Evaluation of the radial pressure distribution of vortex models and comparison with experimental data

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Evaluation of the radial pressure distribution of vortex models

and comparison with experimental data

T Hommes1_{, J Bosschers}2_{, H W M Hoeijmakers}1

1_{Engineering Fluid Dynamics Department, Twente University, Drienerlolaan 5, 7522} NB, Enschede, the Netherlands

2_{MARIN, Haagsteeg 2, 6700 AA, Wageningen, the Netherlands} E-mail: j.bosschers@marin.nl

Abstract. An important input parameter for a recently developed semi-empirical prediction

method for the noise generated by cavitating vortices is the diameter of the cavitating vortex. This diameter may be obtained from models of non-cavitating vortices that describe the radial distribution of the azimuthal velocity, for given vortex strength and size of the viscous core. The present paper discusses several vortex models that suit this purpose and compares the distribution of the azimuthal velocity as well as that of the pressure with detailed experimental data for the tip vortex of a rectangular wing obtained from literature.

1. Introduction

A semi-empirical method is in development for the prediction of broadband hull pressure fluctuations and radiated noise generated by cavitating vortices on marine propellers [1]. The method requires an estimate of the cavity diameter, which can be obtained from a vortex model and a known value for the tip vortex circulation and viscous core size. The circulation will be obtained from a boundary element method and the viscous core size will be estimated empirically. The vortex model consists of the radial distribution of azimuthal velocity from which the radial distribution of the pressure can be computed. Previous research has suggested that a vortex model describing non-cavitating flow can be used to estimate the cavity diameter given the cavitation number [2]. The present paper discusses several vortex models, found in literature, that describe the distribution of the azimuthal velocity of a vortex in non-cavitating flow and that can be used in the semi-empirical method. All models require as input the tip vortex circulation and the viscous core size, while some of the models have tuning coefficients as well. The distribution of the azimuthal velocity and that of the pressure obtained from the models are compared to experimental data for a tip vortex of a half-model of a rectangular wing at ten degrees incidence as measured by Chow et al. [3] in a low speed wind tunnel at a Reynolds number based on chord length of 4.6x106_{. Because the pressure has been measured for this dataset we first investigate if} this pressure distribution can be reconstructed from the measured distribution of the velocity and Reynolds stresses. Data was measured at several cross-flow planes but here only results are shown for the plane at x c0.678 with c the wing chord length and with x0 located at the wing trailing edge and x increasing in downstream direction.

2. Reconstruction of the measured pressure distribution from the measured velocity field

The measured velocity components were given in Cartesian coordinates and, after interpolation to an O-type grid, they are converted to a cylindrical coordinate system preserving the original orientation

9th International Symposium on Cavitation (CAV2015) IOP Publishing

Journal of Physics: Conference Series 656 (2015) 012182 doi:10.1088/1742-6596/656/1/012182

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of the x-axis. The origin of this coordinate system was slightly adjusted until all azimuthal velocities are positive. A colour coded contour plot of the azimuthal velocity is given in figure 1. At x c0.678 most of the trailing vorticity is in the tip vortex, apparent from the distribution being reasonably axisymmetric. However, further upstream where a substantial part of the vorticity is still in the wake shear layer, the distribution is far from axisymmetric.

All velocity components have been made non-dimensional by a reference velocity V_ref  _ 2r_v

with _ the vortex circulation in the far field and rv the radius of the viscous core, defined as the radius at which the azimuthal velocity has its maximum. All coordinates have been made non-dimensional with rv. An overline indicates a non-dimensional parameter with



, , 



T x r

V V V the non-dimensional time-averaged velocity component in axial, radial and azimuthal direction, respectively and



v v vx, ,r _



T the non-dimensional turbulent velocity component in axial, radial and azimuthal direction, respectively. The pressure coefficient Cp, defined as 



 





2 1 2

p ref

C p p V , can be computed from the radial component of the Reynolds-averaged Navier-Stokes equation expressed in cylindrical coordinates, assuming axisymmetric flow and constant molecular viscosity :

 

Re r x r p r r r r r r x v v rV C V V V V v v v V V r r x r r r r x r r x r          _{ }  _  _ _   _ _ _ _ _        __ __  __  __   2 2 2 2 2 1 2 2 1 1 ₍₁₎

in which the Reynolds number Re is defined as ReV r_{ref v}  and the second overline indicates the time-averaged value of a product of turbulent velocity components giving the Reynolds stresses. Equation (1) has been used to investigate how various terms contribute to the pressure distribution in the vortex. For this a distinction has been made between (i) the azimuthal velocity component



V_2 r



, (ii) all terms with a mean velocity component only and (iii) all terms including Reynolds stresses. The measured velocity components were first averaged in azimuthal direction and then numerically integrated in radial direction using a Hermite spline to interpolate between the measurement locations. Figure 2 shows that the pressure distribution is reasonably well predicted from the azimuthal velocity component alone. The contribution of the other terms with mean velocity components in equation (1) is negligible and the result is not shown. Including the contribution of the Reynolds stresses gives some improvement near the vortex centre. The difference in the value of the minimum pressure is 16% using the azimuthal velocity component alone and 7% using all terms including Reynolds stresses. At station x/c= 0.005 the influence of the Reynolds stresses was much smaller and the difference in the value of the minimum pressure was 4% using the azimuthal velocity alone and 5% using all terms including Reynolds stresses.

