Scaling of Tip Vortex Cavitation Inception for a Marine Open Propeller

Transcription

1 27 th Symposium on Naval Hydrodynamics Seoul, Korea, 5-10 October 2008 Scaling of Tip Vortex Cavitation Inception for a Marine Open Propeller C.-T. Hsiao and G. L. Chahine (DYNAFLOW, INC., USA) ABSTRACT The tip vortex flow generated by a marine open propeller was numerically simulated for three different scales using a Reynolds Averaged Navier-Stokes (RANS) solver. The solutions of RANS were further improved by conducting a Direct Navier-Stokes Simulation (DNSS) in a reduced computational domain. This resulted in significant modifications of the minimum pressure coefficients along the tip vortex. These were analyzed and used to deduce a scaling law. Nuclei effects on the scaling law were investigated with the help of a surface-average pressure (SAP) spherical bubble dynamics model with a realistic bubble nuclei distribution. It was found that the Reynolds number power of the scaling law predicted based on the single phase flow solution is quite different from the classical value. However, the nuclei effects tend to adjust the power back to the classical value if the reference inception criterion is not stringent (higher number of events per second). 1. Introduction Scaling propeller tip vortex cavitation inception from laboratory experiments to large scale tests has not always been very successful. For example, according to experimental data by Jessup et al. (1993), the extrapolation of the cavitation inception number, i, from their model test to the full scale condition overestimated i with a factor of 1.5 when the classical scaling laws, i R e with = 0.4 (McCormick 1962), was used. Many recent analytical studies have revealed that = 0.4 used in the classical scaling laws is only suitable for low Reynolds number. Based on a modified boundary layer theory Shen et al. (2003) showed that is a function of the Reynolds number instead of a constant and decreases as the Reynolds number increases. Amromin (2006) used asymptotic analysis and gave a two-range scaling for. He suggested = 0.4 for laminar flows and = 0.24 for turbulent flows. Aside from the problems associated with scaling properly the flow field for a simple one-phase flow, derivation of a reliable scaling law for tip vortex cavitation inception encounters at least two more issues. One of the issues is to appropriately incorporate nuclei effects. Another issue which produces discrepancies between different studies is the actual means used to detect cavitation inception. To address these issues Hsiao and Chahine (2005) proposed a numerical experiment which enables simulation of practical methods to call cavitation inception. They used the surface-average pressure (SAP) spherical bubble dynamics model and a realistic bubble nuclei distribution to predict the cavitation inception for a tip vortex flow. The SAP spherical model was then used to track the nuclei and record the acoustic signals generated by their volume oscillations and deduce from this statistics of acoustics pressure peaks. Hsiao and Chahine (2005) applied acoustic criteria which define the cavitation inception by counting the number of acoustical signal peaks that exceed a certain level per unit time to deduce the cavitation inception number for different scales. They have demonstrated that larger scales tend to detect more cavitation inception events per unit time than smaller scale because a relatively larger number of nuclei are excited by the tip vortex at the larger scale due to simultaneous increase of the nuclei capture area and of the size of the vortex core. However, this scaling study was based on RANS solutions of tip vortex flows generated by a finite-span hydrofoil used as a conical problem to study tip vortex cavitation inception. It has been shown by previous studies (Dacles-Mariani et al. 1995, Hsiao and Pauley 1998, 1999) that the RANS solution is inadequate for predicting the tip vortex flow accurately. Furthermore, the results derived from the conical problem may pose another concern for practical marine propeller applications. Hsiao and Chahine (2006, 2008) found that the RANS solution of the tip-leakage vortex flow

