Date Author
Address
August 2009
Oprea, lulia, and Norbert Bulten Delft University of Technology Ship Hydromechanics Laboratory
Mekelweg 2, 26282 CD Deift
TUDeift
Deift University of Technology
RANS simulations of 3D sheet-vortex cavitation
ByLulia, Oprea and Robert Bulten
Report No. 1666-P
2009
Published in: Proceedings of he 7th International Symposium on Cavitation, CAV2009, University of Michigan, Ann Arbor, USA, 17-22 August 2009
CA V2009
?h International Symposium on Cavitation
Conference Proceedings
August 16th -2O", 2009
Rackham Building,
915, E. Washington St
University of Michiqan, Ann Arbor, USA
ca v/tat/on. engin. umich. edu
Welcome!
On behalf of the local organizing committee and the conference co-chairs, I would like to welcome you to Ann Arbor and the 7th International Symposium on Cavitation: CAV2009.
The aim of the symposia series is to promote the worldwide exchange of cavitation knowledge.
Theinaugural meeting of the series was held in Sendai, Japan, in 1986. Over time, the scope and participation in this meeting has grown to encompass almost every aspect of cavitation. We have accepted 116 papers for the symposium covering a wide range of topics, including fundamental cavitation flow physics, cavitation issues associated with turbomachinery and naval systems, and new applications of cavitation in industrial and biomedical systems. We all will learn about the most recent advancements (experimental, numerical, and theoretical) in the understanding, prediction, and management of cavitating flows. Our six plenary speakers will share their insights on a range of interesting and important subjects.
I would like to thank the Scientific Committee for their help in the paper review process. Their efforts are vital to maintaining the quality of the symposium. Moreover, the technical papers judged by the Scientific Committee to be of the highest quality and interest will be selected for publication in a special issue of the ASME Journal of Fluids Engineering.
Finally, I would like to thank the Local Organizing Committee for all of their effort in bringing this meeting about. I would particularly like to thank the Ms. Jane Ritter, Mr. Harish Ganesh, Dr. Natasha Chang (the
Chair of the Local Organizing Committee), and the UM Conference Services for their tremendous
contribution to the success of the symposium.
On behalf of my conference Co-Chairs, Prof. Joseph Katz and Dr. Georges Chahine, I extend to you a
warm welcome to Michigan.
Prof Steven L. Ceccio
People
Program Chairs Steven Ceccio Joseph Katz Georges Chahine Program CommitteeNatasha A. Chang (Local Chair) David R. Dowling Zoran Filipi J. Brian Fowikes Wei Shyy Armin W. Troesch Scientific Committee
Roger Arndt, University of Minnesota
Francois Avellan, Ecole Polytechnique Federale de Lausanne Goeran Bark, Chalmers University of Technology
Laurence Briancon-Marjollet, Bass/n dEssais des Carenes Tim Colonius, California Institute of Technology
Larry Crum, University of Washington Luca dAgostino, University of Pisa
Mohamed Farhat, Ecole Polytechnique Federale de Lausanne John E. Field, University of Cambridge
Jean-Pierre Franc, Laboratoire des Ecoulements Geophysiques et Industriels Grenoble Toshiaki Ikohagi, Tohoku University
Stuart Jessup, Naval Surface Warfare Center- Carderock Division Hiroharu Kato, The University of Tokio
Valery Kedrinskii, Lavrentyev Institute of Hydrodynamics Ki Han Kim, Office of Naval Research
Kwang-Yong Kim, lnha University
Spyros Kinnas, University of Texas atAustin
Ivan Kirschner, A lion Science and Technology Corporation Gert Kuiper, Consultant
Detlef Lohse, University of Twente
Yoichiro Matsumoto, The University of Tokyo Knud Aage Moerch, Technical University of Denmark
Kirill V. Rozhdestvensky, Saint Petersburg State Marine Technical University
Vladimir Serebryakov, Inst. of Hydromechanics of Ukrainian NationalAcademy of Science Bernd Stoffel, Darmstadt University of Technology
William Straka, Applied Research Laboratory- Pennsylvania State University Shu Takagi, The University of Tokio
Yoshinobu Isujimoto, Osaka University
Tom van Terwisga, Maritime Research Institute Netherlands & DeIft Technical University Yulin Wu, Tsinghua University
Plenary Talks
Cavitation erosion: towards a new approach - Prof Jean-Pierre Franc, University of Grenoble, France
Monday August 172009, 9. 15-10.05 AM About the speaker
Prof Franc is the Research Director (CNRS), Turbomachinery and Cavitation Research Group, Laboratory of Geophysical and Industrial Fluid Flows (L EGI) of the Grenoble University Insitut National Polytechnique de Grenoble (INPG) and Universitd Joseph Four/er, France. He has published extensively in the area of cavitation, and is the author of 'La cavitation. mdcanismes physiques et aspects industriels and the co-author of Fundamentals of Cavitation
Physical and mathematical problems of hydrodynamics for
high speed underwater motion with
supercavitation - Dr. Vladimir V Serebryakov, Institute of Hydromechanics - Kiev, Ukraine
Monday August 172009, 1.00-1.50 PM About the speaker
Dr. Vladimir Serebryakov, Ph.D., leading scientist of Institute of Hydromechanics of National Academy of Sciences of Ukraine, project manager is known expert in the field of High Speed Hydrodynamics including supercavitation, drag reduction and propulsive systems, dynamics and hydro elastics problems, sub-, supersonic flows in water. Double high education: shibuilding
engineering and physics-mathematics sciences. Post graduate 1969-1972 at the Institute of Hydromechanics of NASU After that he for over 25 years has been closely collaborating with Prof Georgy Logvinovich - father founder of the famous Russian torpedo Shkval. Dr. Serebryakov is author of asymptotic theory for ax/symmetric supercavitating flows in incompressible fluid, for subsonic and supersonic speeds. He developed equations which expressed known principle of "Independence of the cavity expansion" introduced by G. Logvinovich. At present these equations are seen as one of the most effective way for practical estimation of supercavitation flows. Over 100 papers, National Award of 2002 on science and engineen'ng, DAAD stiendium -Germany 2002, Brain Power stiendium - South Korea 2007, member of sci. corn. of CA V200 1- USA, 2003-Japan, 2006-Netherlands, High Speed Hydrodynamics scientific school "HSH' HSH2002, 2004, 2006, 2008, SuperFAST2008 - Russia.
