numerical prediction of sheet cavitation on · pdf filelifting flows assuming...

NUMERICAL PREDICTION OF SHEET CAVITATION ON RUDDER AND PODDED PROPELLERS USING POTENTIAL AND VISCOUS FLOW SOLUTIONS

Vladimir I. Krasilnikov, Aage Berg,

MARINTEK – Norwegian Marine Technology Research Institute, Trondheim, Norway and Ivar J. Øye,

CFD Norway AS, Trondheim, Norway Abstract This paper presents some recent validation results with velocity-based source BEM analysis program as applied to prediction of cavitation on rudder and podded propellers. The RANS solver is also applied to modelling of the rudder behind the propeller and the both BEM and RANS numerical predictions are compared with the experimental data. An example of prediction of cavitation on podded propeller in non-homogeneous hull wake at shaft inclination and heading angle is considered. 1 Introduction

Last years propeller researchers address their efforts to numerical simulation of non-cavitating and cavitating flows around rudder and podded propellers (Ghassemi and Allievi 1999), (Achkinadze et.al. 2002), (Achkinadze et.al. 2003), (Kinnas et.al. 2003). Estimation of cavitation on rudder is important from the point of view of noise and erosion of the rudder surface. At high speeds and high rudder angles the developed cavitation can effect on rudder forces. Prognosis of cavitation on podded propellers is especially interesting at high heading angles when propeller is subject to strong oblique flow and even successful “open-water” design can experience very unfavourable conditions. The both problems are related in the sense that they imply accurate modelling of geometries and accounting for the interaction between the components of propulsor, which, to significant extent, has viscous nature.

At the same time available experimental data on such configurations are quite few and detailed verification of the numerical methods is still challenging. In the present work the authors’ boundary element method (BEM) analysis program (Achkinadze and Krasilnikov 2003), (Achkinadze et.al. 2003) is applied to modelling of the propeller/rudder system and the pod drive with a focus on estimation of cavitation performance. The validation of numerical procedure is performed by comparing the numerical predictions with experimental data from the cavitation tunnel including propeller-induced velocity field, rudder forces and cavitation picture on the rudder surface in a wide range of rudder angles and at the different propeller loadings. In parallel, the RANS program is applied to the same configuration in order to estimate the possibilities of the direct “viscous” simulation of such arrangement. For the case of podded propeller the comparison of BEM-predicted cavity patterns with experimental observations is performed at shaft inclination/heading conditions in hull wake. 2 Calculation methods 2.1 Outline of the velocity-based BEM with MTE in steady and unsteady formulations

The authors’ BEM belongs to the group of so-called source velocity based BEMs that is defined by a manner of obtaining of the main integral equation from the Green’s third identity which can be written as follows

( ) dsRn

UUdsRn

UnUPkU

SS∫∫

∂∂′−+

∂′∂

−∂∂

−=11)( 1 . (1)

Proc. of the 5th Int. Symposium on Cavitation - CAV'2003, Osaka, Japan - Nov 1-4, 2003

This formula generalizes the common form of the Green’s identity in the case when the two flow domains are considered: the realistic flow outside the body characterized by the harmonic function U and the imaginary flow inside the body characterized by the harmonic function U ′ (Lamb 1944). In the formula (1) R represents the distance between the current point P and fixed control point 1P , n is the surface normal facing outward (to

the domain of the external flow), and the surface S consists of the wetted body surface WETS , cavity surface

CAVS , trailing vortex surfaces UFVSS and L

FVSS leaving from the upper and lower sides of the thick trailing edge,

respectively, and imaginary vortex surface VS inside the body. The coefficient k in the right-hand part can take

the following values in dependence upon the location of the control point 1P : π4=k if the control point is

located in the external domain, π2=k if it is located on the boundary surface S , and 0=k if it appears in the

domain of the internal flow. Choosing the different representations for the internal flow potential U ′ we can obtain integral equations, which correspond to the different kinds of BEM. For example, if we require 0=′U

and 0/ =∂′∂ nU and accept ϕ=U , π2=k , nVVnU Errr

⋅−=∂∂ Ψ )(/ everywhere on the surface WETS we

come to the formulation of perturbation potential and Fredgolm integral equation of the 2nd kind common in potential based BEM. In velocity based BEM, requiring UU =' and nUnU ∂∂=∂′∂ // on the whole surface S , we can obtain the unique expression for the potential either through the source or through the doublet strength. However, in order to come to the integral equation the differentiation of (1) along the surface normal is necessary. When done we have the following identity

( ) dsRnn

UUdsRnn

UnU

nU

nU

nPU

SS∫∫

∂∂∂′−+

∂∂

∂′∂

−∂∂

−

∂′∂

−∂∂

=∂

∂ 112)(42

1 ππ , (2)

which in the case of UU =' satisfied for the points on the WETS surface (source distribution) leads us to the well-

known integral equation obtained by Hess and Smith for the non-lifting flows (Hess and Smith 1964). For the lifting flows assuming nUnU ∂∂=∂′∂ // on the whole surface S we can obtain the model which uses only doublets (vortices) and results in the singular integral equation of the 1st kind regarding unknown doublet strength, which meets the so-called vortex velocity-based BEM. In the source velocity-based BEM we keep the

condition of UU =' only on the wetted surface WETS and cavity surface CAVS while on the vortex surfaces UFVSS , L