2 in a ducted propulsor can be significantly improved by conducting Direct Navier-Stokes Simulation (DNSS) in a reduced computational domain. The interaction between the tip-leakage vortex and the trailing-edge vortex was better captured by their RANS-DNSS simulation and the resulting value and location of the minimum pressure agreed much better with experimental observations than those predicted by RANS alone. In this study the same RANS-DNSS approach is applied to improve the RANS solution of tip vortex flow generated by an open propeller. Numerical simulations were conducted for three different propeller scales to derive scaling laws for the single phase flow solution. Nuclei effects on the scaling law were then investigated with the help of the SAP model and a realistic nuclei size distribution. 2. Numerical Approach 2.1 Domain and Grid for RANS Computations In this study we consider the David Taylor Propeller 5168, an open five-bladed propeller with a inch (0.40 m) diameter (see Figure 1). The flow field around this propeller was previously simulated by Hsiao and Pauley (1999) for different advanced coefficients, J U / nd, where U is the inflow velocity, n is propeller rotational speed, and D is the propeller diameter. They used an H-H type grid with a total of 2.4 million points for a computational domain, which considered a one blade-to-blade passage (see Figure 2). This computational domain has two periodic side-boundaries, one for the suction side and one for the pressure side, formed by following the inlet flow angle and an outer boundary located 2 propeller radii away from the hub center, an inlet boundary located at 1.8 propeller radii upstream and an outlet boundary located two propeller radii downstream of the propeller mid plane. The first grid spacing was specified as chord length on the blade surface to ensure that the first grid point is at y + 2 for J = 1.1 which corresponds to a Reynolds number of based on: 2 2 U (0.7 nd) C0.7R R e, (1) where C 0.7R is the propeller blade chord length at the 0.7 radius section. n is the rotation speed and and is the viscosity. In the present study, we adopt the same geometrically scaled computational domains and grid topology to simulate two other different propeller sizes with the same advance coefficient (J = 1.1). One of the propellers is one half of the original reference size and the other is twice the reference size. This results in three different scales, i.e. small, medium and large, which correspond to three different Reynolds numbers, R e = , and , respectively, if the propeller rotational speed is adjusted to conserve the advance coefficient while keeping the same inflow velocity. Table 1 shows key detailed characteristics of the three propellers that were studied. Small Medium Large D 0.2 m 0.4 m 0.8 m C 0.7R 0.09 m 0.18 m 0.36 m U 10 m/s 10 m/s 10 m/s n s s s -1 R e Table 1. Characteristics of the three propeller scales. Figure 1. Front and side views of Propeller Figure 2. 3-D view of the computational grid, which considers a one blade-to-blade passage. 2.2 Domain and Grid for DNSS Computations

3 To improve the numerical solution from the RANS computations, we construct a reduced computational domain starting from the blade tip and which encompasses the whole tip vortex roll-up region. This computational domain has a square cross area and extends spirally to the end of the original RANS domain. Figure 3 illustrates the location of the reduced computational domain relative to the propeller. We consider a 7-block grid system with a total of 245 grid points in the streamwise direction and grid points in the cross flow plane. All grid points are evenly distributed without stretching in the first five blocks but gradually stretch out in the streamwise direction in the 6 th and 7 th blocks. Numerical investigations on the effects of the domain size and grid resolution have been conducted to ensure that the final domain and grid setup have the least impact on the solution. Figure 3. A view of the reduced computational domain used in the computations. Figure 4. The interpolated pressure field of the reduced computational domain near the propeller tip region. To conduct numerical computations in this reduced domain, the results of the RANS computations were interpolated to provide the initial conditions at the grid points of the reduced domain. Figure 4 shows the interpolated pressure contours at different streamwise locations near the propeller tip region to illustrate the initial roll-up of the tip vortex. 2.3 Navier-Stokes Computations The present computations are conducted on a rotating frame which is fixed on the propeller blade. The steady-rotating reference frame source terms, i.e. the centrifugal and Coriolis force terms are added to the Reynolds-Averaged Navier-Stokes equations derived in the inertial frame to account for the rotation. The resulting unsteady incompressible continuity and Navier-Stokes equations written in nondimensional vector form and Cartesian notations are as the following: u 0, (2) Du p u r 2u, (3) t Re where u ( u, v, w) is the velocity, p is the pressure, r is the radial position vector, is the angular velocity, Re u * L * / is the Reynolds number, u* and L* are the characteristic velocity and length, is the liquid density, and is its dynamic viscosity. To solve Equations (2) and (3) numerically, a three-dimensional incompressible Navier-Stokes solver, INS3D, developed by Rogers et al. (1991), is applied to calculate the rotating propeller flow. INS3D is based on the artificial-compressibility method (Chorin 1967) in which a time derivative of pressure is added to the continuity equation as 1 p u 0, (4) t where is the artificial compressibility factor. As a consequence, a hyperbolic system of equations is formed that can be solved using a time marching scheme. This method can be marched in pseudo-time to reach a steady-state solution. To obtain a timedependent solution, a Newton iterative procedure needs to be performed at each physical time step in order to satisfy the continuity equation. In this code, the first-order Euler implicit difference formula is applied to the pseudo-time derivatives. The spatial differencing of the convective terms uses a fifth-order accurate flux-difference splitting based on Roe's method (1981). A second-order central differencing is used for the viscous terms. The resulting system of algebraic equations is solved by a Gauss-Seidel linerelaxation fully implicit method in which several linerelaxation sweeps through the computational domain