Numerical aspects of the collapse of non-spherical bubbles- Prof Hiroyuki Takahira, Osaka Prefectural University Japan
Tuesday August 18 2009, 8.30-9.20 AM About the speaker
Hiroyuki Takahira is currently a Professor of the Department of Mechanical Engineering at the Osaka Prefecture University. His current research interests are bubble dynamics, cavitation, gas-liquid two phase flows, and computational fluid dynamics. Hiroyuki Takahira received his B.S. and MS. degrees in Mechanical Engineering from Kyoto University in 1985 and 1987, respectively. He received his Doctor of Engineering degree from Kyoto University in 1992. He joined Kyoto University in 1988 and subsequently worked about 8 years as an instructor and lecturer of the Department of Mechanical Engineering. In 1995, he joined Osaka Prefecture University as an associate professor of the Department of Energy Systems Engineering. He was promoted to a full professor of the Department of Mechanical Engineering at the Osaka Prefecture University in 2004. He was awarded the JSME Young Engineers Award in 1993, the JSME Medal for Outstanding Paper in 1999, and the Frontier Award ol JSME Fluids Engineering Division in 2008.
Naval Propeller Cavitation: Historical Development of Design, Evaluation and Prediction- Dr. Stuart Jessup, Naval Surface Warfare Center, Carderock Division, LISA
Tuesday August 182009, 1,25-2. 15 PM About the speaker
Dr. Jessup attended MIT from 1970-1976 receiving his BS and MS/n Ocean Engineering. He then began his career at the Naval Surface Warfare Center Carderock Division as a member of the Propulsor Branch within the Hydromechanics Department. In
1989 he received his PhD from The Catholic University of America.
Dr. Jessup developed as a propeller designer and an experimental scientist conducting research related to improving the design process and the overall quality of naval propulsors. In 1982 Dr. Jessup developed Laser Doppler Velocimetry (LDV) for use in measuring detailed propeller blade flows, including blade boundary layers. In 1988, he began the development of arbitrary propeller blade section technology for the improvement of propeller cavitation performance. This led to installation of an advanced blade section propeller on the DDG-79 Flight I/a class. In 2002 Dr. Jessup was promoted to the position of Senior Scientist for Hydrodynamics for the US. Navy. In recent years lie has investigated unsteady flows related to the ASDS, UUV docking, and propellers operating in crashback. Presently he is working on the DDG- 1000 SOE development and investiqating propeller operation/n heavy seas.
Dr. Jessup received The Washington Academy of Science Engineering Science Award in 1986, the NSWCCD David W Taylor Award for Scientific Achievement in 1996, the Navy Meritorious Civilian Service Award in 2000, the ASNE American society of Naval Engineers Gold medal award/n 2004 and the SNAME Davidson Medal in 2008. Dr. Jessup was also inducted into the NAE, NationalAcademy of Engineers in 2007.
Nozzle-geometry-dependent breakup of diesel jets by ultrafast x-ray imaging: implication of in-nozzle
cavitation - Dr. I/n Wang Argonne National Lab, USA
Thursday August 202009, 8.30-9.20 AM About the speaker
Dr. I/n Wang, Physic/st and Group Leader for Time-Resolved Research at the Advanced Photon Source (APS) of Argonne National Laboratory (ANL), earned li/s doctoral degree in physical chemistry from The Ohio State University in 1994. After so, he was appointed a post-doctoral fellow at Exxon Research and Engineering Company. He continued his research at ANL in 1995 as a post-doctoral fellow, and was promoted to assistant physicist/n 1997, physicist/n 2001, group leader in 2003. H/s research interest includes emerging science and engineering on advanced combustion of conventional and alternative fossil and bio-fuels, structure-function relationships in dynamical systems. His /s currently working on dynamics and structure of pressure, high-speed fuel sprays for energy applications, kinetics and dynamics of metal/polymer nanocomposites and interaction between high-power and short-pulse laser and solid state surfaces. Wang has co-authored or authored more than 100 journal article publications including those in Nature, Science, Nature Physics, Advanced Materials, and Physics Review Letters. Wang received numerous awards, including the Best Paper Presentation A ward of the ASME Internal Combustion Engine Division in 2006, the University of Chicago Distinguished Performance Award in 2005, the US Department of Energy National Laboratory R&D Award in May2002, the Finalist, Discover Magazine Technology Innovation Awards in 2001.
Cavitation Modeling: bridging the gap between micro- and macro-scales- Dr. Georges Chahine, Dynafiow,
LISA
Thursday August20 2009, 1.25-2. 15 PM About the speaker
Dr. Georges Chahine, President and founder of Dynaflow has acquired a very broad academic background - civil engineering in 1970 from University St Joseph, Be/rut Lebanon (ES/B), naval architecture, 1972, and Engineering Doctorate/n Fluid Mechanics, 1974 (from ENS TA, Paris) and Doctorat d'Etat es-Sciences in Applied Mathematics, 1979 (U. Pierre and Marie Curie, Paris). He spent eqht years in academia and led a research group on the study of interface phenomena (ENS TA, Paris), then another eight years with the engineering firm, Tracor Hydronautics Inc., directing the Fluid Mechanics and Materials Science Department before founding Dynaflow in 1988. He has published more than 300 technical papers and reports and has three patents - two on decontamination of liquids with the DynaJets cavitating jets and one on a cross flow filtration system. Dr. Chahine has very actively contributed to the field of cavitation and bubble dynamics and has directed numerous investiiations on cavitating and
vortical flows, on waterjet technology and in various acoustic and hydrodynamics fields.
