip to thetip is finite. So when the trailing vortex is dis-cretized, the strength of the last element at thetip remains finite. An example of the develop-ment of a trailing vortex behind an ellipticalwing is given by Moore [48].

Figure 9.8: Spanwise distribution of trailingvortices behind a wing (Moore 1974)

He divided the vortex sheet in 60 discretevortices of equal strength γ. The distance ofthese vortices can then be found from Rx =γ

ω(x). Initially these vortices are widely spaced

near the center, and become more denselyspaced towards the tip. Figure 9.8 shows theposition of these vortices in a plane at 5% ofthe half wing span downstream of the initialposition. An approximation of the tip vortexstrength due to roll-up was already given byKaden [29] as

Γ(tip) = gy1/3Γ0 (9.6)

for y smaller than 0.1. Here y is the distancebehind the wing, made non-dimensional withthe semi-span. The value of g is close to 1. Aty=0.1 this means that 46% of the total trailingvorticity is present in the tip vortex. Moorecalculated complete roll-up at about y = 10,but at y = 1 already 80% of the circulation wasrolled-up. In case of a propeller the semi-spanof the wing is about 0.7R (R is the propellerradius). So with a similar loading as an ellip-tic wing 80 percent of the trailing vorticity isrolled-up within a distance of 0.7R behind thetip. In terms of a rotating propeller blade thisis within a blade rotation of 40 degrees.

9.2 Modeling the veloci-

ties in a cylindrical

vortex

In a vortex the main velocity component isperpendicular to the vortex center. Therefore


two-dimensional vortex models are used todescribe the velocity field. Some classicalsimple models will be discussed now. The aimis to show the main properties of a vortex.The accuracy of these models is very limited.

Assume a vortex with all vorticity concen-trated in an infinitely small circle in a planeperpendicular to the vortex. The strength ofthe vortex is Γ. Now let the radius of the vor-tical region go to zero while maintaining thestrength of the vortex. This leads to a singularbehavior in the center of the vortex, becausethe velocities become infinite. Also the notionof vorticity and its distribution loses its mean-ing since al vorticity is contained in one point.Such a velocity field is called a Rankine vor-tex (although sometimes a vortex with a solidcore of finite diameter is called the same). Theflow field around such a vortex is now deter-mined by Γ alone. Since the circulation at ev-ery radius is the same, the tangential velocityaround the vortex is

Vr =Γ

2πr(9.7)

The flow away from the center has a moder-ate velocity gradient, so the effect of viscositycan be neglected, at least over a short timespan. Since there is no vorticity, the outerflow of a vortex is a potential flow field. Thismeans that the flow fields of several vorticescan be added and the outer flow of Fig. 9.3can thus be described by a distribution ofRankine vortices.

It is clear that close to the vortex this modelis not physically realistic, as is indicated bythe fact that the flow velocity in the core of aRankine vortex is infinite.

Consider now a situation in which the vor-ticity is distributed in the center of the vor-tex, as in Fig.9.9. Symmetry requires that inthe center of the vortex the tangential velocityshould be zero.

Figure 9.9: The velocity distribution in acylindrical vortex

In the outer region there is no vorticityand the Rankine velocity distribution (eq.9.7)exists. The strength of the vortex is theintegral of the vorticity, so the strength of thevortex is the circulation at radius B. Insideradius B the velocity gradient becomes sohigh that viscosity becomes more important.It will slow down the tangential velocity. Thisresults in a certain distribution of vorticity inthe center of the vortex, which correspondswith a velocity distribution inside radiusB. When the vortex core is defined as theregion with vorticity, the vortex core radius isradius B. From radius B inward the velocitydistribution will deviate from the Rankinedistribution because the circulation decreasesat smaller radii (because there is vorticityin the radius between A and B the integral


Figure 9.10: The velocity distribution in a vor-tex with a solid core rotation

decreases with decreasing radius). This leadsto zero tangential velocity at the vortex center.

At a radius of about 1.12√

4νt the tangen-tial velocity will have a maximum (radiusA). This radius is often defined as the vis-cous core radius. Strictly speaking this isconfusing, because the vorticity distributionbetween radius A and B is also generated byviscosity. But radius A is used as the coreradius because it can be determined from themeasured velocity field.