Figure 1. Measured V_ interpolated to an O-type grid.

Figure 2. Reconstruction of the pressure distribution from various measured velocity terms in equation (1).

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3. Comparison of vortex models

For comparison of the results of different vortex models only the azimuthal velocity is considered, presented in non-dimensional form with V_ V_



_ 2r_v



and rr r_v as





 









 





 

. . . . . Rankine ( ) Lamb-Oseen [4] e ( ) Burnham-Hallock[5] ( ) e e Proctor[6] ( ) e Proctor-Winckelmans[7] exp i r B r r r B r B i r r V r r V r r V r r V r r V r                  _       _ _{ } _     _ _      2 0 75 ₂ 0 75 1 2526 2 1 1 2526 1 4 2 1 2 1 1 1 ₁ ₃ 4 1 1 1 1 5 1 ₁ ₁ 1 ₁



₁

_{ }



 





. . . ( ) Jacquin[8] ( ) Vatistas[9] ( ) p o p r B n n r r _r V r r r r r r r r V r        _ _    _ _        _  _    _    1 1 2 0 5 0 0 5 0 0 0 1 2 6 1 1 1 7 1 8 1

In the vortex core model of Proctor and Proctor-Winckelmans the parameter B corresponds to the non-dimensional span of the wing. The parameter  in the Proctor model is a tuning parameter with suggested value 10. The suggested ranges of tuning parameters in the Proctor-Winckelmans model are p 2 4,010,i400 800. The parameter r 0 in the Jacquin model corresponds to

the radial location at which the roll-up region ends and the potential flow part of the vortex starts with . .

r00 1 The parameter n in the Vatistas model is also a tuning parameter for which nB 2 gave a good fit with experimental data obtained for a vortex chamber.

A comparison with the experimental data is given in figure 3. The pressure distribution inside the vortices was obtained by numerical integration (or analytical integration if possible) of the azimuthal velocity distribution using _p





r

C V_ r d r 

 



₂ 2 _{. For the non-dimensionalisation we have to}

determine _ and r . The parameter _v r could directly be obtained from the maximum in the v

azimuthal velocity but for the O-type grid shown in figure 1a the circulation was not yet constant at the outer radii. The measurement grid was, however, larger for one segment of the O-type grid and at these outer radii the circulation no longer changed. The value for  at these radii was about 5% larger than that obtained at the outer radius of the O-type grid.

The tuning parameters were adjusted by fitting the azimuthal velocity to the experimental data. The most accurate fit is obtained with the vortex core model of Proctor-Winckelmans, which also has the largest number of tuning parameters, closely followed by the Vatistas vortex core model, which has

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only one tuning parameter. Both the Rankine and Lamb-Oseen model give a very significant overprediction of the azimuthal velocity and therefore much too low values of the pressure coefficient. The fit of the vortex core models was applied to five cross-flow planes located between

0.005

x c and x c0.678 and the tuning parameters were adjusted for each plane. Results for the different vortex models were similar. The tuning parameter of the Vatistas model showed an almost linear variation with the reference velocity that varied between the planes while the parameters of the Proctor and Proctor-Winckelmans model showed quite some scatter. Application of the Vatistas and Proctor model to experimental data of wings of elliptical planform, published in [10], confirmed the presence of a distinct trend of the parameter of the Vatistas model with reference velocity and also confirmed the scatter of the parameter of the Proctor model.

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Figure 3. Comparison between different vortex core models and experimental data [3] at x c0.678 for the azimuthal velocity distribution (left) and the pressure distribution (right).

References

[1] Lafeber F H, Bosschers J and van Wijngaarden E 2015 Computational and Experimental prediction of propeller cavitation noise, Conference Oceans’15 MTS/IEEE Genova, Italy

[2] Bosschers J 2010 On the influence of viscous effects on 2-D cavitating vortices, 9th International Conference on Hydrodynamics, Shanghai, China

[3] Chow J S, Zilliac G G and Bradshaw P 1997 Mean and turbulence measurements in the near wake of wingtip vortex, AIAA Journal Vol. 35, No. 10, pp. 1561-1567

[4] Lamb H 1932 Hydrodynamics, Cambridge University Press

[5] Burnham D C and Hallock J N 1982 Chicago monostatic acoustic vortex system system, Report No. DOT-TSC-FAA-79-104.IV

[6] Proctor F H 1998 The NASA-Langley wake vortex modeling effort in support of an operational aircraft spacing system, AIAA 98-0589, 36th Aerospace Sciences Meeting & Exhibit, Reno, Nevada

[7] Winckelmans G S, Thirifay F and Ploumhans P 2000 Effect of non-uniform wind shear onto vortex wakes: parametric models for operational systems and comparison with CFD studies, Proc. 4th Wakenet Workshop on Wake Vortex Encounter, NLR, Amsterdam, The Netherlands

[8] Jacquin L, Fabre D, Geffroy P and Coustols E 2001 The properties of a transport aircraft extended near field: an experimental study, AIAA Conference Proceedings 2001-1038

[9] Vatistas G H, Kozel V and Mih W C 1991 A simpler model for concentrated vortices, Experiments in

Fluids 11, pp 73-76

[10] Fruman D, Dugue C, Pauchet A, Cerruti P and Briançon-Marjolet L 1992 Tip vortex roll-up and cavitation, 19th Symposium on Naval Hydrodynamics, Seoul, Korea