Using the Proceedings Flash drive
The proceedings of the conference are made available to the participants in an electronic form rather than a hard bound book. In order to use the electronic version, make sure that the computer used for viewing the proceedings has Adobe Acrobat Reader. The document is in a pdf form.
Once the flash drive is loaded, a text document titled Instructions and two folders CAV2009-papers and Proceedings would be visible in the explorer window of the removable flash drive. There are three ways to access any paper The papers can be directly accessed (by paper number) by getting in to the CAV2009-papers folder.
Another way of accessing the paper is through the day wise schedule located in the Proceedings folder. There are four pdf files one for each day of the conference. In these files the paper number in bold font has the link to the paper. This link will open up the corresponding paper in the CAV2009-papers folder.
Yet another was of accessing is through the book of abstracts. The book of abstracts can be found in the
Proceedings folder. The author index can be found from the book of abstracts.
After noting the papernumber of the author, the paper wise index in the book of abstracts can also be used to get the appropriate paper. In the paper index, the bold faced number is a link to that corresponding paper.
Symposium Tour and Banquet
CAVS 2009 attendees will spend an evening touring the Ford Rouge plant, followed by a banquet reception at the Henry Ford Museum. The tour and banquet will be held on Wednesday, August 1 gth
Transportation has been arranged to take conference participants and guests to and from the venues. Buses will leave from Rackham Auditorium at 1:40 pm (after lunch). The tour of the Ford Rouge Complex will take place from approximately 3:00 pm to 5:00 pm. Then, we will return to the buses for transportation to the Henry Ford Museum. Dinner will take place from 6:00 pm to 8:00 pm, and them participants will be brought back to Rackham Auditorium.
ABSTRACT
On marine propellers, cavitation appearance and development is critical for performance and erosion considerations. Behind a ship, the propeller experiences all kinds of cavitation types, varying from sheet and bubbles to tip vortex cavitation. When a cavitation analysis is required, two methods are available: experimental or numerical.
To find the optimum propeller that fits into different
configurations and requirements, designers need accurate predictions within reasonable time. The experimental method is
typically used at
the end of a design process to verify
perfoniiance. Therefore, quick and accurate numerical
predictions are essential at different stages in the design process, to evaluate performance and cavitation patterns.
Mathematical methods range from basic panel codes to the more complex ones, derived from the Navier-Stokes equations. Methods like DES and LES require large meshes and small
time steps wlìich makes their usability limited. The most
practical viscous numerical method available at the moment in industry is Reynolds Average Navier-Stokes (RANS).
The current paper will present the results ofa RANS simulation of a 2D sheet cavity and a 3D sheet-tip vortex cavitation. Accurate results of these basic simulations are steps towards the end goal, cavitating propeller simulations. In this method the viscous effects are taken into account with aid of a two equation turbulence model, which results in a reasonably fast approach due to reasonably grids requirements.
It is concluded that the RANS method can predict complex 3D sheet-vortex cavitation development and shedding. In addition,
it is appropriate for industrial use because it achieves
reasonably quick and accurate results. As a next step in the research project, the cavitation development on a propeller will be analyzed with this method.
INTRODUCTION
Cavitation is the vaporization of a liquid when pressure drops below the saturation pressure of the liquid. Many engineering machineries deal with appearance and disappearance of
Proceedings of the7thInternational Symposium on Cavitation CAV2009 - Paper No. 49 August 17-22, 2009, Ann Arbor, Michigan, USA
RANS simulations of a 3D sheet-vortex cavitation
lulia Oprea Norbert Bulten
Wärtsilä Wärtsilä
The Netherlands The Netherlands
cavitation that causes noise, vibrations and erosion. The present paper deals with the interaction between sheet cavitation and vortex cavitation prediction, using numerical modeling. This
interaction is an important issue for marine propeller design. Cavitation is a design issue for all propellers. The type of
cavitation can be divided in
sheet, bubble and tip vortexcavitation. In
this paper the emphasis will be put on the
interaction of sheet-tip vortex cavitation. This type starts along the leading edge of the wing/blade and develops in to a vortex towards the tip, where a low pressure region is foniied. When the pressure gets below the vapour pressure, a clear sheet-vortex cavitation can be observed in the experimental tunnel. The cavitating tip vortex is a source of noise and vibrations. For specific propeller designs, this type of cavitation is to be avoided to reduce the noise. Evaluation of the propeller design is generally based on model scale tests. However, due to costs involved in the model tests, scaling effects (see [I]) and the fact that the design process involves numerous intermediate steps to be analysed, a numerical approach is desired.
An industrial alternative for the experimental investigation of the flow around a propeller is the use of numerical methods. These methods can be split into three different groups; potential flow methods, Euler methods and Reynolds averaged Navier-Stokes (RANS) methods. A potential flow method neglects viscosity and vorticity in the flow. Since the flow phenomena in the tip region are governed by both viscosity and vorticity, it is
concluded that potential flow methods are not capable of
analysing complex sheet-tip vortex phenomena. Euler methods neglect viscous effects, but can take vorticity into account. This vorticity is prescribed and not affected due to the missing viscous effects. A RANS method takes both viscosity and vorticity into account. Such a method is suitable, in principle, to investigate sheet-vortex flows.