A simplified case is when outside radiusA there is no vorticity. Such a case is givenin Fig. 9.10. With a linear distribution ofvorticity inside radius B the core rotates likea solid body. It is clear that this situationis only a crude approximation of the real.The velocity distribution as in Fig. 9.10 issometimes also called a Rankine vortex.

A more realistic velocity distribution is thevelocity distribution which is found from theNavier Stokes equation for laminar flow, start-ing from a Rankine vortex ([43]). In the center

of the vortex the velocity gradients are hugeand viscosity will cause an outward dissipa-tion of vorticity. After some time the tangen-tial velocity distribution which is obtained hasthe shape of Fig. 9.9 and is formulated as:

v(r) =Γ

2πr[1− e−

r2

4νt ] (9.8)

This illustrates the asymptotic nature of thevorticity distribution in radial direction.

In the foregoing the outer flow of a vortexhas been considered as irrotational (no vor-ticity). The question is if that corresponds tophysical flows because there is a velocity gradi-ent in radial direction in the outer flow, whichwill generate vorticity. Only when the viscos-ity is negligible, irrotational flow is possible.Irrotational flow means that the flow particlesdo not rotate, and a fluid particle will thusmove around the vortex center without chang-ing its position. This is illustrated by the fluidparticles A and B in Fig. 9.11. When theseparticles are in position A1 and B1 their ori-entation is the same as in position A2 and B2.This seems counterintuitive, because a bodyis expected to rotate around the vortex core.This, however, is still the case, because the ro-tation refers to infinitely small fluid particles.A body with extension AB will still rotate dueto the velocity gradient between the two radiiof particles A and B. So even though the ir-rotational outer flow is an approximation (thereal fluid is viscous and a very slight rotationwill occur), the approximation of the transla-tion velocities is still accurate.

9.3 The pressure distribu-

tion in a vortex

The pressure distribution in the center of avortex is lower than in the surrounding fluidbecause of the centrifugal effects of the rotat-ing fluid. In a cylindrical vortex this can easilybe derived from the force equilibrium on a fluid


Figure 9.11: The motion of fluid particles andof bodies in an irrotational flow around a vor-tex

particle in the rotating flow. A rotating parti-cle with follows a cylindrical path around thevortex core is subject to a centrifugal force ona fluid element ρrdrdφ, which has to be com-pensated by the pressure force in radial direc-tion

∂p

∂r= ρ

v2(r)

r(9.9)

In the case of a Rankine vortex as in eq. 9.7,integration over r from a radius a r = a tor =∞ results in

p∞ − p(a) =ρΓ2

4π2a2(9.10)

In the center of the vortex a = 0 and thepressure difference goes to infinity, so the min-imum pressure of a simple rankine vortex ismeaningless. With a solid core rotation, as inFig. 9.10 the velocity distribution inside thecore is

v(r) =Γr

2πa2v

(9.11)

the pressure drop in the core is

pmin − pa =ρΓ2

4π(9.12)

and thus independent of the viscous core ra-dius.The minimum pressure in the vortex center is

p∞ − pmin =ρΓ2

4π2av2(9.13)

in which av is the core radius, which isuniquely defined in this case.

In case of a velocity distribution as inFig. 9.9 the radial velocity distribution is givenby equation 9.8. In a similar way the minimumpressure in the vortex core can be calculatedanalytically to be

p∞ − pmin =ρΓ2

4π2

ln(2)

4νt(9.14)

When the maximum velocities are kept thesame (av = 1.12

√4νt) the minimum pressure

of eq. 9.14 differs from eq. 9.13 only by a factorln(2) ∗ 1.122 = 0.87. Because of the presenceof the time in eq. 9.14, it also gives a measureof the increase of the minimum pressure withtime.

9.4 The structure of a cav-

itating vortex

The reason for the focus on the minimum pres-sure in the vortex core is cavitation inception.Let us assume that when the pressure is lowerthan the vapor pressure water changes into va-por. The volume of vapor is so much largerthan the volume of water that the amount ofwater which evaporates can be neglected. Thesituation of the cavitating vortex is then as inFig. 9.12. The vortical fluid is moved outwards


Figure 9.12: The velocity distribution in a cav-itating vortex

Figure 9.13: The velocity distribution in a cav-itating vortex with larger cavitating core

and a vapor region occurs in the vortex centerwith a radius ac.