4 are performed before the solution is updated at the new pseudo-time step. For the RANS computations, the INS3D code used the Baldwin-Barth one-equation turbulence model (Baldwin and Barth 1990). 2.4 Boundary Conditions All the boundary conditions are treated in an implicit manner in the RANS computations except the outlet boundary. The boundary conditions on each of the boundaries are as follows: free stream conditions are specified for all variables at the inlet and outer radial boundaries; the no-slip condition is applied on the blade and the hub/shaft surfaces; a periodic boundary condition is specified on the sideboundaries. For the outlet boundary, a mass and momentum weighted extrapolation method as suggested by Chang et al. (1985) is adopted. The velocities and pressures obtained by this method maintain the proper shape of the streamlines and at the same time conserve the mass and momentum fluxes. For the reduced domain DNSS computations, the boundary conditions are deduced from the RANS solution. The initial values of the pressure and velocities interpolated from the RANS solution are imposed at all boundaries except at the inlet and outlet boundaries. At the inlet boundary the method of characteristics is applied with all three components of velocities specified from the RANS solution. For the outlet boundary all the variables are extrapolated from the inner grid points. 2.5 Bubble Dynamics Models To investigate nuclei effect on cavitation occurrence, a multi-bubble dynamics and trajectory code, DF_MULTI_SAP, is applied to track the bubble nuclei and predict their acoustic signals during cavitation events. In DF_MULTI_SAP the bubble transport is modeled via the motion equation described by Johnson and Hsieh (1966), while the bubble dynamics is simulated by solving a Surface Averaged Pressure (SAP) Rayleigh-Plesset equation (Hsiao and Chahine, 2003). In the present study we have considered the following form of the equation for the bubble radius, R(t), which accounts for liquid and gas compressibility, liquid viscosity, surface tension, and non-uniform pressure fields, and is based on Gilmore s approach (Gilmore, 1952). R 3 R 1 R R d (1 ) RR (1 ) R (1 ) c 2 3c c c dt u u 2 2 R enc b pv pg penc 4, R R 4 (5) where c is the sound speed, is the liquid density, is the liquid viscosity, p v is the vapor pressure, p g is the gas pressure, is the surface tension, p enc and u enc are the liquid pressure and velocity respectively, and ub is the bubble travel velocity. In Equation (5), which we dub the Surface-Averaged Pressure (SAP) bubble dynamics equation (Hsiao and Chahine 2003), we have accounted for a slip velocity between the bubble and the host liquid, and for a non-uniform pressure field along the bubble surface. Penc and u enc are defined as the average of the liquid pressures and velocities over the bubble surface. The use of Penc results in a major improvement over the classical spherical bubble model which uses the pressure at the bubble center in its absence. The gas pressure, p g, is obtained, as described in the next section, from the solution of the gas diffusion problem and the assumption that the gas is an ideal gas. The bubble trajectory is obtained using the following motion equation (Johnson and Hsieh 1966): dub 3 3 CD P enc b enc b dt u 4 R u u u (6) 3 uenc ub R, R where C D is the drag coefficient given by an empirical equation such as that of Haberman and Morton (1953): 24 ( CD Reb Reb ), Reb (7) 2R uenc ub Reb. The last term on the right hand side of Equation (7) is the force due to the bubble volume variations, which is obtained by solving Equation (5). The pressure at a distance l from the bubble center generated by the bubble dynamics is given by Fitzpatrick and Strasberg (1958). R 2 pa RR 2R l. (8) 3. Results and Discussion 3.1 RANS Liquid Phase Flow The RANS solution of the flow field for J = 1.1 and Re = was obtained previously by Hsiao and Pauley (1999) and extensively compared with the experimental measurement by Chesnakas and Jessup (1998). The RANS solutions for R e = and cases were computed in this study with the same numerical setup except for the first grid spacing which was adjusted to maintain the same y + for the different R e. The resulting pressure coefficients along the tip vortex core for these three Reynolds numbers

