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CAV2001:lecture.007
NEW DEVELOPMENTS AROUND SHEET AND TIP VORTEX CAVITATION ON SHIPS' PROPELLERS G.Kuiper Marin, The Netherlands. A concept of tip vortex cavitation on propellers is described qualitatively, leading to the distinction of trailing vortices, local tip vortices and leading edge vortices. Improvements of the inception behaviour using this distinction are presented. Observations on developed tip vortex cavitation are given to show that the concept of vortex bursting seems inadequate.

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NEW DEVELOPMENTS AROUND SHEET AND TIP VORTEX CAVITATION ON SHIPS'PROPELLERSG.Kuiper
Marin, The Netherlands.
A concept of tip vortex cavitation on propellers is described qualitatively, leading to the distinction of trailing vortices, local tip vortices and leading edge vortices. Improvements of the inception behaviour using this distinction are presented. Observations on developed tip vortex cavitation are given to showthat the concept of vortex bursting seems inadequate. The problem of broadband vibrations due to acavitating tip vortex is illustrated. Arguments are given for the fact that a three dimensional approachis necessary to describe shedding of cloud cavitation at the trailing edge of a sheet cavity.
1.
Introduction.
New developments are possible when the concepts which are used to describe the phenomena arechanging. An example of a changing concept in the field of cavitation inception was the inclusion of viscous effects in the description of cavitation inception. Before that inclusion, cavitation inceptionhad beeen considered from the viewpoint of bubble dynamics only. Although bubble dynamics and boundary layer effects are not yet fully integrated, investigations on cavitation inception have had thesame basic background in the last decades.The problem of inception of tip vortex cavitation is classical and very difficult. It is still important for navy propellers, but progress in this field has been very limited and only some simple empirical scalingrelations and a basic two-dimensional modeling is available. CFD can potentially bring the predictionof tip vortex cavitation on a new level, but this is still under development and only for the non-cavitating case. In this field new physical concepts are being developed and in this paper some resultsare presented.The behaviour and collapse of a developed tip vortex is increasingly a subject of concern in practice.Here the question is which parameters are important for e.g. rudder erosion and for pressurefluctuations on the hull. One such a parameter is the occurrence of vortex bursting. Good observations, both at model and at full scale, can help to develop an understanding of the phenomena and lead toadequate modeling of the problem. In this paper observations are presented and discussed.Finally there is the problem of erosion on propellers due to cavitation. Erosion is the final stage of inception, development and collapse of cavitation. Bubble cavitation is generally avoided on ship propellers, so the problem of erosion is most apparent on sheet cavitation. This occurs always in highlydynamic conditions, where shedding of clouds of cavitation occurs at the trailing edge of the sheetcavity. To connect the calculations with impact calculations of a bubble cloud it is necessary toestimate a length scale of these structures. This has not yet been possible, due to the complicatedstructure of the clouds. The problem is often investigated in a two-dimensional way, e.g. in CFDcalculations. In this paper it is argued that this simplification makes that the most important parameter of cloud shedding may be lost.
2. The model of a tip vortex.
Inception of vortex cavitation is still one of the most complex phenomena on a ship propeller. Bothnuclei content, probably the total gas content, the strength of the vortex, the size of the viscous core andthe vorticity distribution outside the viscous core determine the inception conditions. In this list thevortex is thought of as a predominantly transverse velocity field, so that a two dimensional descriptioncan be used. Without cavitation inception the model as described by Rule and Bliss (1992), as shown inFig. 1, is representative for this approach.
CAV2001:lecture.007
Fig. 1. Vortex model as used by Rule and Bliss(1998)
This model can be summarized as follows. The change in bound circulation on the wing generatestrailing vorticity. This vorticity rolls up from the tip. Conservation of circulation and of the first andsecond moment of vorticity provides enough information to determine the span location and the radiusof the trailing vortex at every axial position. Conservation of axial flux of angular momentum leads tothe determination of the axial induced velocity. The roll-up process can be considered as taking placeon a series of nested, contracting circular tubes. Rule and Bliss show that the axial velocity leads to astronger singularity in the invicid vortex core than the classical fully two-dimensional approach. Thesingularity in the core is eliminated by the viscous core, which is taken as a solid body rotation up to acertain radius and connected to the outer flow using a logarithmic velocity distribution. Such alogarithmic velocity distribution is a solution of the fully 2 dimensional Navier-Stokes equations.It is important to note that Rule and Bliss show with this model that rollup is a three-dimensional process and that an axial velocity is required, which in turn has a significant effect on the roll-up process. The importance of the axial velocity or pressure distribution was already stressed by Batchelor (1964)
3. Determination of tip vortex inception.
Still a cavitating tip vortex is generally modeled as a predominantly two dimensional flow. Anexample of such a nicely cavitating tip vortex as shown in Fig. 2.