Details about the numerical approach and cavitation modelling will be discussed in the Numerical Background paragraph. In order to quantify the CFD accuracy of a 3D cavitating flow, results of a model test are used for validation purpose. The experimental setup consists of an Elliptic 11 Rake hydrofoil
which is a specially designed wing to exhibit this special type of cavitation similar to a propeller. The angle of attack varies with the span, starting with 3 degrees at the tunnel walls and reaching 11 degrees at the tip. This arrangement is specially developed to produce a steady sheet cavitation at the leading edge that develops into tip vortex cavitation towards the tip of the foil. Experimental values of the forces on the foil and the velocity field are available for the cavitating flow case and can be found in [2], [3] and [4].
To validate and quantify the cavitating flow predictions of a given RANS code, simulations similar to the experiments on Elliptic Ii Rake are performed both in wetted and the cavitating
cases. Before that, a 2D NACA profile is analyzed as a test case. For certain conditions this profile exhibits a shedding cavity, which has to be captiircd with the numerical cavitation model as well. In the second step, the 3D case is analyzed and
the capability
of a
sheet-vortex cavitation prediction isinvestigated. Velocities and forces are compared with the experimental values for the cavitating case. Moreover,
development and shedding of the sheet-tip vortex cavity are captured by the current method. The ability of unsteady RANS simulations to capture sheet-vortex cavitation development is
analyzed.
Two topics of the model scale calculations vill be discussed in more detail: (i) influence of the applied turbulence model and (ii) local mesh refinement in the vortex core on the cavitating results. Cavity visualization results of the Elliptic II Rake foil are compared with the pictures of experimental results of the cavity found in [4]. The validation of the numerical model is addressed and the calculated forces and velocity distributions are compared with experimental data as well. Finally, detailed results of the tip vortex cavitation formation, velocities and vorticity are analyzed.
EXPERIMENTAL SET UP
The experiments were conducted in the cavitation tunnel at Delft University of Technology. Due to the fact that the interest in the measurements and CFD is the sheet-tip vortex cavitation interaction, a special foil that gives such a cavitation pattern is
investigated. Van der I-lout [4] has carried out cavitating experiments on a 3D elliptic skewed hydrofoil with a finite span to investigate the interaction between sheet cavitation and
vortex cavitation.
The investigated finite-span hydrofoil with tip rake and increasing angle of attack to II degrees at the tip is named
Elliptic 11 Rake foil.
The focus of the experiments was to visualize the tip vortex cavitation at different angles of attack and to measure the forces to get a better understanding of the physics of the sheet and vortex cavitation interaction. Experimental results will be used and presented further when compared with the CFD results. NUMERICAL BACKROUND
Ihc commercial code STAR-CD version 4.02 [5] is used for all
flow simulations.
Flow motion equations and
cavitationmodeling used are presented in the current paragraph. In the present paper the numerical approach used for flow simulations are the incompressible RANS equations. In this case system of equations is formed by the mass conservation equation (I) and impulse conservation equations (2).
(1)
-
-)p+puVu=Vp+pV2u+pg (2)
In the conservation equations ii is the velocity tensor, p is fluid density, g is the gravitational force tensor and l is the viscosity
of the fluid.
The turbulence models used are
either the two-equation standard k-c turbulence model or the RNG k-c turbulencemodel in conjunction with the algebraic law-of-the-wall
approach.
The discretization schemes are second order MARS in space and first order Euler implicit in time. The solver procedure is a steady (wetted conditions) or transient (cavitating conditions) flow calculation with SIMPLE.
Cavitation modeling available and used within the following simulations is described next. The solution methodology used can handle cavitating flows and belongs to the class of so-called interface-capturing or fixed-grid methods, also known as VOF methods. It deals with a single continuum whose
properties vary in space according to its composition. The solution of the transport equations for the component fluids determines the composition.
The transport of vapor is computed according:
+ V
(clt) = Sa
(3) In equation (3) S represents the source of volume fraction ofvapor. And the volume fraction of vapor is
defined as:=
V is the fraction of the control volume V occupiedby vapor.
The initial volume fraction of vapor is defined by the number of seed bubbles n0 and their initial radius R by:
(4/3R3)n0
1+(4/3irR3)n0
The source term in equation (3) is defined as:
4,rR2n0
dR
a
1+(4/3nR3)n0 di
lii
equation (5) the rate of change of a bubble radius
isestimated using a simplified Rayleigh-Plesset equation:
2
(4)
=sign(p -i4
21p%, -p13PL
where p
is the saturation pressure, p the local pressure
around the bubble and PL the density of the fluid,
The volume fraction of the components is determined from the
condition:
(7)
And the properties of the effective fluid vary in space according to the volume fraction of each component. Density is defined
by:
p = ap +
(8)and viscosity:
/1 = ap +
(9)All the transport equations are the same for the effective fluid as in the single phase flow case, with the exception that density and viscosity vary sharply across the cavity surface.
NUMERICAL RESULTS 2D TEST CASE: NACAO015
Numerical modeling of cavitating flows is difficult, due to the coexistence of two fluids, water and vapor. The main issues are the treatment of the surface between the two phases and the mass transfer from one phase to another. Therefore a simple 2D test case is used as a first step for cavitation assessment. The cavitation model described previously is analyzed in terms of grid resolution and turbulence modeling.
A NACA 0015 profile is used for wetted and cavitating flow simulations, being a benchmark test case for many researchers and therefore results for comparison purpose are available for this 2D case.
The analyzed NACA 0015 profile has an angle of attack of 6 degrees and a chord of 200 mm. The domain size is 1400 x 570 mm, therefore 2 chords at the inlet and 4 at the outlet. Profile mesh is a multi-block structured grid, with an 0-grid type around the profile (including a small round trailing edge) for a good control over the y+ values, see figure 1.