5 are shown in Figure 5. It is seen that for all three cases the minimum pressure coefficients occur at x/r prop = where R prop is the propeller radius and x/r prop = 0 is the propeller mid plane. This is very close to the blade tip; the initial vortex rollup due to the flow crossing over the tip from the pressure side to the suction side is responsible for this minimum pressure. The corresponding minimum pressure coefficients for the three scales are 3.08, 3.86 and 4.26 (see Table 2). Figure 5. Pressure coefficient along the tip vortex center obtained by RANS computations for three different scales. If the cavitation inception number, i, is assumed to be -Cp min, then the scaling law between two different scales can be deduced by i,2 Cpmin,2 R e2, (9) i,1 Cpmin,1 Re1 where subscripts 1 and 2 represent two different scales. As a result, = 0.33 and 0.15 when comparing medium to small scale cases and large to medium scale cases, respectively. It is obvious that the predicted is quite different from the classical value 0.4 since the current propeller flow is in a turbulent flow regime even for the small scale case. Based on a modified boundary layer theory, Young et al. (2001) gave as: log R e2 5.16log log Re1, (10) R e2 log Re1 According to Equation (10) is 0.35 for medium to small scale and 0.30 for large to medium scale. It is seen that our predicted for medium to small scale is quite close to that predicted by Equation (10) but is higher than the 0.24 value suggested by Amromin (2006) for the turbulent flow regime. However, the predicted for large to medium scale is much lower than both theoretical values. It is known that the tip vortex initial rollup is very sensitive to the boundary layer resolution. Although we have made efforts to reduce the grid dependence of the solution by adjusting the first grid spacing to maintain the same y + for different Reynolds numbers, further grid refinement is still required to ensure a well-resolved turbulent boundary layer especially for the high Reynolds numbers before we can draw a solid conclusion to confirm such deviation. Cp min Small Medium Large RANS DNSS Table 2. Summary of the scaling results with RANS and DNSS computations. 3.2 DNSS Liquid Phase Flow Except for the tip vortex initial rollup, the RANS solution appears to not be able to resolve well the continuous rollup process due to the averaging nature of RANS and to inadequate turbulence modeling. To address this issue we also conducted RANS - DNSS computations for all three scales. The simulation of the vortex rollup and its interaction with the vortex sheet in the reduced computational domain was conducted using Navier Stokes solution with no turbulence model applied. Figure 6 shows the resulting pressure coefficients, Cp, along the vortex core for the three cases considered and compares them to the RANS solutions. Major fundamental differences are seen between the RANS and the DNSS results. The DNSS predicts a lower Cp min, which occurs much further downstream than the RANS minimum. This is probably due to excessive vortex diffusion and dissipation in the RANS computation. Figure 6. Comparison of the pressure coefficient variation along the vortex center between the RANS and DNSS solution for the three different scales.

6 Figure 7 shows the streamwise vorticity contours on grid planes perpendicular to the vortex trajectory of both the RANS and the DNSS solutions for J = 1.1. It is seen that the interaction between the tip vortex and the wake (vortex sheet) is much weaker for the RANS solution due to excessive vortex diffusion and dissipation in the RANS computation. On the other hand, the tip vortex is seen to gain strength as the vortex sheet continuously rolls up into the tip vortex. It is seen that continuous rollup of the vortex sheet enhances significantly the strength of the tip vortex and results in a much low pressure region further downstream as shown in Figure 6. RANSS J=1.1 solution. This issue will be addressed further in our future work. 3.3 Area of Bubble Capture To investigate nuclei effects on the occurrence of cavitation inception, small nuclei coming from upstream the propeller need to be taken into account. However, following all upstream nuclei may not be efficient in the numerical simulations. From previous studies of Hsiao and Chahine (2003, 2004) it is known that only nuclei which pass though a window of opportunity will be captured by the tip vortex and encounter the minimum pressure. With the knowledge of the location and size of this small window, we are able to distribute and follow nuclei more efficiently. Furthermore, near inception the size of the window of opportunity is strongly related to the probability of cavitation occurrence. Release Area DNSS Figure 7. Comparison of the vorticity contours between solutions obtained by RANS and DNSS for J=1.1. The minimum pressure coefficients for the three scales from the DNSS solution are 3.56, 4.13 and 4.38 (see Table 2). According to Equation (9) = 0.22 and 0.11 when comparing medium to small scale and large to medium scale, respectively. It is seen that while the predicted for medium to small scale is much smaller than that obtained by Equation (10), it is very close to the 0.24 value suggested by Amromin. However, similar to RANS solution, the predicted of the DNSS solution for the large to medium scale is relatively small. Since the RANS solution is used to specify boundary conditions for the DNSS computations, the quality of the RANS solution inherently influences the DNSS solution. This means that even though the DNSS can improve the continuous rollup of tip vortex, it still carries the RANS boundary layer solution. If RANS does not resolve properly the initial rollup, it will also contaminate the DNSS Figure 8. Location of the release area used for establishing the window of opportunity. To establish the window of opportunity a rectangular release area was specified ahead of the tip leading edge of the propeller in the x-r plane with 588 nuclei released from a 2128 grid point array. Figure 8 illustrates the location of the release plane relative to the propulsor blade. The cavitation number was specified high enough such that the maximum growth size of a nucleus was less than 10 %. Each nucleus was tracked and the minimum pressure it encountered during its travel was recorded and assigned to the release grid point. This enables us to plot a contour of the minimum encountered pressure coefficient for the release grid points and to obtain the window of opportunity for each case. Several iterations to adjust the location and size were required to ensure that the release area covers all low pressure coefficient regions. Since larger nuclei are easier to be captured by the tip vortex and cover a larger low pressure region, the location and size of the release