Fig. 2. Cavitating tip vortex
Cavitation occurs in the center of the vortex. This makes it possible to locate the vortex core, but it alsoshows that cavitation removes the viscous core. Arndt and Keller (1992) have made some exerciseswith a simple two-dimensional Rankine vortex with a solid cavitating core, showing that twosimultaneous solutions are possible, a cavitating and a non cavitating one, for the same vortex strengthand angular momentum. Are we looking for something which is undefined when we look for theinception conditions of a vortex? In practice inception occurs in a flashing way. The temporal variationof visual inception of cavitation is generally averaged over time, so that inception is called when e.g.cavitation is visible during 50% of the time. This is very difficult to determine and the percentage isoften a very rough estimate. An alternative and better way is to call inception when you see it at leastonce per unit time. When the time is chosen as e.g. 10 seconds it means that after observation of acavity it is checked if within 10 seconds the cavity re-appears. If not inception is not yet called.The observation of Fig.3 shows some discrepancies with the model as described above. The roll-up process predicts an increase in the vortex strength with increasing distance to the tip. In the invicid twodimensional model this would increase the diameter of the cavitating core. It does not, on the contrary,the diameter has the tendency to slowly decrease (Kuiper, 1981). This illustrates that the cavitydiameter is not directly related with the total vortex strength of the tip vortex. Some efforts have beenmade to redefine the inception conditions from the diameter of the cavitating core as a function of the pressure or cavitation index (vanTerwisga et al, 1999). An example is given in Fig.3, in which a
c
is thecavitating core radius of the tip vortex and
σ
is the cavitation index. From two-dimensional modelssuch as described above the relation between the core radius and the pressure can be expressed as a power law. When the power is known, measurements of the core radius versus the pressure can replaceinception measurements at very small core radii, where the inception condition may be undefined or very difficult to determine because it depends on Reynolds number and nuclei distribution. It has beenfound that the cavitating condition is not dependent on those parameters (vanRijsbergen and Kuiper,1997). The definition of a minimum diameter is sufficient to define the inception condition in Fig. 3 ina repeatable and unique way, independent of the scale!
Fig. 3 Cavitating core diameter of a vortex versus pressure coefficient.
The next problem arises when this approach is followed: the determination of the diameter of thecavitating core. In the case of a strong tip vortex in steady conditions the core may be a smooth tube.But this tube is very sensitive to variations in pressure along the core. When e.g. in a cavitation tunnel a smooth cavitating vortex core is generated, the actual shape of the cavity is strongly different from atube with constant diameter. A typical observation is given in Fig. 4. from a high speed video(vanRijsbergen and Kuiper, 1997).
Fig. 4 Cavitating tip vortex.
So it requires a statistical analysis of many pictures to determine the average diameter of the cavitatingcore of the tip vortex. The disturbances of the cavitating core are probably caused by pressuredisturbances in the tunnel. In a non-uniform inflow velocity, such as occur with propellers in a wake,this effect is strongly amplified. This, in combination with the fact that the slope of the curve in Fig. 3 becomes very small when the radius is small makes that this approach for the determination of theinception conditions is not an easy way out. For strong tip vortices, such as behind an elliptic wing, thisapproach may however be more accurate and repeatable than simple visual observation of inception.Moreover, when it is true that the cavitating vortex is independent of the Reynolds number, thisapproach is an alternative for the usual scaling of tip vortex inception by the so called McCormick rule:
( )
35.0
modmod
el shipel ship
Rn Rn
=
σ σ
in which the exponent of 0.35 is more or less based on experience.However, the problem of tip vortex inception on Navy propellers is
not
a problem of a strong tipvortex, as in Fig. 2. The loading of the tip of a Navy propeller in the design condition is close to zero!Cavitation inception occurs when the tip is slightly loaded and unloaded during the revolution of the blade in a wake. This is expressed in the well know inception diagram, in which the pressure is plottedagainst the propeller loading (Fig. 5).
Fig. 5 Inception diagram of the tip vortex of a propeller.
This diagram gives the relation between the inception pressure of the tip vortex on the vertical axis andthe propeller loading at the horizontal axis. Because of the shape of the inception curves this diagram iscalled an inception bucket . A strong suction side tip vortex occurs in the upper right side if thecurve. In this condition the roll-up mechanism is dominant and the description as given above mayhave some relevance. A strong pressure side vortex will occur in the upper left part of the curve. It isgenerally assumed that the roll-up model is also applicable in that case.

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