Applied boundary conditions are an inlet boundary type with inlet velocity of 6 m/s and turbulence intensity of 1%, pressure boundary of 0 Pa at the outlet and slip walls at the outer domain
sides.
(6)
Figure 1: NACA 0015 Profile mesh
Three flow conditions are analyzed: (I) wetted flow, (2) steady cavitating flow at c=1.6 and (3) shedding cavitating case at
o=l .0. In computation values for the water density of 998 kg/m3 and vapor density of 0.023 kg/nY are used.
I. Wetted/low case. In this case RANS wetted flow simulations are performed over the NACA 0015 profile with the previous described settings. Results obtained for pressure coefficient and lift and drag coefficients are presented in the following for grid resolutions and turbulence modeling influence assessment. Wetted flow pressure coefficient results, for grid GI (250 profile vertices) and grid G2 (418 profile vertices) with the standard k-c turbulence model and grid G2 with the RNG k-c model are shown in figure 2.
The pressure coefficient Cp is defined by:
c=
Preiatjve° O.5pu
The Cp distribution along the chord for the mesh and
turbulence variation is presented in figure 2.
NacaOOlS, Gdeg., GmIs, wetted flow
Figure 2: Pressure Coefficient NACAOOI5, wetted flow
Figure 2 shows little difference for
all three investigated arrangements. Only near the leading edge differences can be seen at pressure and suction side. At the stagnation point the Cp3
(10)
dR
is 1.053 for grid GI, 1.054 for grid G2 and 1.016 for grid G2
with the RNG turbulence model. Over-prediction of the
pressure coefficient in the stagnation point is a well known issue of the turbulence modeling; see Bulten & Oprea [1] and Moore & Moore [7]. Therefore, the best estimation in figure 2 is obtained with the grid G2 with RNG model due to the fact that is the closest to exact value of the Cp in the stagnation point, which is unity.
Non-dimensional lift and drag coefficient are defined in the by the equations (II).
CL
Lfl
CDDrag
(II)
O.5pu2cS
O.5pucS
Lift and drag coefficients variation with mesh and turbulence are presented in table I.
Table I: Li t and Drag Coefficients for NACAOO 15 profile for wetted flow condition
Lift coefficient predictions are in
the same range for all
analyzed cases, while drag coefficient predicted with the RNG model is lower than the predictions made with the standard k-e. Over-prediction of the pressure in the stagnation point will result in an over-prediction of the pressure drag of a profile (see [I ]). Due to the fact that stagnation pressure is better predicted with the RNG model, the corresponding drag is consequently lower. Lift coefficient is less affected by the over-prediction of the stagnation point pressure.
2. Steady cavitating case. In this case the cavitation model is enabled and time dependent RANS simulations are performed over NACA 0015 profile.
Using the Bernoulli equation, the cavitation number is defined
as:
a
-O.5pu2
From equation (10) and (12) the relation between pressure coefficient and cavitation number is defined by equation:
a =
Cpmin (13)The steady cavitating case results at =l.6 are presented and analyzed next for the same 3 cases as for the wetted flow by means of pressure and lift and drag coefficients.
The pressure coefficients
for grid and turbulence model
variation are shown in figure 3.
(12) 4 9 04 02 02 -00
NacaOOl5, fldeg., 6m1., cavitating flow stgma.1.S
Figure 3: Pressure Coefficient NACAOO15, steady cavitating
flow
The figure shows a constant Cp of -1.6 in the cavitating cases. Tlìis corresponds to the leading edge cavity sheet presence and confirms that the cavitation appears when the vapor pressure is reached, see equation (13). In this case the mesh resolution and turbulence model influence is limited. The RNG model predicts a slightly longer steady cavity.
Lift and drag coefficients corresponding to the steady cavitating case are presented in table 2.
Table 2: Lift and Drag Coefficients for NACAOOI5 profile for steady cavitating flow condition
From tables I and 2 the influence of the cavitation over the profile performance is assessed. Lift coefficient is decreasing and drag coefficient is increasing compared with the wetted flow case, when cavitation is present. Grid and turbulence variation have limited influence over lift and drag prediction results in this case, like in the wetted flow case.
3. Shedding cavitating case. The third case in this 2D study is represented by a time dependent shedding cavitating flow case at o=1.0. A lower cavitation number implies more cavitating fluid and appearance of instabilities and shedding of cavity
clouds.
When analyzing the unsteady cavitating case at o=l.0 the
differences between the turbulence models used are noticed. The mesh refinement, time steps and inlet values for turbulence have no influence on results using the standard k-c model. The cavity is slowly increasing and decreasing periodic in time without shedding. When the RNG model is used the cavity becomes highly unsteady and the cavity starts shedding. In figure 6 the cavity volume variation in time for grid resolution and turbulence modeling influence is shown.
Name 01 k-c G2 k-c 02 RNG Cl Cd 0.644 0.019 0.638 0.020 0.667 0.014 Name Gi G2 G2 RNG Cl Cd 0.630 0.022 0.629 0.022 0.642 0.019
00156 041314 0.0032 0001 0 00301 0.0004 00302
NACAOOI5, alfe6, V.6mis, Cn. 3
1000 0000 0800 0100 0800 .05o0 0400 0330 0200 0700 o too 04 0.45 05 050 06 065 01 31,00 s.i
Figure 5: Lift coefficient variation in time
The lift coefficient is varying with the cavitation forniation, detaching and shedding. A frequency analysis of the shedding simulation case lift and drag coefficients results in a first order frequency of 14Hz and a second order frequency of 27Hz. These high frequencies obtained for lift and drag coefficients are related to the collapse of shedding vapor structures. This simple 2D case proves that the current method is capable of capturing complex cavitating flows: attached sheet cavity, unsteady shedding cavity, reentrant jets and vortices, when the
RNG turbulence model is used. Therefore, the RNG turbulence model performs better in cavitating flows predictions and it will be used on the 3D case simulations.