7 area was determined by using the largest size of nuclei to be released. Figure 9 shows contours of minimum encountered pressure coefficient for the medium and small scales for R 0 = 100 m. It is seen that the area of the window of opportunity virtually scaled up with the propeller size. This implies that the medium scale has 4 times higher probability for cavitation events to occur than the small scale. measurements to generate the bubble population. Figure 10 shows typical measured bubble density distributions and the one used in the present study. The bubble number density distribution function, n(r), is defined as the number of buble per unit volume having radii in the range [R, R+dR]: dn ( R) n( R), (11) dr where N(R) is the number of nuclei of radius R in a unit volume. This function can be expressed as a discrete distribution of M selected nuclei sizes. Thus, the total void fraction,, in the liquid can be obtained by M 3 4 Ri Ni, (12) i1 3 where Ni is the discrete number of nuclei of radius R i used in the computations Medium R0 = micron Void Fraction = 1e-7 Acquisition Time =0.2 Sec Number of nuclei (N/m^3) Bubble Released Figure 9. Contours of the encountered Cp min for medium and small scales for R 0 =100 m nuclei Nucleus Size (m) N u m b e r D e n s ity (m -4 ) 1.0E E E E E E E E E E+03 Radius (micron) Figure 10. Measured bubble number density curves (from Franklin, 1992) and the curve used in the present study. 3.4 Modeling of a Real Nuclei Field In order to simulate a realistic nuclei flow field such as what exists in nature or in the waters of a cavitation tunnel, we have selected a bubble number density curve from one of the experimental Figure 11. The number of nuclei released versus nuclei size for the nuclei size distribution considered in this study. In our model the nuclei are considered to be initially distributed randomly in a fictitious volume feeding the release area of the computational domain. The bubble population to be released is generated according to the selected number density curve for given void fraction, discrete bubble sizes and sought physical duration of the computation. We have selected a set of discrete bubble sizes ranging from 10 to 100 µm and a void fraction of The fictitious volume is determined by product of the release area, the sought physical duration of the simulation, and the characteristic velocity. Figure 11 shows the selected discrete bubble size distribution and the total number of nuclei in each discrete bubble size bin which is released into the computational domain from the release area during 0.2 sec acquisition time for the medium scale. The total number of bubbles released for the small scale is one quarter of that shown in Figure 11.