FVSS and VS the identity nUnU ∂∂=∂′∂ // is required. As above, ϕ=U everywhere in the

domain of the external flow and nVVnU Errr

⋅−=∂∂ Ψ )(/ on the wetted surface. Thus, we obtain the flow

model, in which the sources of strength n

UnUq

∂′∂

−∂∂

= are placed on the body and cavity surfaces and

represent thickness effect while doublets of strength UU ′−=σ are placed on the vortex surface inside the body and on the trailing vortex surfaces behind it and simulate lift effect. The latter surfaces merge into one thin vortex surface, which consists of the mean surface and free vortex surface behind the trailing edge. As a result,

the following integral equation can be written for the control points on the surface WETS :

( ) nVVdsRnn

dsRn

qqE

SSQ

rrr⋅−=

∂∂∂

+

∂∂

− ∫∫ ψ

σ

σππ

1411

41

2

2. (3)

It expresses no flow condition on the wetted surface and, as one can see, becomes Fredgolm integral equation of the 2nd kind regarding unknown source strength q if the doublet strength σ is known. It has to be noted that in

this kind of method the distribution of doublets can be, in principal, arbitrary since all doublet effect on external flow is defined by the value of circulation around considered section of the body. It allows us to apply the simplest linear distribution of doublets along the chord and find unknown circulation from the Kutta-Jowkovski condition at the trailing edge (Achkinadze and Krasilnikov 2003). The implementation of the latter condition requires additional comments. The most widespread approach to this problem in different BEMs implies satisfaction of the equal pressure condition on the upper and lower side panels closest to the trailing edge. Usually it is performed in iterations either by means of auxiliary correction to potential in the potential based methods (Lee 1987) or via additional velocity at the trailing edge normal to the mean surface in the velocity

based methods (Ando et.al. 1998). However, if we consider the inviscid flow around 3D trailing edge with non-zero angle between upper and lower sides we meet the known paradox since the total velocity vector has to be tangential to the both sides, if it is not zero (Mangler and Smith 1970), (Mishkevich 1997). The presence of the thin vortex surface in the wake behind the trailing edge allows, in principal, to satisfy the mentioned condition if the trailing vortex surface is tangential to the upper or lower side at the trailing edge. However, it never takes place in practice. In order to overcome the mentioned paradox the authors introduce an additional short panel behind the realistic trailing edge and call it as Modified Trailing Edge (MTE) (Achkinadze and Krasilnikov 2003), (Achkinadze et.al. 2003). This panel is used for the direct satisfaction of the Kutta-Jowkovski condition which is written as follows

MTEMTER SonnV 0=⋅rr

, (4)

i.e. like common “no flow” condition on the MTE. The length of the MTE panel is taken as equal to 2 per cent of local chord length and its orientation is defined by the tangent to the mean surface at the trailing edge of the body. From the physical point of view it allows us in a local domain to simulate the flow around 3D trailing edge with zero angle between suction and pressure sides, or, in other words, to obtain the flow with reciprocal point at the given spanwise section, and by this meet the correct formulation of the problem within the frameworks of the ideal fluid theory.

Using the well-known identity between doublet and vortex strengths (Artyushkov et.al. 1988) σγ σ gradn ×=

rr, (5)

where σn is the normal to the vortex surface and γ is the surface vortex strength, and the Biot-Savart identity we

can rewrite the main integral equation of no cavitation problem (3) as follows

[ ] [ ] ( ) MTEWETEL

L

SS

SSonnVVR

nRldds

RnRds

RRnqq

Q

+⋅−=×⋅Γ

+×⋅

+⋅

+ ∫∫∫rrrrrrrrrrr

ψ

σσπ

γππ 333 4

141

41

2. (6)

This equation is solved numerically by reducing it to a linear system. In order to do it the blade surface is represented by the curvilinear quadrilateral source panels of constant strength, which form a non-uniform cosine grid with condensation towards the blade edges and blade tips (or hub in a case of propeller). Respectively, the mean surface is divided into stripes in the spanwise direction, which correspond to the grid on the blade surface and define the span of the MTE panels. Initially the bevel angle of the MTE is defined by the tangent to the mean surface while at the following steps it is corrected accounting for the viscous effects on circulation using the simple condition of the equal lift coefficients of the blade section in the equivalent 2D viscous and considered 3D potential flows. The details of the MTE algorithm accounting for the viscous effects and curvilinear approximation of the blade surface can be found in (Achkinadze and Krasilnikov 2003), (Achkinadze et.al. 2003).

In the unsteady flow around propeller the both left-hand and right-hand parts of (6) become time or, in the propeller-fixed coordinate system, angular dependent, and numerical solution is performed in a step-wise manner

for a certain number of blade angles Pθ . At the first step, which we call as quasi-steady, we omit the effect of

radial shed vortices and consider that the vortex system consists only of radial bound and helical free vortices, while in the fully unsteady solution the radial shed vortices are included and their strength is defined through the circulation as follows:

( )

−Ω−

ΓΩ−=

RM

MP

PRMPr V

rdd

Vr ξξ

θθ

θξγ ,,, , (7)

where Ω is the rotational speed of propeller, Mξ is the helical coordinate of the control point M on the MTE

panel, RMV is the projection of the total velocity at the M point on the tangent to the free vortex surface. On each

unsteady time step the values of circulation are taken from the previous iteration. Let us note that in the present method the shed vortices are generated only in the propeller wake behind the TE and not on the mean surface. The corresponding effect is compensated by the source strength obtained from the solution. The introduction of the shed vortices on the mean surface is also possible, but in this case only the source strength would be different while total effect remains the same.