3D CASE: ELLIPIC It RAKE WING
The geometry used for three-dimensional case is an Elliptic 11 Rake wing, with a NACA0009 profile, root chord of 0.15 in and tip chord 0.05m as described in [4].
The numerical domain sizes are: inlet location at 2 root chords upstream of leading edge, outlet at 3 root chords downstreamof
trailing edge and a normal test sectionof2 by 2 root chords, as in the experimental setup, see figure 6.
Figure 6: Computational domainofElliptic II rake hydrofoil The mesh for the wing is created with a structured multi-block hexagonal mesh generator. The mesh near the blades is based on a C-grid type (sharp trailing edge), to maintain control over the quality of the mesh near the blade. Development of the boundary layer along the blade surface is taken into account using wall functions. The requirements for the y+ values on the foil surface can be met with an acceptable number of cells in the normal direction.
Moreover, the aspect ratio and the
differences in cell sizes can be kept low. Figure 7 shows the surface mesh of the Elliptic II Rake hydrofoil.
-.S,0%t4t650WRh//.Z0#/.Yf/Mt//4t2' 20000045 00(4510 0600/////4415'flYhtt .,,/,'#/il/,v,/I/t///////,'I,IfIflMh,J,/oci /4O9W4W#/I/fi///II/l/IIJflIOhhIIflhj2106 .20/.W//IllI/F/f&//jIJI///I//Ifflh/J1fljIlfll,/ ,iW.t4WI///iNh/I//#h11!//!hWIflpp1flj0400i ':-%nat,Inu,nnn,on,,,nw,,nnunba,.. 22'//IJIW11I/II!I/III1II/1ffluhIIIf$I!ThN40' .4///f/flI#I/IJffffffJJJJllI1Iff!flIffhIliftflJU 1/I/III1#hIIInIflflflIflhIrnflflUflflhIwO'tt (I/I/IIIIJilihIII!iIIIfIIlmIIIIflfIIfluug;poio '/I/llIffII!IIImIIflIflhIIIIUIIIIIflljIIflffl73t' 1/41HUjffIl1INIflJflhIIf,fl,flflflIflJ,fflJ41,. -111f#I0lilI77IIflffSIIflIIUbflhhIIIltU3IIfff00$ 10flhIlIUhIdIflhIIIIIIIflhIIftIffIIUtIJIflh7IOt 'iflhl,uhI,IuIuhIIIIIItSj,bflI.ft'flfl'I'Jj,fi0i0, t0731l13fIIIIIIIflIflhJItftI0flIflflJhIJflJf33870 I 021097114.' .tIltIIlIIIIlUtIUIlllIItflflflJflhJfthfggJIgIgJgot: lltIOhIIIlihlIlIlIIIIJ'1113111t0110011ntUliffhli hlfItI7l 140760 IUhIliWiIiiiaflhiIliiiiIflI 1103101. .ttIWlIIIIItIII111IIIIhlItIIUUIlllUfIJlllflfloJlIll 680863 I 8 I I I I 6 2 I I 0711 6UINIII 0tall4
ilIL'iffIffJflJffEffi
Figure 7: Surface mesh of Elliptic 11 Rake hydrofoil
5
04 045 09 ass os 064 07 075 08 0.03
hole (s.e.]
Figure 4: Cavity volume variation in time
The RNG model produces less turbulent viscosity at the
water-vapor interface, leading to a strong reentrant jet which is
capable ofdetaching the tail of the cavities. Therefore, more unstable and faster shedding of the cavities can be seen in figure 4.
Figure 4 shows that the cavity oscillations are insensitive to different grid resolution with the standard k-c model (dark blue and red lines). When the RNG model is used for the same grid, highly unsteady and faster periodic variations in the cavity volume appear (grcen line). The RNG turbulence model
predicts a time period of 0.07s resulting in a frequency of
14Hz, contrary to 4l-lz as obtained with the standard k-c model. In the literature for NACA 0015 frequencies from II to 24 are reported, see [8].
The corresponding lift coefficient variations for the unsteady cavitating case are presented in figure 5.
NACA 0015,141 Coofficiont wetted flow 00. cavitOling So gm1.0).
StOfldard 05. R74G torbolonoa wodel
The C-grid around the profile is shown in figure 8. This type of meshing is efficient for y+ control, which is kept constant for
all grid variations, with values between 15 and 100, as recommended for the wall function approach [5].
0
I,/!II,II4gIIsg,
r,,,, III,a,I1,i,11111111111n,jiiiiiiiiiiiiiuiuiiiiiiiiinuiui,i I IIfl 1111111
ph,II!III,jIIIIIh lIIIIlHIlluflIIuIIIflhIIuIuuIIuuIflI IIUIHllI'l IlillIuIflu
lIlj,,,:1Illg,iluIlIl8huuuuIuuuuuIIuIIIuuuuIuIInuuuu 11111 IllIti lillIllIllIl I, 111111 I IlIllIluIll
r ' 11111, 1iIIIIIIIIIIIIIIIIIIIIIIIIIIIIIlIIIIIIIIIIIIIIIIHllIMI IlilIllIllIll
l!IIII,,,l,ihIIIIItIIIIIIUiiiiiJiiiiIiiiiiiiiiiiliiIiiiIiiiHHhlhlui I'iJIIIIIIIIII - i.pui,llliiiiiuiiuiiiuuiuii,iiiiu.uuiiiuiuupu,,iuuuu,iiiuiiii IIuIriIIuuuui
Figure 8: Cross-sectional mesh of Elliptic II rakc hydrofoil Applied boundary conditions are at the upstream an inlet boundary condition, which requires the prescription of the velocity components and additional values for the turbulence.