When the cavitating radius increases the ra-dius of the vortical region increases also. Be-cause the total amount of vorticity is constantthe thickness of the vortical region decreasesrapidly, as sketched in Fig. 9.13.

Ultimately the vortical region will becomesmall and the cavitating vortex has the sameshape as the Rankine vortex, but with a va-porous core. Using eq. 9.9 a relation betweenthe cavitating radius, the vortex strength andthe outside pressure is found for that case:

Figure 9.14: The cavitating core of a tip vortexbehind a propeller

p∞ − pv =0.5ρΓ2

4π2a2c

(9.15)

Note that a simple replacement of the non-cavitating vortex core with a vaporous core,as in eq. 9.15, does not conserve energy. Itassumes that the energy of the non-cavitatingvortex core simply disappears. To conserveenergy the circulation of the vortex aroundthe cavitating core has to increase.

Equation 9.15 shows that at a certain out-side pressure P∞ the cavitating radius is pro-portional to the vortex strength. However, ob-servations of a cavitating vortex core behind apropeller blade show that the cavitating vortexcore changes very little with distance to the tip(Fig. 9.14 with the flow from right to left) de-spite a strong increase of the vortex strengthdue to the roll-up of the trailing vorticity be-hind the blade.

This shows two important things. Firstthat a significant amount of vorticity hastop be present outside the cavitating core,so that the vortex strength can increase withdistance to the tip, but the cavitating coreradius remains constant or even reduces. Thesecond observation is that there is a verysmall decrease of the cavitating core over thedistance to the tip. This shows that viscousdissipation is very small in the case of a cav-itating vortex and inviscid theory can be used.


9.4.1 The relation between thecavitating core radius andthe pressure

For a better understanding of the cavitatingvortex we take a simple inviscid approachwhich dates back to Betz in 1931 (from[55]).The bound circulation at a certainposition of the wing rolls up into a vortexwith a certain radius, as shown in Fig 9.8.Close to the wing tip an arbitrary circula-tion distribution can be approximated byΓb(x) = c1x

1/m where Γb is the bound circula-tion and c1 represents the wing loading, x isthe distance to the tip. For m=2 this is theelliptic loading distribution. Betz assumedthat the trailing vortex sheet from a locationx to the tip rolls up into a vortex with radiusλx and he took λ as a constant. The radialdistribution of the circulation can be writtenas Γ(r) = c1(λx)1/m = c1(r)1/m for r < av.This gives a distribution of the red vorticalregion in Fig. 9.9. Note that in the inviscidcase the circulation at the vortex center canbe finite. The location and strength of thetrailing vortex at some distance from the wingtip can be found from conservation laws suchas conservation of vorticity and of momentsof vorticity ([55]).

When cavitation inception occurs and thevorticity is maintained the circulation is movedoutwards by the vaporous core, and the circu-lation distribution in the vortical region out-side the cavitating core with radius ac becomesΓ(r) = c1(r − ac)1/m for ac, r, av.. As an ex-ample we will consider an elliptical tip loadingwith m = 2. This results in the velocity dis-tribution around the vortex core as:

v(r) = Γ/2π ∗ r r > av (9.16)

v(r) =Γ√r − ac√

av − ac2πrac < r < av (9.17)

(9.18)

where Γ is the maximum circulation in the vor-

tex at radius av. Inside ac there is vapor at thevapor pressure pv. This is the situation as inFigs. 9.12 and 9.13.

Using eq. 9.9 this gives after some algebrathe relation

p∞ − pv =ρΓ2

8π2avac(9.19)

The important aspect of this is that eq. 9.19gives a rather simple relation between the pres-sure p∞ and the cavitating core radius ac.There is no viscous core in this model, soac cannot go to zero. With increasing pres-sure p∞ the cavitating core radius goes to zeroasypmtotically. Therefore this relation cannotgive an inception pressure. But a relation likethis can be used to define cavitation inceptionas the occurrence of a finite minimum cavitat-ing radius, as will be discussed later.