Having source and doublet(/vortex) strengths defined we can calculate corresponding induced velocities at the control points on the body surface and then obtain pressure distribution from the Bernoulli identity:

( )2

)(*22

ERA

VVt

ghpp−

−∂

∂′−=+−

ρϕρρ , (8)

where absolute total pressure *p includes gravity forces, Ap is the atmospheric pressure, h is the

submergence of the considered control point under the free surface and the time derivative of the absolute total potential is calculated in the coordinate system fixed on body. A method of computation of this derivative in the case of rotating propeller blades is described in (Achkinadze et.al. 2003). After that the potential part of the blade forces is calculated by simple integration of pressure distribution along the blade surface.

In order to include the influence of viscosity at least two following effects have to be taken into account: viscous section drag and viscous effect on section lift. The first effect is estimated via empirical formula proposed in (Bavin et.al. 1983) for NACA,a=0.8/66mod sections and used earlier in the authors’ propeller design and analysis programs:

1458.0Re/)3.21(05808.0 SDC δ+= , (9)

where ct /0=δ is the maximum relative thickness of the section and SRe is the local Reynolds number. It has

been found that the mentioned formula allows reasonable estimation of DC not only near shock-free entrance

condition (for instance, propeller design point, for which this identity was originally obtained), but also in a quite wide range of angles of attack, which correspond, for instance, to the different propeller loading. However, in the cases of high and low loadings we observe that another effect begins to prevail – viscous effect on lift. In order to account for this effect the algorithm of MTE is employed. As it has been mentioned above the initial orientation of the MTE panel as tangential to the mean surface at the TE allows correct solution for the purely potential flow. One can notice that the effect of this additional panel is analogous to effect of small flap, which deviation towards the upper side leads to the reduction of the section lift and, oppositely, deviation downward increases the section lift. The magnitude of the MTE bevel angle is defined from the identity of the lift coefficient obtained in the 3D potential solution and lift coefficient in the equivalent 2D flow around given body section. The latter can be obtained from the direct viscous calculation of the section profile or, as it has been done in the present algorithm, taken from the empirical data on 2D profiles. The following estimation of the lift coefficient proposed by Mishkevich (Bavin et.al. 1983) for NACA,a=0.8/66mod sections has been used

( )[ ]( )

( )

−−

+=

−+−−+=+=

,4378.0Reln04664.0

05.01015.1

,)Reln1855.046.120691.0exp(187.01),2(2

2S

S

CLC

δδκ

δδµκδαπµ

(10)

where cfC /0=δ is the maximum relative camber and α is the angle of attack.

The equivalence between 3D and 2D flows around blade section is achieved using conditional non-homogeneous velocity field around 2D profile, which provides the same pressure distribution as one obtained from the 3D analysis. As one can note such pressure distribution automatically guarantees the equivalence in terms of section forces. The velocity components of the equivalent 2D flow are defined from the solution of an unconstrained minimization least-square problem using the Levenberg-Marquardt algorithm with a finite-difference Jacobian.

An example of use of the MTE tool for viscous correction in a case of 2D foil has been considered in (Achkinadze and Krasilnikov 2001) while its application to propeller analysis is considered in (Achkinadze et.al. 2003). The introduction of the mentioned corrections allows us to obtain a better agreement with experimental data in terms of propeller forces at high and low propeller loadings. However, the ultimate result of such technique consists in that it allows correction of pressure distribution in response to change of the lift coefficient and, consequently, circulation.

In the present study the described BEM is applied to modelling of propeller and rudder (or strut in the case of podded propulsor). The interaction between the components of propulsor is treated via method of velocity field iterations. Besides, propeller-induced velocity field is always considered as circumferentially averaged and an infinite-bladed propeller model similar to one developed by Hough and Ordway (Hough and Ordway 1965) is

used. The differences from the original Hough and Ordway theory are as follows. The pitch of the trailing vortices in the authors’ method corresponds to the Generalized Linear Model (Achkinadze and Krasilnikov 1997), i.e. it is different from the pitch of lifting line trajectory in the purely linear theory described in (Hough and Ordway 1965) and meets simulated propeller loading. In addition, the correction factors accounting for the friction in the blade boundary layer and propeller wake contraction are included (Achkinadze et.al. 2003). At the same time the velocity fields induced by the rudder(/strut) and pod(/hub) are spatially variable. So, the propeller is always subject to non-homogeneous flow. 2.2 Finite-volume method for the RANS equations

The method used for the solution of the incompressible Reynolds-averaged Navier-Stokes (RANS) equations is a finite-volume method based on central differences in space coordinates and an explicit three-stage Runge-Kutta method for time integration (Øye 1997), (Eriksson 1993). This method is marching in time from an initial field until a stationary condition is reached and the conservation equations for mass and momentum are satisfied. The k -ε model by Chien (Chien 1982) with additional length-scale corrections near the wall (Øye 1996) was applied for the simulation of turbulence.

The flow solver facilitates multi-block grids with a general and flexible specification of boundary conditions. The propeller slipstream is modelled through the additional source terms specifying the axial, tangential and radial velocity profiles in the propeller plane. At the outflow boundary non-reflecting boundary conditions are applied.