8 3.5 Nuclei Effects on Cavitation Scaling In the previous sections we have shown the scaling of the tip vortex cavitation inception based on single phase flow solutions. Here, we demonstrate how accounting for the nuclei can modify the scaling law based on the small and medium scales. As pointed by Hsiao and Chahine (2005), although many measurements including optical and acoustic methods can be applied to detect the cavitation inception and deduce the scaling of cavitation inception, scaling based on the acoustic signals generated by cavitating bubbles seems to be the best choice. With DF_Multi_SAP the resulting acoustic signals due to bubble volume variations can be recorded as the nuclei travel in the computational domain. Figure 12 shows an example of the superposed shape of a visible cavitating bubble in a time sequence and the corresponding acoustic signals when a cavitation event occurs near inception. From the results shown in Figure 13, we can define a cavitation inception number based on the number of cavitation events per unit time exceeding a certain value. One cavitation event is counted when the acoustic pressure peak exceeds a given level. Here, as an example, the acoustic pressure of 10 pa, which corresponds to 70db at 1m, is used to define a cavitation event. To deduce the cavitation inception number a curve for the number of pressure peaks higher than the selected acoustic pressure level is created for each cavitation number and for both scales. Figure 14 shows such curves for the selected acoustic pressure level and for both scales. It is important to note that the slope of the curve of the medium scale is about four times that of small scale. This is consistent with the probability for the cavitation event we found in the window of opportunity study. Figure 12. Example of a visible cavitating bubble contours superposed in a time sequence and the corresponding acoustic signals when a cavitation event occurs. A series of computations were conducted at different cavitation numbers for the same nuclei distributions to obtain acoustic signals for conditions above and below cavitation inception. The acoustic signals were acquired at one (1) meter away from the propeller tip for both scales. Figure 13 illustrate the acoustic signals for the small scale at three different cavitation numbers near cavitation inception. Figure 13. The acoustic signals for the small scale at three different cavitation numbers.

9 Cavitation Event Per Second Small > 10 pa at 1m Medium > 10 pa at 1m Cavitation Number Figure 14. Number of events per second versus cavitation number at two criteria of acoustic level for both small and medium scales. Small Medium -Cp min (DNSS) i (one peaks/sec) i (50 peaks/sec) Table 3. Cavitation inception numbers obtained from the numerical study using various criteria and the scaling power deduced from these results. For a selected criterion based on the number of peaks and acoustic pressure level, one can determine the cavitation inception number from Figure 14. The deduced cavitation inception numbers of both small and medium scales for the criteria: one peaks/second and 50 peaks/second, are shown in Table 2. The values of by comparing small and medium scale results are also shown in Table 3. It is seen that different criteria for defining a cavitation inception event lead to different cavitation inception numbers and different coefficients. The scaling effects due to the nuclei can be demonstrated by comparing the deduced inception number with -Cp min. Table 3 shows that cavitation inception scaling deviates more from -Cp min when the reference inception criterion becomes less stringent (higher number of peaks per second). This trend is more significant for smaller scales. Furthermore, the predicted is closer to the classical value ( = 0.4) as the reference inception criterion becomes less stringent. This agrees with many experimental studies usually established in laboratory conditions where background noise and detection techniques lead to high values of the pressure amplitude for inception detection, but is not good for advanced detection techniques and silent propellers. 4. Conclusions The scaling of the tip vortex cavitation inception was numerically studied for an open marine propeller. The flow field was first computed using a RANS code and then was improved by a DNSS simulation in a reduced computational domain. For the RANS solutions, the minimum pressure due to the initial rollup of the tip vortex was found to occur very close to the propeller tip. However, from the DNSS solution the continuous rollup of tip vortex due to the wake sheet was found to strengthen up the tip vortex and resulted in a much lower pressure to occur further downstream. Based on the three scales studied, we have used the resulting minimum pressure coefficients to deduce the scaling law parameter. We found that the Reynolds power varies with the Reynolds number and is not a constant. The predicted value for scaling from small to medium scale was found to deviate significantly from the classical scaling law derived based on laminar boundary layer flow, but matches better with the recent analytical results which have account for turbulent boundary layer effects. However, the predicted value for medium to large scale was found to have a much smaller value than predicted in previous analytical studies. Further grid refinement is required before we can confirm this result. Comparison of the size of the bubble capture area or window of opportunity showed that the area of the window of opportunity virtually scales with the square of the length scale. This implies that larger scales has higher probability for cavitation events by a factor of the square of the scale ratio. As a result, nuclei effects tends to adjust the power back close to the classical value if the reference inception criterion becomes less stringent (higher number of required events per second to call cavitation). ACKNOWLEDGMNETS This work was conducted at DYNAFLOW, INC. ( and was supported by the Office of Naval Research under contracts No. N C-0110 and N C-0448 monitored by Dr. Ki-Han Kim. REFERENCES [1] Amromin, E., Two-Range Scaling for Vortex Cavitation Inception, Ocean Engineering, 33, pp , [2] Baldwin, B. S., Barth, T. J., A One-Equation Turbulence Transport Model for High Reynolds

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