9.5 Inception of tip vortex

cavitation

Inception of a vortex is a complicated affair,because it involves a vortex with a low pressureregion in the core, but also nuclei to expandin that vortex core. The pressure gradient of avortex causes nuclei to move towards the vor-tex center. When such a nucleus reaches acritical pressure it will rapidly expand, but itwill also take an elongated shape. But thecavity also moves with the flow and is thusmoved away quickly. As a result cavitation in-ception of a tip vortex begins with flashes ofcavitation and is very difficult to discern. Thecloser we look, the earlier cavitation inceptionis called. Since one reason to determine cavi-tation inception of a tip vortex is the radiatednoise, it seems reasonable to detect inceptionacoustically. That is a good method, as longas there are no other cavities or noise sourcesaround. In that case acoustic cavitation incep-tion is found when occasional bursts of noisefrom the expanding and collapsing nuclei arefound. With decreasing pressure the frequency


Figure 9.15: Video:Inception of a tip vortex athigh tip loading with stroboscopic illumination

of these bursts increases and again a ratherarbitrary criterion has to be formulated: thenumber of bursts per second [23]. Acousticcavitation inception occurs generally slightlyearlier than visual inception. If it occurs later,gaseous cavitation may occur without a violentcollapse and thus without noise.

Cavitation inception of a vortex dependsstrongly on the nuclei content. At model scaleit is generally determined with a maximumof nuclei in the cavitation tunnel, but eventhen inception occurs incidentally, as shownin video 9.15 for a highly loaded tip. In thisvideo the pressure decreases gradually until acavitating vortex is formed. In such a case thevortex is strong and cavitation develops in awell defined core.

In case of a low tip loading the tip vortex isweak and it requires a much lower pressure toreach inception. The result is a less coherentand more flashy cavitation in the tip vortex, asshown in Fig. 9.16 in which video the pressureagain decreases gradually.

To avoid the dominant influence of nuclei,cavitation inception can also be determined

Figure 9.16: Video:Inception of a tip vortex atlow tip loading with stroboscopic illumination

when the cavitation disappears. This is calleddesinent cavitation. In that case a stable tipvortex is first created, as shown in Fig. 9.17.

But when the pressure increases the cav-itating core breaks again into small parts,which disappear in different positions andtimes. In case of steady tip vortex cavitationdesinent cavitation has the risk of beinggaseous, because the cavitating tip vortexcontinuously collects nuclei from the fluidtogether with diffusion from the fluid which islocally supersaturated.

When the inception pressure differs from thepressure at desinence this is called hysteresis.Desinent cavitation is shown in Fig. 9.18 (flowfrom right to left).

Although desinent vortex cavitation is oftenmore reproducible than incipient cavitation,the determination of the precise inceptionpressure remains difficult and has a consider-able error margin.

http://youtu.be/UbJK8zOgW3k

http://youtu.be/sOM699P4zgI


Figure 9.17: The cavitating core of a tip vortexbehind a propeller

Figure 9.18: Desinent tip vortex cavitation

9.5.1 The cavitating radius as acriterion to define incep-tion

The relation between the cavitating vortex ra-dius and the pressure for a given vortex can be

Figure 9.19: The radius of the cavitating tipvortex as a function of the pressure

explored to define inception in an alternativeway. When the relation between the pressureand the cavitating radius can be measured adiagram like Fig. 9.19 is found.

When inception is now defined as theoccurrence of a certain minimum radius acithe inception pressure or cavitation index σican be read from the regression curve throughthe measured points.

When the regression curve is known, e.g.as eq.9.19, only one or a few measured pointsare necessary to find an inception pressure.This approach has been explored ([41],[54]),but theoretical relations were not yet veryconsistent with measured data.

Note that the definition of a certain mini-mum radius at inception for both model andfull scale does not remove the differences ininception index between model scale and fullscale. In eq.9.19 the cavitation index at modeland full scale would be the same when bothvortex radius and inception radius scale withthe scale ratio. For the vortex radius this isplausible, byut not for the inception radius.


Figure 9.20: Distortions on a cavitating vortexof constant diameter

Figure 9.21: Determination of the diameter ofa cavitating vortex by image recognition

There is another difficulty in this approach.The cavitation radius is often not very con-sistent. Even in a seemingly stable vortex lo-cal instabilities and small pressure fluctuationscause disturbances in the cavitating vortex, asillustrated in Fig. 9.20.

So in case of a propeller the radius of the tipvortex has to be determined using automaticimage recognition, as shown in Fig. 9.21.