The cell-centred finite-volume discretisation is obtained from the integral form of the Navier-Stokes equations. For the finite control volume Ω it can be written as follows

[ ]ε,,,,,, 2 kwvucpdSdt

T

S

==⋅+Ω∂∂

∫∫Ω

UHnFU , (11)

where the time derivative term in the continuity equation including the static pressure (p) and the artificial speed of sound (c) originates from the artificial compressibility method by Chorin (Chorin 1967). The variables u, v, and w are the Cartesian velocity components in x -, y-, and z-directions, respectively. The variables are assumed to be averaged over each control volume. In equation (11) F refers to the common flux vector containing both inviscid and viscous transport terms while the vector n is the outward unit normal to the surface of area S. At each cell face the characteristic variables are obtained using a 3rd-order accurate upwind-biased interpolation stencil. After transformation back to primitive variables the fluxes are assembled. The artificial dissipation terms are not necessary to achieve stability. The source terms in H are treated in a point-implicit manner to enhance stability and robustness. 2.3 Actuator disk model in the RANS solver

In order to simulate the effect of swirling propeller slipstream without modelling of the realistic propeller geometry in the RANS solver the authors employed a simplified propeller model based upon the actuator disk with a prescribed radial distribution of the axial and tangential velocity components. The latter are taken from the BEM propeller analysis as estimated at the propeller plane and then the propeller wake progress is considered. From the momentum theorem the pressure jump across the propeller disk is expressed as follows:

)1(2 CC uup +=∆ , (12)

where Cu is the axial induced velocity at the propeller plane.

In the considered flow domain we define a grid plane, which coincides with the propeller plane assuming that the thrust is aligned with the x-axis. For each cell the local radius from the propeller axis is defined and if the cell is located within the propeller disk the prescribed velocity field data are interpolated into the grid centre. The propeller thrust in x-direction can be expressed as follows

VOLtSpH X

∆∆=2

, (13)

where p∆ is obtained from the equation (12), XS is the cell face area, t∆ is the time step and VOL is the cell

volume. By creating the forcing function θH in the form given below

)( θθθ uuCH −= ) (14)

we ensure that the desired tangential velocity θu is satisfied at steady state.

3 Numerical modelling of rudder behind operating propeller. Calculation/experiment comparisons

The experiments with propeller/rudder system have been performed in the cavitation tunnel at MARINTEK. The principal data on tested arrangement are given in Table 1. The program of the experiments included open-water testes with single propeller, measurements of propeller-induced velocity field at the site of rudder, rudder force measurements behind operating propeller and observation of cavitation on the rudder at different conditions. The same arrangement was later a subject to numerical analysis using BEM and RANS programs. The comparisons with BEM code in no cavitation case were considered in details in (Achkinadze et.al. 2003). Therefore, below we will reproduce only the main results. It was of a high importance to see how predicted propeller forces and induced velocity field agree with experimental data since in BEM method this velocity field is used directly in iterations with rudder while in RANS code the propeller was simulated as an actuator disk using thrust value and velocity field from the BEM analysis. Table 2 shows calculated and measured values of the thrust and torque coefficients for single propeller at J values close to the ones selected later in the cavitation tests.

The comparison with propeller-induced velocity field has been done for the two J values of 0.6 and 0.8 at the measuring section of 0.5D that corresponds to the location of rudder axis. In the Figures 1 and 2 the calculated axial and tangential velocity components as predicted by BEM and RANS codes are compared with those measured in the cavitation tunnel. In the BEM analysis the full system of correction factors accounting for viscous effects on circulation (via MTE panel), friction on the blade surface and wake contraction was applied. The wake contraction factor at the calculation section amounted KW=RW/R=0.96 for J=0.8 and 0.945 for J=0.6.

The both numerical predictions are in satisfactory agreement with the measurements, although in the RANS solution the both axial and tangential components are underestimated compared to the experimental data and BEM results. The reason for this is not clear, but artificial diffusion introduced by misalignment of the grid and the lack of spatial grid resolution are believed to be critical factors. However, it is interesting to note that the wake contraction factor in RANS appeared to be very close to those empirically estimated in the potential code and observed in the tests.

The numerical/experimental comparisons in terms of rudder forces presented below embrace two loading conditions (J=0.8 and J=0.6) and the range of rudder angles from minus 30 (starboard) to plus 30 (portside) degrees. The direction of propeller rotation was “top of propeller to starboard” (right-handed propeller). The axis of rectangular rudder was located at half propeller diameter downstream of the propeller plane. The lower rudder tip was levelled with the propeller blade tip in the lower vertical position. The Figures 3 and 4 show the measured and BEM-calculated values of rudder transverse force coefficient KY and rudder axial moment coefficient MZAX versus rudder angle δR. As one can see the BEM program demonstrates reasonably good agreement with experiment in terms of transverse force almost in the whole range of rudder angles. We can also notice a certain asymmetry in transverse force in dependence upon rudder angle caused by the direction of propeller rotation observed in the experiment and indicated by the code. However the difference between calculated values at corresponding positive and negative rudder angles is smaller than between experimental figures. For example, at J=0.6 the experimental values were KY=−0.191 (δR=−15°) and KY=0.221 (δR=+15°) while calculation predicted KY=−0.178 (δR=−15°) and KY=0.202 (δR=+15°). This can be presumably explained by the averaging of propeller-induced velocity field applied in the BEM algorithm. The range of satisfactory agreement with experiment in terms of rudder moment narrows to ±10-15 degrees becoming noticeably worse at higher rudder angles where the viscous effects prevail.