At the
outlet boundary downstream, a constant pressure boundary condition is applied. This condition enables both inflow and outflow at the outer surface. At the domain sidesslip wall boundaries are applied. Figure
6 shows
thecomputational domain and the
location of the
boundaryCOfl(litiOflS.
The reference Cartesian coordinate system has the X axis on chord-wise flow direction, Y axis in span-wise direction and Z axis is normal to the inflow.
A detailed numerical investigation is made on the Elliptic Ii Rake geometry in wetted flow case and cavitating case, with different grid resolutions and angles of attack. Also, based on the available data, predicted forces and velocities are compared with the experiment results.
WETTED FLOW RESULTS
The Elliptic 11 Rake foil is first computed in a wetted flow
condition for an inlet velocity of 7.43 rn/s and an outlet
pressure value of 21700 Pa. Grid influence is assessed from four generated meshes, see table I. Grids G2 and G3 are build based on an over-all mesh refinement of the grid GI with a grid ratio ofh. Then from grid G3 a fourth mesh is created based
on the cells that correspond to the
tip vortex location, determined with the Q-factor criterion (see equation (14)) and refined in all X, Y, Z directions with 2 by 2 by 2.Table 3: Grid cells number for different meshes
From grid GI to G3 the over all mesh refinement influence is analyzed, while from grid G3 to G3_ref the effect of the locally mesh refinement at tip vortex location is evaluated.
The non-dimensional lift and drag coefficient are defined as in the equations (11) and their variation with grid resolution quantified in table 4.
Table 4: Lift and drag coefficients for different meshes for wetted flow condition
Refining the overall mesh (grids Gl, G2 and G3) decreases lift and drag coefficients. Local vortex refinement has limited effect on drag and slightly increases lift.
Variation of lift and drag with angle of attack 13 are also investigated and there results are shown figure 9. Lift increases linear with , while drag increases parabolic with 13. The
variation of the
lift and drag with the angle of attack is
agreement with experience, see [9].
Ci & Cd. Elilpilo II ,U., V-T.43..d.. O0ii.i pr*..'2l?OEPL w.Ii.d fl20o
6 014 012 020 E '000. 004 002 -'-ci Cd
Figure 9: Cl and Cd variation with angle of attack, wetted flow
TIP VORTEX FLOW
The main interest of the current investigation is the tip vortex flow region. To asses the grid resolution influences over the tip vortex simulations, pressure, velocity and vorticity (Q-factor) are analysed, downstream of the foil at x=0. I Sm (see figure 10)
through the vortex core.
Cells Gi G2 G3 G3_ref Fluid Profile 287392 96 569200 136 1180928 192 1851892 192 Name GI G2 G3
G3ref
Cl Cd 0.1279 0.00981 0.1268 0.00901 0.1266 0.00899 0.1274 0.00898 0.014 0012 0.01 0000 0004 0002U
x=0.15
Figure 10: Plane x=O.15 location, downstream of the foil
When dealing with
vortex topics like vortex definition,detection and visualization are important to be addressed. Along the years many vortex definitions have been attempted, but not with much success. Its definition remains vague and therefore its predictions and detection arguable. One of the first and most general definitions is made by Lugt and states that "a vortex is the rotation motion of a multitude of material particles around a common centre" see [101. Due to this uncertainty there arc numerous vortex detection methods more or less
successful but not a definitive one. Still, one of the most
successful methods are the Galilean invariant methods, and the Q-criterion being one of the most simplest in definition and implementation among the other two, X2 and A, see [II]. The Q factor criterion is implemented in the CFD and used in the present paper to capture high swirling flow regions/vortices. The Q factor is defined by:
1
(u1
2
2 1.xj
ax JxWhen Q>0 the rotaton is dominant and the region determines a vortex tube. Note that the local vortex refinement is made using cells with high Q-factor values, higher than a certain positive values chosen by the user.
The vortex core pressure coefficient values are influenced by the grid resolution as shown in figure 11.
(14) 7 o o, 0. I
Cp p.nwI. vo,,. throogh th. vort.o o. wok. .t r.O.l5 lri.I I000tloo
Oooro,N. . V H
Figure 11: Cp variation with mesh, wetted flow
For increasing mesh density, the minimum pressure within the vortex core is decreasing. Overall mesh refinement (grids Gl, G2 and G3) improves the Cp prediction but a locally refined mesh (G3_ref) gives the most important step in the prediction of the low pressure within the tip vortex core.
Accurate predictions of velocity components at tip vortex location are important to determine the vortex location by
means of the
Q-factor criterion.Intluence of the mesh
resolution over axial and normal velocity components are presented is figures in 12 and 13.
Non.dlnwn.I000I roIodty000p000nt UIUO. .4 0.0.150 ptho., U07.l3rol.
Vt-I
Figure 12: Non-dimensional velocity U in flow direction for 4
meshes
The U-component velocity (in flow direction) through the vortex core is lower than in the rest of the field. Higher mesh resolution is decreasing further the minimum value within the vortex core. The local mesh refinement gives the highest
0
t
Nendke.n.Ion.I W, thrrn.gh the vo,t.e oo. t e-5
!pnw. *.Gte.,. yl.l
Figure 13: Non-dimensional velocity W in normal direction for 4 mesh variants
The minimum and maximum values of the velocity component normal to the flow are improved when local mesh refinement is applied, as shown in figure 13.
The Q-factor prediction is also influenced by mesh density as presented in figure 14.