This was tried in [58],but the repeatabilityof the inception pressure was still not betterthan with visual observations. This is mainlydue to the asymptotic nature of the cavitatingradius close to inception. Small deviations ofthe inception radius lead to large variationsof the inception pressure and these variationsin radius occur especially when the radius issmall. At lower pressures the cavitation radiusincreases, but near the propeller tip becomesincreasingly non-cylindrical, as illustrated inFig. 9.22

It can therefore be concluded that no stan-dard method for inception of tip vortex cav-itation is available yet. Visual observationat high air content is the current practice incavitation tunnels, acoustic measurements aremost appropriate at full scale. The repeata-

Figure 9.22: Non cylindrical tip vortex atlower pressures

bility and accuracy of both methods is still in-sufficient.

9.6 Origins of a tip vortex

Until now the origin of the tip vortex has beentaken as the roll-up of the trailing vorticitybehind a wing or propeller blade. This is thecommon source of tip vortices at some distancebehind the tip, although inception occurs closeto the tip (Fig. 9.17).

In special cases the cavitating vortexremains detached from the tip, as shown inFig. 9.23. However, this may also be due toa lack of nuclei, because addition of nucleioften connects the cavitation with the tip


Figure 9.23: Detached tip vortex cavitation

and the origin of the vortex still is vortexroll-up at the tip. Detached cavitation canstill be expected because the roll-up takestime to decrease the minimum pressure to theinception pressure. This is especially the casewhen the propeller tip is unloaded.

The cavitating tip vortex which consists ofrolled-up vorticity from the trailing edge of thepropeller blade is called a trailing vortex cavi-tation.

Another source of a tip vortex is local sepa-ration at the tip of the blade. This separationdepends on the tip loading and on the tip ge-ometry. This tip vortex will be called a localtip vortex. On moderately or heavily loadedtips this vortex coincides with the trailing tipvortex and thus causes attached vortex cavi-tation. On unloaded tips the trailing vortexand the local tip vortex may occur as separatevortices.

In cases of unloaded tips the inner radii ofthe propeller blade are heavily loaded. Sepa-ration of the flow at the leading edge at innerradii of a propeller blade also causes a vortex.This vortex moves towards the tip at somedistance from the leading edge of the bladeor foil and when cavitating this is thereforecalled leading edge cavitation. Leading edgeseparation occurs in streamwise direction, assketched in Fig. 9.24.

An example of a cavitating leading edge vor-

Figure 9.24: Streamlines at leading edge sep-aration

Figure 9.25: Leading edge vortex cavitation

tex is given in Fig. 9.25.

An example of simultaneous occurrence ofleading edge vortex cavitation and local tipvortex cavitation is given in video 9.26.

It is not always possible to make a sharpdistinction between the different types of vor-tex cavitation. However, when the inceptionof tip vortex cavitation has to be delayed itis important to distinguish when possible,because when inception is caused by trailingvortex cavitation the tip loading has to bemodified. When inception is caused by localtip vortex cavitation the shape of the tip hasto be modified. And when inception is causedby leading edge cavitation the shape of theleading edge at inner radii has to be modifiedor the radial loading distribution has to beadjusted.


Figure 9.26: Video:Local and leading edge vor-tex cavitation. (stroboscopic illumination of apropeller in uniform inflow)

9.7 Effects of the nuclei

content on inception of

vortex cavitation

The influence of the nuclei content in the fluidon cavitation inception of vortices is strong.This is mainly because the vortex occurs inthe flow, not at a surface. The presence of asurface may generate additional nuclei or dis-turbances in the pressure, such as in incep-tion of sheet or bubble cavitation. But vortexcavitation depends fully on nuclei in the bulkflow and a lack of nuclei will delay the incep-tion pressure to far below the vapor pressure.From visual observations such a delay can berecognized, because with a lack of nuclei theradius of the cavitating core at inception willbe large. Similar as with bubble cavitationthe few nuclei which reach the critical pressureand expand, will reach a larger size. With anabundance of nuclei the inception radius willbe smaller and inception will occur less ran-domly. Note that for steady vortex cavitation

such an abundance of nuclei can cause a collec-tion of gas in the core. On a ship propeller in-ception generally takes place every revolutionand this effect of gas collection is less severe.

