Chao-Tsung Hsiaoe-mail: email@example.com
Georges L. Chahinee-mail: firstname.lastname@example.org
Dynaflow, Inc.10621-J Iron Bridge Road, Jessup, MD 20794
Scaling of Tip Vortex CavitationInception Noise With a BubbleDynamics Model Accounting forNuclei Size DistributionThe acoustic pressure generated by cavitation inception in a tip vortex flow was simulatedin water containing a realistic bubble nuclei size distribution using a surface-averagedpressure (SAP) spherical bubble dynamics model. The flow field was obtained by theReynolds-averaged NavierStokes computations for three geometrically similar scales ofa finite-span elliptic hydrofoil. An acoustic criterion, which defines cavitation inceptionas the flow condition at which the number of acoustical peaks above a pre-selectedpressure level exceeds a reference number per unit time, was applied to the three scales.It was found that the scaling of cavitation inception depended on the reference values(pressure amplitude and number of peaks) selected. Scaling effects (i.e., deviation fromthe classical s i}Re
0.4! increase as the reference inception criteria become more stringent(lower threshold pressures and less number of peaks). Larger scales tend to detect morecavitation inception events per unit time than obtained by classical scaling because arelatively larger number of nuclei are excited by the tip vortex at the larger scale due tosimultaneous increase of the nuclei capture area and of the size of the vortex core. Theaverage nuclei size in the nuclei distribution was also found to have an important impacton cavitation inception number. Scaling effects (i.e., deviation from classical expressions)become more important as the average nuclei size decreases. @DOI: 10.1115/1.1852476#
1 IntroductionScaling of the results of a propeller tip vortex cavitation incep-
tion studies from laboratory to large scales has not always beenvery successful. Aside from the problems associated with properlyscaling the flow field, existing scaling laws as derived or used byprevious studies, e.g., @16#, lack the ingredients necessary toexplain sometimes major discrepancies between model and fullscale. One of the major aspects which has not been appropriatelyincorporated in the scaling law is nuclei presence and nuclei sizedistribution effects. Another issue which may cause scaling prob-lems is the means of detection of cavitation inception. Practically,the flow condition is considered to be at cavitation inception wheneither an acoustic criterion or an optical criterion is met@7,8#. These two detection methods are known to provide differentanswers in the most practical applications. Furthermore, for prac-tical reasons inception may be detected by one method at modelscale and by another at full scale. To address this issue in a moreconsistent manner for different scales, the present study considersan acoustic criterion which determines the cavitation inceptionevent by counting the number of acoustical signal peaks that ex-ceed a certain level in unit time.
To theoretically address the above issues in a practical wayspherical bubble dynamics models were adopted in many studiesin order to simulate the bubble dynamics and to predict tip vortexcavitation inception @810#. In our previous studies @8,11#, animproved surface-averaged pressure ~SAP! spherical bubble dy-namics model was developed and applied to predict single bubbletrajectory, size variation and resulting acoustic signals. This modelwas later shown to be much superior than the classical sphericalmodel through its comparison to a two-way fully three-dimensional ~3D! numerical model which includes bubble shape
Contributed by the Fluids Engineering Division for publication on the JOURNALOF FLUIDS ENGINEERING. Manuscript received by the Fluids Engineering DivisionSeptember 13, 2003; revised manuscript received August 13, 2004. Review Con-ducted by: S. Ceccio.
Copyright 2Journal of Fluids Engineering
deformation and the full interaction between the bubble and theviscous flow field @11#. In the present study we incorporate theSAP spherical bubble dynamics model with a statistical nucleidistribution in order to enable prediction of cavitation inception ina practical liquid flow field with known nuclei size distribution.This is realized by randomly distributing the nuclei in space andtime according to the given nuclei size distribution. According toprevious studies @12,13# the number of nuclei to use in the com-putation can be reduced by considering only the nuclei that passthrough a so-called window of opportunity and are captured bythe tip vortex.
In order to study scale effects in a simple vortex flow filed weconsider the tip vortex flows generated by a set of three geometri-cally similar elliptic hydrofoils. The flow fields are obtained bysteady-state NavierStokes computations which provide the ve-locity and pressure fields for the bubble dynamics computations.The SAP spherical model is then used to track all nuclei releasedrandomly in time and space from the nuclei release area and torecord the acoustic signals generated by their dynamics and vol-ume oscillations.