Table 1: Main elements of propeller/rudder arrangement used in the experimental/numerical study at MARINTEK:

Prop diameter D=0.25m

Number of prop blades Z=4

Blade area ratio 0.61

Pitch ratio at r/R=0.7 P/D=1.095

Rudder span/Prop diameter L/D=1.251

Rudder aspect ratio L/c=1.916

Distance between prop plane and rudder axis XR/D=0.5

Location of rudder axis, % from rudder LE 28.8

Vertical position of rudder regarding prop axis Asymm.

Test conditions, J=0.8, 0.6, 0.4

Rudder angles, δR∈[−30°;+30°]

Table 2: Experimental and calculated values of the thrust and torque coefficients of propeller.

J KT (exp) KT (calc) KQ (exp) KQ (calc) 0.391 0.392 0.3910 0.0625 0.06296 0.583 0.295 0.2987 0.0505 0.05043 0.838 0.161 0.1699 0.0330 0.03248

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3-0,1

0,0

0,1

0,2

0,3

0,4

0,5J=0.8, x/R=-1.00

EXP BEM calc, Kw=0.96 RANS calc

axia

l vel

ocity

, -W

x/V

r/R

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3-0,05

0,00

0,05

0,10

0,15

0,20

0,25

0,30J=0.8, x/R=-1.00


tang

entia

l vel

ocity

, Wθ/V

r/R

Figure 1: Comparison between measured and calculated propeller-induced velocity components in the wake behind propeller. J=0.8. CTh=0.754. Section x/R=−1.0.

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3-0,2

0,0

0,2

0,4

0,6

0,8

1,0J=0.6, x/R=-1.00


axia

l vel

ocity

, -W

x/V

r/R

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,6J=0.6, x/R=-1.00


tang

entia

l vel

ocity

, Wθ/V

r/R

Figure 2: Comparison between measured and calculated propeller-induced velocity components in the wake behind propeller. J=0.6. CTh=2.052. Section x/R=−1.0.

-35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35-0,6

-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,6

solid lines - EXP.dash lines - CALC.

MZA

X

K Y

Rudder angle δR, [deg.]

-0,024

-0,020

-0,016

-0,012

-0,008

-0,004

0,000

0,004

0,008

0,012

0,016

0,020

0,024J=0.8, CTh=0.754

MZAXKY

-35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5J=0.6, CTh=2.052

solid lines - EXP.dash lines - CALC.

MZAX

KY

MZA

X

K Y

Rudder angle δR, [deg.]

-0,025

-0,020

-0,015

-0,010

-0,005

0,000

0,005

0,010

0,015

0,020

0,025

Figure 3: Experimental and BEM calculated values of rudder transverse force KY and rudder axial moment MZAX. J=0.8.

Figure 4: Experimental and BEM calculated values of rudder transverse force KY and rudder axial moment MZAX. J=0.6.

Table 3: Cavitation tests with propeller/rudder arrangement. Test conditions.

Cond. 1 Cond. 2 Advance ratio, J 0.8 0.6 Prop rate of revolution, n [Hz] 15 15 Water speed, V [m/s] 3 2.25 Tunnel pressure, P0 [kPa] 18.6 18.6 Cavitation number, σ(V) 3.756 6.677 Reynolds number, Re=V⋅c/ν 0.49×106 0.37×106

Observation of cavitation on rudder has been carried out for the two conditions specified in Table 3. The photographs of observed cavitation are shown in Figures 5 – 8 along with numerical prediction of cavitation domains obtained using BEM and (for a number of selected rudder angles) RANS programs. It can be seen from the experimental photos that cavitation observed on the rudder surface in these cases can be hardly related to pure sheet caviation and is rather of mixed type. The domains of sheet caviation are present along with clearly seen cloud and vortex zones. Therefore, it would not be quite correct to treat the problem by means of the ideal sheet cavitation model, and the estimation of cavitation domains presented below has been done using a simple condition of σ≥− Cp , where the pressure coefficient is taken from fully wetted solution.

Considering a rudder in propeller wake and deriving the pressure on the rudder surface from the Bernoulli identity in BEM algorithm we have to account for the “jump” of the Bernoulli constant, which takes place in the propeller disk due to power delivered to the propeller. The simple correction factor applied in the present algorithm provides an absence of propeller-induced pressure in the infinitely far wake section that meets the well-known experimental fact (Achkinadze et.al. 2003). The change of the Bernoulli constant is estimated from this condition at one radius which corresponds to the maximum axial propeller-induced velocity in order to obtain the maximum possible value of the pressure “jump”. The corrected pressure value on the rudder surface is obtained as follows: ∞∆−= CpCpCp* , 2)(1 ER VVCp ∞∞ −=∆ , where Cp is the pressure coefficient defined

from the Bernoulli identity without correction and ∞RV is the total relative velocity estimated in the far wake

behind the propeller at the radius of the maximum propeller-induced velocity. From the presented comparisons we can conclude that the BEM program gives a reasonable prognosis of

cavitation at the both studied loading conditions. At the rudder angles of +15, +20, +25 deg. the location and

Exp. J=0.8, Starboard, δR=+15° Analysis, BEM Analysis, RANS




Figure 5: Cavitation on rudder behind operating propeller. Cond 1. J=0.8. Starboard.

Exp. J=0.8, Portside, δR=−20° Analysis, BEM Analysis, RANS



Figure 6: Cavitation on rudder behind operating propeller. Cond 1. J=0.8. Portside. extent of cavitation domains on the starboard side of the rudder is in a good agreement with experimental observations. At +30 deg. on the starboard side and for the negative rudder angles on the portside side cavitation domains are somewhat wider than those shown on the photos. Although it has to be noted that the photos show only instant snapshots of realistic unsteady cavitating flow, while in the analysis rudder is simulated in steady formulation. For example, comparing the results for the rudder angle of –30 deg. on the portside (J=0.8) we can clearly see that the whole rudder span along the leading edge is embraced by cavitation, which is also indicated by the BEM code.