9.8 Effects of viscosity on

inception of tip vortex

cavitation

As shown the minimum pressure in the centerof a vortex depends on the strength of thevortex and on the radius of the viscouscore. The radius of the viscous core dependsstrongly on the Reynolds number. There isno specific length scale or velocity related toa vortex, so the Reynolds number is generallybased on the properties of the foil whichgenerates the vortex, such as Rn = V.c/ν .A crude way of estimating the dependency ofthe viscous core of a tip vortex is to relateit with the boundary layer thickness on thefoil. Using flat plate properties of a laminarboundary layer the thickness of the boundarylayer is proportional to 1

Rn−0.5 . For a turbulentboundary layer the boundary layer thicknessis proportional to Rn−0.2.

If we take the simplest Rankine vortexmodel the minimum pressure in the vortexis inversely proportional to a2

v (equation9.13), and the inception index σi is thereforeproportional to Rn0.4 in the turbulent caseand to Rn in the laminar case. Measurementson foils indicated a Rn-dependency of theinception index with a power of about 0.35([47]). So the relation with the turbulentboundary layer thickness is not very strong.

The inception of propeller tip vortices atmodel scale can be compared with full scaledata. This results in an empirical exponent forthe Reynolds dependency of vortex inceptionfor different facilities. The values are mostlybetween 0.25 and 0.4. Note that the full scale

http://youtu.be/Pm8rujIhSxA


data have an even higher uncertainty thanthe model data.

It has to be kept in mind that in model ex-periments the variation in Rn number is muchsmaller than required for the extrapolation.In model tests the maximum variation in Rnnumber is about factor 3. When the Froudenumber is maintained the Reynolds numberscales with α1.5, in which α is the scale ra-tio. With a scale ratio of 25 this means thatRnship/Rnmodel = 125! So a single exponent ofthe Reynolds number is a far shot indeed.

9.9 Leading edge rough-

ness and tip vortex

cavitation

When the boundary layer thickness affects theminimum pressure in a tip vortex, the stateof the boundary layer will also have an affect.When the boundary layer is laminar on a wingtip, the boundary layer will be thinner and alower minimum pressure is expected and thusearlier cavitation inception . However, due tothe same laminar boundary layer there willbe a lack of nuclei in the vortex core, whichis delaying cavitation inception. The latterresults in detached cavitation at inception(Fig. 9.23), probably because the nuclei inthe bulk fluid need time to move towards thevortex core.

Application of leading edge roughness inthe tip region makes the boundary layerturbulent, resulting in an increase of the min-imum pressure and thus a delay of inception[28]. This applies only when in both caseswit and without roughness the boundarylayer is turbulent. When the boundary layeris laminar at the propeller tip, cavitationinception may be strongly delayed by theabsence of nuclei and a main effect of theroughness is the generation of nuclei in the

tip vortex region, which also often results inattached tip vortex cavitation (Fig. 9.17) [40].

The precise structure of a cavitating tipvortex at inception is still unclear, and the in-ception conditions remain a topic for research.Experiments are very difficult because it isnot possible to directly measure the pressurein the tip region due to its sensitivity fordisturbances, while the region of minimumpressure is very small. It is expected thatCFD can give more insight in the parameterscontrolling tip vortex inception.

9.10 Calculations of incep-

tion of tip vortices us-

ing CFD

The rapidly increasing application of CFDhas also been applied to the tip vortex ofa non-cavitating propeller. The advantageof CFD is that the minimum pressure canbe calculated in the vortex center and thusthat it can be correlated with inceptionmeasurements. The validation of the CFDcalculations has to be done by correlating thevelocities around the core region.

Tip vortices are calculated in the non-cavitating state. Application of CFD to cavi-tating tip vortices is still not done. An exam-ple of such a calculation on a rectangular foiltip is given in Fig. 9.27.

A typical graph of the velocities in a crosssection of the vortex is given in Fig. 9.28.

In the validation with experiments the focuswas on the maximum or peak velocity aroundthe vortex and a good correlation was found,provided a proper grid resolution and turbu-lence model was used. The grid resolution isvery important for the calculation of the vor-tex flow, because the region of low pressure isvery small. This is illustrated in Fig. 9.29. It


Figure 9.27: Calculated Vortex Flow on thetip of a rectangular foil [11]

Figure 9.28: Velocity distributions in a planeperpendicular to the vortex. [11]

is obvious that the low pressure region has tocontain sufficient elements to predict the pres-sure there correctly.