2 Numerical Models
2.1 NavierStokes Computations. To best describe the tipvortex flow field around a finite-span hydrofoil, the Reynolds-averaged NavierStokes ~RANS! equations with a turbulencemodel are solved. These have been shown to be successful inaddressing tip vortex flows @14# and general propulsor flows@15,16#. The three-dimensional unsteady Reynolds-averaged in-compressible continuity and NavierStokes equations in nondi-mensional form and Cartesian tensor notations are written as
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1u j]ui]x j
]t i j]x j
where ui5(u ,v ,w) are the Cartesian components of the velocity,xi5(x ,y ,z) are the Cartesian coordinates, p is the pressure, Re5ru*L*/m is the Reynolds number, u* and L* are the charac-teristic velocity and length selected to be, respectively, the freestream velocity, V` and root chord length, C0 . r is the liquiddensity, and m is its dynamic viscosity. The effective stress tensort i j is given by
t i j51
ReF S ]ui]x j 1 ]u j]xi D2 23 d i j ]uk]xk G2uiu j (3)
where d i j is the Kronecker delta and ui8u j8 is the Reynolds stresstensor resulting from the Reynolds averaging scheme.
To numerically simulate the tip vortex flow around a finite-spanhydrofoil, a body-fitted curvilinear grid is generated and Eqs. ~1!and ~2! are transformed into a general curvilinear coordinate sys-tem. The transformation provides a computational domain that isbetter suited for applying the spatial differencing scheme and theboundary conditions. To solve the transformed equations, we usethe three-dimensional incompressible NavierStokes flow solver,DFIUNCLE, derived from the code UNCLE developed at Missis-sippi State University. The DFIUNCLE code is based on theartificial-compressibility method @17# which a time derivative ofthe pressure multiplied by an artificial-compressibility factor isadded to the continuity equation. As a consequence, a hyperbolicsystem of equations is formed and is solved using a time marchingscheme in pseudo-time to reach a steady-state solution.
The numerical scheme in DFIUNCLE uses a finite volume for-mulation. First-order Euler implicit differencing is applied to thetime derivatives. The spatial differencing of the convective termsuses the flux-difference splitting scheme based on Roes method@18# and van Leers MUSCL method @19# for obtaining the first-order and the third-order fluxes, respectively. A second-order cen-tral differencing is used for the viscous terms which are simplifiedusing the thin-layer approximation. The flux Jacobians required inthe implicit scheme are obtained numerically. The resulting sys-tem of algebraic equations is solved using the Discretized NewtonRelaxation method @20# in which symmetric block GaussSeidelsub-iterations are performed before the solution is updated at eachNewton interaction. A k2 turbulence model is used to model theReynolds stresses in Eq. ~3!.
All boundary conditions in DFIUNCLE are imposed implicitly.Here, a free stream constant velocity and pressure condition isspecified at all far-field side boundaries. The method of character-istic is applied at the inflow boundary with all three componentsof velocities specified while a first-order extrapolation for all vari-ables is used at the outflow boundary. On the solid hydrofoil sur-face, a no-slip condition and a zero normal pressure gradient con-dition are used. At the hydrofoil root boundary, a plane symmetrycondition is specified.
2.2 Statistical Nuclei Distribution Model. In order to ad-dress a realistic liquid condition in which a liquid flow field con-tains a distribution of nuclei with different sizes, a statistical nu-clei distribution is used. We consider a liquid with a known nucleisize density distribution function, n(R). n(R) is defined as thenumber of nuclei per cubic meter having radii in the range @R ,R1dR# . This function has a unit m24 and is given by
n~R !5dN~R !
where N(R) is the number of nuclei of radius R in a unit volume.This function can be obtained from experimental measurementssuch as light scattering, cavitation susceptibility meter and ABSAcoustic Bubble Spectrometer measurements @21# and can beexpressed as a discrete distribution of M selected nuclei sizes.Thus, the total void fraction, a, in the liquid can be obtained by
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where Ni is the discrete number of nuclei of radius Ri used in thecomputations. The position and timing of nuclei released in theflow field are obtained using random distribution functions, al-ways ensuring that the local and overall voi