Figure 7: Cavitation on rudder behind operating propeller. Cond 2. J=0.6. Starboard.




Figure 8: Cavitation on rudder behind operating propeller. Cond 2. J=0.6. Portside.

For the positive rudder angles of 20 and 25 deg. the cavitation domains predicted by the RANS program are in a satisfactory agreement with those indicated by the BEM analysis for both loading conditions. Entirely, RANS predicted cavitations domains are smaller than BEM predicted. However, at +30 deg. on the starboard side and at –25 deg. on the portside we observe that the RANS calculations are somewhat different from both BEM and experimental results. For example, at +30 deg. RANS program showed the stripe of cavitation at the lower part of the rudder and did not indicate cavitation at the upper part where in the tests it was the most intensive. Presumably, the premature leading edge separation of laminar nature clearly seen from the RANS results “cut off” the pressure peak in the mentioned domains and did not allow cavitation to develop. At the same time looking at the experimental photos we can notice that in these cases the observed cavity patterns, indeed, belong rather to unsteady separated formations. The further study of the effect of the employed turbulence model on computation results is necessary.

4 Application of BEM to prediction of 3D cavity patterns.

When we consider cavitation problem we keep equation (6) on the wetted surface and MTE panel and apply the dynamic boundary condition in cavitation domains. In general case the dynamic boundary condition

requires equal values of the absolute pressure *p and vapour pressure Vp on the cavity surface CAVS . Using

the unsteady Bernoulli identity (8) we can obtain the following relation for the total relative velocity

( )t

VV ZER ∂∂′

−+=ϕσ 2122 , (15)

where the local cavitation number is written as ( ) )/(2 2EVAZ Vpghp ρρσ −+= . The relative velocity is

connected with the induced velocity W , transferal velocity EV and inflow velocity ψV as follows

ER VVVWrrrr

+=+ ψ . Assuming that the relative velocity vector lies in the plane tangential to the body surface and

choosing direction of the reference tangent vector τir

in the plane of considered section we obtain the linearized

form of the dynamic boundary condition

tViViViW ZEE ∂

∂′−++⋅−⋅=⋅

ϕστψττ 2)1(2rrrrrr

. (16)

Thus, the integral equation expressing the dynamic condition can be written in a form analogous to equation (6) for the kinamatic condition:

[ ] [ ]CAVZEE

L

L

SS

Sont

ViViVR

iRlddsR

iRdsR

RiqQ

∂∂′

−++⋅−⋅=×⋅Γ

+×⋅

+⋅

∫∫∫ϕσ

πγ

ππ τψττττ

σσ

2)1(41

41

41 2

333

rrrrrrrrrrrr

. (17)

Considering equations (6) and (17) we can note that both of them meet mathematically correct formulation of the cavitation problem since equation (6) satisfies the Fredgolm’s alternatives as a Fredgolm integral equation of the 2nd kind, and a singular integral equation of the 1st kind (17) is not restricted by these alternatives due to its singularity (Gachov 1977), (Muskhelishvili 1968). The cavity closure and detachment conditions define class of functions in which numerical solution of (17) is performed.

Equations (6) and (17) are to be solved simultaneously and since cavitation domains are unknown the iterative procedure is necessary. In their method the authors employ the Iterative Cavity Alignment procedure

with free cavity length which implies prescribed value of the section cavitation number Zσ while cavity length

is a quantity on search. This procedure consists of the following consequent steps: 1) Fully wetted calculation; 2) Definition of the first-step cavitation domain from the condition σ≥− Cp ;

3) Linearized calculation of the cavity planforms (within cavitation domain singularities are placed on the body surface, and the boundary conditions are satisfied on the body surface); 4) Non-linear calculation of the cavity planforms (within cavitation domain singularities are placed on the cavity surface, and the boundary conditions are satisfied on the cavity surface). At the 3rd and 4th steps the cavity closure and detachment conditions are imposed in order to resolve equation (17) and define cavity shape from the streamline equation. From the point of view of cavity closure condition the authors’ scheme belongs to the miscellaneous type since it implies the use of the closed and open models and the intermediate model with assumed cavity tail height in dependence upon location of the cavity termination point

CTx . The following three quantities are the free parameters in this model: 1Cx , 2Cx - chordwise coordinates

which define the change of the cavity closure condition, CTh - height of the cavity tail in the intermediate model.

Then, the algorithm is build as follows:

if 1CCT xx ≤ , closed model is used;

if 21 CCTC xxx ≤< , intermediate model is used and the cavity tail height CTh in this range is defined by its

end values: 0)( 1 =Cxh (closed model) and )( 2Cxh (open model) with linear approximation in between;

if 2CCT xx > , open model is used.