Figure 9.29: Grid refinement in the vortexregion (courtesy of National Aerospace Lab.NLR in the Netherlands)

This requires such a fine grid density thatlocal grid refinement is necessary for sufficientresolution. This requires knowledge of the po-sition of the vortex core, so the grid refinementis therefore applied in steps, following the po-sition of the vortex.

The final validation of such calculations isthe prediction of the minimum pressure, whichwould mean the prediction of cavitation incep-tion. Until now, however, the correlation ofthe experimental inception pressure with thecalculated minimum pressure in a tip vortexis still bad, even in steady conditions and onsimple forms like foils. This may be due to thecalculations ([60]), but also due to the nucleicontent in the experiments. This requires fur-ther investigations before reliable predictionsof tip vortex inception can be made. In un-steady conditions, such as occur on a propellerin a wake, calculations of the inception pres-sure of the tip vortex are still to be developed.


Figure 9.30: Cavitating tip vortex near therudder in a cavitation tunnel (courtesy VWSBerlin)

9.11 Implosion of vortex

cavitation

Tip vortices are considered harmless for ero-sion, although they often hit the rudder(Fig.9.30).

The cavitating tip vortex folds itself aroundthe leading edge of the rudder. The part alongthe rudder collapses in a distributed way andis generally not erosive. The cavitating tipvortex continues perpendicular to the ruddersurface and the two parts on each side of therudder reconnect downstream of the rudder.The reconnection is often with a knuckle,because circulation around the rudder movesthe flow upwards on one side and downwardon the other side of the rudder.

Still vortices may implode on the rudder.When a cavitating vortex core implodes thevortex becomes unstable and the cavitatingvortex core breaks up into elements of acertain length. When these elements aresmall, the erosive energy is limited. Whenthese elements are large the implosion canbe erosive.This seems especially the case onvortices generated on the rudder itself, suchas vortices from corners and from cornersat a rudder horn. Cavitating hub vortices

Figure 9.31: Cavitating hub vortex on a rud-der at full scale (rudder to SB)

may also collapse on the rudder, especiallywhen the rudder is at an angle. An exam-ple of a cavitating hub vortex at full scaleis given in Fig. 9.31. In that picture thecavitating hub vortex is stenghtened by alow pressure region on the propeller and acollapse near the trailing edge of the propeller.

Erosivity of a cavitating vortex is evenexplored in erosion studies using a vortexcavitation implosion. The vortex is generatedperpendicular to the eroded surface and im-plosion is then studied by rapidly increasingthe pressure. ([4]). In that case long elementsof the cavitating core collapse on the surfaceand cause erosion.

Tip vortex implosion also generates noise.This can be a problem in case of excessivetip vortex cavitation, where the cavitating tipvortices collapse in the vicinity of the hull.A special type of noise generation was foundby Maimes. In his case it seems that waveson the surface of the cavitating vortex causenoise with discrete frequencies [46]. For lower


Figure 9.32: Cavitating tip vortex in the topregion of a propeller in a wake

frequencies such a mechanism has been heldresponsible for tip vortex induced pressurefluctuations [63]. In certain circumstancestip vortex cavitation can cause ”broadbandnoise”, which is an increase in noise levelin frequencies between about the third andfifth blade harmonic. The cause is still un-known, but Bosschers is also looking at waveoscillations on the cavitating vortex surface [9]

In case of a propeller behind a ship the tipvortex increases and decreases in strengthduring a revolution. This causes dynamicbehavior of the cavitating tip vortex, oftenincluding implosion. An example is given inFig.9.32)

This behavior is sometimes related tovortex bursting, but it is not. In case ofvortex bursting the vorticity disappears dueto viscous dissipation. In this case the vortexcavitation and thus also the vortex continuesdownstream. Only the strength of the vortexvaries with the blade position. An illustrationof the dynamics of such a varying vortexstrength on the cavity dynamics is given onthe high speed observation on a containership.

Figure 9.33: Video:Vortex cavity dynamicswhen a propeller blade passes a wake peak

http://youtu.be/pcqyelhH_c4

chapter 9: vortext cavitation

Education

vortex cavitationobjective

body of uid

tip vortex cavitation

cavitating tip vortex

inviscid uid

uid accelerations

terms of vorticity

noncavitating vortex