Q facto, C e-O 15 th,rn,gh vnrte Core
Sp,*I, dfr.thO,, V II
Figure 14: Q-factor for 4 mesh variants
From equation (14) it is known that a vortex is defined as a flow region where the Q is positive. Therefore in figure 14 the high peaks of the Q-factor values downstream of the foil indicate a vortex region, as seen also in pressure coefficient and velocity components. Local vortex refinement applied to grid G3 is very important, obtaining a Q-factor value of 106000 for G3_ref instead of 36000 for the grid G3. Therefore the vortex is better predicted with a fine mesh. Still downstream of the analysed plane (x=0.15), the vortex is fast decaying in strength. Further refinement may be needed to capture the accurate behaviour of strong downstream vortices.
When dealing with tip vortices local refinement is crucial for accurate predictions of pressure and velocity components with
its core.
CAVITATING FLOW RESULTS, cr=0.68
When cavitation is present on the Elliptic II Rake foil, lift and
drag coefficients, pressure coefficients and velocities are
influenced as presented in this section. Results of the RANS
cavitation simulation
are shown and
validated with theavailable experimental results.
The influence of cavitation over the foil on
lift and drag coefficients is shown in table 5, where the first column is the wetted flow, second column is the cavitating flow and third the experimental cavitating result.Table 5: Lift and drag coefficients
for wetted flow and
cavitating flow compared to experimental data
Compared with the wetted flow case the lift and drag are
increasing when cavitation is enabled. Cavitating results are close to the experimental ones for lift and slightly higher for
drag.
The pressure coefficient for the wetted flow and the cavitating flow is presented in figure 15.
Cp, wettOd end nevlteting ceo, U7.43 ni.
8
I
Sp.n*ie. d,.n&,. VEI
Figure 15: Cavitation influence over Cp
From figure 15 can be seen that the pressure coefficient is
decreasing due to the cavitation modelling within the vortex core. The minimum pressure coefficient is -0.68 in agreement with the analysed cavitation number, see equation (13).
The velocity in flow direction through the vortex core is
decreasing when cavitation is enabled as shown in figure 16.
Within the highly swirling region of the vortex, the axial
velocity is
decreasing, while the normal components are
increasing. This is enhanced when cavitation is present, as shown in figures 16 and 17.
The differences found in pressure distribution and velocity components can be explained by a reduction of the viscosity in the cavitating vortex core, according to equation (9).
CFD CFD EXP
Name Wetted o=0.68 o=0.68
CI 0.1274 0.1311 0.1297
0.50 00 005 0.75 0.7 085 06
Figure 17: Non-dimensional velocity component \V in normal direction for wetted and cavitating flow
Tip vortex cavitation is decaying fast downstream of the x=O.15 plane. For strong cavitating vortices downstream, very fine local meshes are required.
TIP VORTEX CAVITATION VISUALIZATION
Experimental visualization results for the Elliptic Ii Rake foil are available in [3]. In figure 18 the case with angle of attack I=3, velocity inlet 7.43 m/s, r=O.68 is shown.
The current RANS simulations for the same condition as in the experiments in figure
18 are show for comparison and
validation purpose in figure 19.
Figures 18 and 19 show the visualization of the cavity based on experiments and on computations. For a volume fraction iso-surface of 1%, the simulated cavity shape and volume are comparable with the experimental ones. Also the starting point of the re-entrant jet (1) and of the tip vortex formation and detachment (2), see figure 19, is very well in agreement with the experimental observations, see figure 18. Changes in the volume fraction iso-surface, have limited influence on cavity shape visualization, only small variations of the volume can be
noticed.
Figure 18: Experiments cavitation visualization at ==O.68,
=3, u=7.43 mis, taken from [3]
Figure 19: RANS simulations, particle tracks and cavitation
volume fraction iso-surface=O.007, at c=O.68, f3=3, u=7.43 mis
CONCLUSIONS
The scope of the present paper is to give an approach to
successful predictions of the complex cavitation phenomena using a CFD method for industrial use.
At the end of the current study, two main conclusions should be
outlined.
First, the k-c RNG turbulence model proves to be the most important ingredient in successful shedding
cavitation simulations for two-dimensional NACA
9 01 075 0.2 025
SpnoIs.drnoton.VH
Figure 16: Non-dimensional velocity component U in flow direction for wetted and cavitating flow
0015 profile with good results
for the cavitation shedding pattern and frequencies.Second, with an appropriate mesh, focused on the sheet-tip vortex cavitation region, the exhibited tip vortex is well captured. Results of the simulations for the foil forces, velocities and cavitation pattern are in agreement with the measurements and experimental observations for a complex 3D Elliptic 11 Rake foil. Thus, the k-c RNG turbulence model in combination with a local refined mesh of the tip vortex region proves to be a good recipe for successful simulations of cavitation prediction when using CFD approach.
In a next step all the knowledge acquired within the present research is going to be applied on propellers for sheet-tip vortex cavitation assessment. NOMENCLATURE c Chord length Cd Drag coefficient Cl Lifi coefficient Cp Pressure coefficient
Cr Chord length at the root
Ct Chord length at the tip
no Number of seed bubbles
p Pressure Vapour pressure
R Bubble radius
S Span
Say Source of volume fraction of vapor [Its]
T Time [s]
Q Second invariant of the velocity
gradient tensor
U Velocity component in X direction
V Volume
Vv Fraction of control volume V
w
Velocity component in Z directionx, y, z
Flow field coordinate systemx
Flow directionY Span-wise direction
z
Normal directiona
Volume fractionVolume fraction of vapor
10
Angle of attack [deg]
II Viscosity [Pas]
p Density [kg/rn3]
0 Cavitation number [-]
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