In the case of closed cavity model the condition for search of the termination point consists in equality of the calculated streamline ordinate and the ordinate of the body surface at the point CTx (exactly, control point on

the cavity end panel): ),(),( , CTLUCTC xrYxrY = . During iterations we check this condition going through the

panels. Besides, if CY on the current panel happens to be less than LUY , then we move to the next panel

upstream, and if LUC YY ,> − to the next panel downstream. The closure condition in the open scheme is

similar to described above, but concerns derivatives of the streamline and surface ordinates:

),(),( ,CT

LUCT

C xrx

Yxr

xY

∂

∂=

∂∂

, i.e. it ensures parallelism of the cavity boundary and body surface in the case

of the partial cavity, or parallelism of the upper and lower cavity boundaries in the case of super cavity. The cavity detachment point is searched from the Brillouin-Villat condition, which numerically leads to satisfaction of the following two conditions (Young and Kinnas 2002):

1) ),(),( , CDLUCDC xrYxrY > at the cavity detachment point CDx ;

2) the pressure value on the first panel upstream of the cavity detachment panel is higher than vapour pressure. When treating equations (6) and (17) under described conditions the first of them (no flow) is applied on

the wetted surface of the body, MTE panel and cavity detachment panel, while the second equation (dynamic condition) is applied everywhere within cavitation domain except for the cavity termination panel. Thus, there are no conditions applied on the cavity termination panel. On the cavity detachment panel the both kinematic and dynamic conditions are applied.

The numerical implementation of the authors’ method in 2D case has been considered in (Achkinadze and Krasilnikov 2001) where a number of validation examples with 2D foils are given and the results of prediction of cavity ordinates on a 3D wing using closed cavity model are presented. An example of prediction of cavitation on highly skewed propellers can be found in (Achkinadze et.al. 2002). Below we will demonstrate some recent results of prediction of cavitation on podded propeller in oblique flow. 5 Example of modelling of cavitation on podded propeller.

The model tests with pulling podded thruster have been carried out in the cavitation tunnel at MARINTEK. The tested arrangement presents the portside of the hull afterbody with mounted thruster with single pulling left-handed propeller. The axial component of the hull induced velocity field measured at the propeller plane is given in Figure 9 and the general view of podded unit is shown in Figure 10. The simulated conditions are given in Table 4. Table 4: Cavitation tests with podded thruster. Test conditions.

Advance ratio, J 0.486 Rate of revolution, n [Hz] 19

Cavitation number, )(8.0nDσ 1.565

Shaft inclination, ϕ° -3.0 Heading angle, χ° -20.0 Thrust coefficient, KT Exp:0.226 Calc:0.213 Torque coefficient, KQ Exp:0.0267 Calc:0.0272 As one can see in this case we deal with both shaft inclination (3 degrees downward) and heading angle (20 degrees outward) and non-homogeneous flow as prescribed at the propeller plane. Only axial wake velocity component was available from the measurements. Since the interaction between propulsor and hull is not taken into account in the present method all this can be a source of inaccuracy when comparing numerical and experimental predictions. Instead of full 3D calculation of cavity patterns on propeller blades the simplified approach has been applied. The cavity alignment procedure is performed for the cylindrical blade sections in an

equivalent 2D flow which ensures the pressure distribution from the 3D fully wetted solution. This equivalent flow is defined in a same way as in the MTE algorithm described in section 2.1, i.e. from the solution of the minimization least-square problem. The full 4-step cavity alignment procedure is then carried out for each blade section.

Figure 9: Axial wake picture behind hull dummy (seeing from astern).

Figure 10: General view of podded thruster.

The values of thrust and torque coefficients predicted by the BEM program and measured in the tunnel are

compared in Table 4 for the tested conditions. Figure 11 shows the comparison between predicted and observed cavitation domains on propeller blades. The blade turn angle shown on the cavity pictures corresponds to Figure 9. θ=45°, Experiment BEM Analysis

θ=30°, Experiment BEM Analysis


Figure 11: Observed and predicted cavitation domains on podded propeller in oblique flow.





Figure 11: Observed and predicted cavitation domains on podded propeller in oblique flow. (continue). Entirely, numerical algorithm reflected correct progress of cavitation, which agreed reasonably well with experimental observations, although predicted cavitation patterns were somewhat shorter than those observed in the experiment. At the blade positions of 315 and 330 degrees the program showed the cavitation at the root sections which did not reveal in the experiment. At the same time the domains of the start of cavitating tip vortex at 15 – 345 degrees were indicated correctly. Obviously, the reasons of possible inaccuracy are as follows: only axial wake component was available from the tests; “effective” interaction between propulsor and hull was not taken into account; the calculation was carried out without accounting for tunnel wall effects. Conclusions

The velocity based source BEM analysis program has been applied to prediction of cavitation on rudder behind operating propeller and podded propeller in oblique flow. The developed method allows the direct satisfaction of the Kutta-Jowkovski condition and the approximate account for viscous effects on pressure distribution using an algorithm of Modified Trailing Edge. In the case of cavitating flow the main integral equations meet mathematically correct formulation of the boundary problem.

The results of validation study of rudder behind propeller have shown that the BEM program allows trustworthy prediction of propeller-induced velocity field and transverse rudder force and satisfactory estimation

of cavitation domains in a wide range of rudder angles (from –30 to +30 deg.). Significantly worse agreement has been observed for the rudder axial moment.

The same configuration has been modelled using the RANS solver and the predicted velocity fields in propeller wake and cavitation domains on rudder at rudder angles of +20, +25 deg. are in a reasonable agreement. At the same time the RANS analysis revealed domains of the laminar flow separation for the high rudder angles, which may cause the mentioned discrepancy in terms of such a sensitive quantity as rudder moment as well as cavitation domains at +30 and –25 deg.

Simulation of cavitating podded propeller in hull wake at shaft inclination of 3 deg. and heading angle of 20 deg. has demonstrated potential possibilities of the BEM method in handling this complicated case. At reasonably close values of the measured and calculated propeller forces the program has shown a correct progress of cavitation with blade turn angle. Although estimated cavitation patterns were shorter compared to those observed in the tests.

Future efforts should be directed to the full 3D calculation of the cavity patterns and more detailed study of the viscous effects on propeller/rudder and propeller/pod interaction. When comparing numerical results with experiments done in the cavitation tunnel accounting for the tunnel wall effects is, apparently, important. The more profound study of the effect of turbulence model on rudder performance in the RANS method is necessary. References ACHKINADZE, A.S., BERG, A., KRASILNIKOV, V.I. and STEPANOV, I.E. “2002 Numerical Prediction of Cavitation on Propeller Blades and Rudder Using the Velocity Based Source Panel Method with Modified Trailing Edge.” Proceedings of the International Summer Scientific School “High Speed Hydrodynamics”, Cheboksary, Russia, June 16-23.

ACHKINADZE, A.S., BERG, A., KRASILNIKOV, V.I. and STEPANOV, I.E. “2003 Numerical Analysis of Podded and Steering Systems Using a Velocity Based Source Boundary Element Method with Modified Trailing Edge.” Proceedings of the Propellers/Shafting’2003 Symposium, Virginia Beach, VA, USA, September 17-18.

ACHKINADZE, A.S. and KRASILNIKOV, V.I. “2001 A Velocity Based Boundary Element Method with Modified Trailing Edge for Prediction of the Partial Cavities on the Wings and Propeller Blades.” Proceedings of the 4th International Symposium on Cavitation CAV2001, Pasadena, CA, USA, June 20-23

ACHKINADZE, A.S. and KRASILNIKOV, V.I. “2003 A New Velocity Based BEM for Analysis of Non-cavitating and Cavitating Propellers and Foils.” Oceanic Engineering International, Vol. 7, No. 1.

BAVIN, V.F., ZAVADOVSKI, N.Y., LEVKOVSKI, Y.L. and MISHKEVICH, V.G. “1983 Marine Propellers. Modern Methods of Calculations.” Sudostroenie, Leningrad. (In Russian).

CHIEN, K.Y. ”1982 Predictions of Channel- and Boundary Layer Flows with a Low-Reynolds Number Turbulence Model,” AIAA Journal, 20, pp. 33-38.

CHORIN, A.J. ”1967 A Numerical Method for Solving Incompressible Viscous Flow Problems,” Journal of Computational Physics, Vol. 2, No. 1.

ERIKSSON, L.-E. ”1993 A Third Order Accurate Upwind-Biased Scheme for Unsteady Compressible Viscous Flow.” VFA Report 9370-154, Volvo Flygmotor AB, Sweden.

GACHOV, F.D. “1977 Boundary problems.” NAUKA, Moscow, 540 p. (in Russian).

GHASSEMI, H. and ALLIEVI, A. “1999 A Computational Method for the Analysis of Fluid Flow and Hydrodynamic Performance of Conventional and Podded Propulsion Systems.” Oceanic Engineering International, Vol. 3, No. 1. HESS, J.L. and SMITH, A.M.O. “1964 Calculation of non-lifting potential flow about arbitrary three dimensional bodies.” Journal of Ship Research, 8(2), September. HOUGH, G.R. and ORDWAY, D.E. “1965 The Generalized Actuator Disk Theory.” Developments in Theoretical and Applied Mechanics, Pergamon Press. KINNAS, S., NATARAJAN, S., LEE, H., KAKAR, K. and GUPTA, A. “2003 Numerical Modelling of Podded Propulsors and Rudder Cavitation.” Proceedings of the Propellers/Shafting’2003 Symposium, Virginia Beach, VA, USA, September 17-18.

LAMB, G. “1947 Hydrodynamics.” OGIZ, GIT-TL, Moscow-Leningrad, p.928 (translation, in Russian). LEE, J.-T. “1987 A Potential Based Panel Method for the Analysis of Marine Propellers in Steady Flow.” Report 87-13, Dept. of Ocean Engineering, MIT, July.

MANGLER, K.W. and SMITH, J.H.B. “1970 Behavior of the Vortex Sheet at the Trailing Edge of a Lifting Wing.” Aeronautical Journal, vol.74, No.719.

MISHKEVICH, V. “1997 Flow Around Marine Propeller: Non-Linear Theory Based on Vector Potential.” Proceedings of the Propellers/Shafting’97 Symposium, Virginia Beach, VA, USA, September 23-24.

MUSKHELISHVILI, N.I. “1968 Singular integral equations.”, NAUKA, Moscow, 511 p. (in Russian).

YOUNG, Y.L. and KINNAS, S.A. “2002 A BEM Technique for the Modeling of Supercavitating and Surface-Piercing Propeller Flows.” Proceedings of the 24th Symposium on Naval Hydrodynamics, Fukuoka, Japan, July 8-13.

ØYE, I.J. ”1996 On the Aerothermodynamic Effects of Space Vehicles,” Dr.ing. thesis, MTF-rapport 1996:140 (D), NTNU, Trondheim, Norway.

ØYE, I.J. “1997 Evaluation of Turbulence Models for Flow about Wings and Bodies.” In ECARP-European Computational Aerodynamics Research Project: Validation of CFD Codes and Assessment of Turbulence Models (W. Haase / E. Chaput / E. Elsholz / M.A. Leschziner / U.R. Müller, Eds.), Notes on Numerical Fluid Mechanics, Vol. 58, Vieweg.

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