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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 423 Validation of High Reynolds Number, Unsteady Multi-Phase CFD Modeling for Naval Applications J.Lindau R.Kunz D.Boger D.Stinebring H.Gibeling (Penn State Applied Research Laboratory, University Park, Pennsylvania 16802) ABSTRACT Unsteady, high Reynolds number validation cases for a multi-phase CFD analysis tool have been pursued. The tool, designated UNCLE-M, has a wide range of applicability including flows of naval relevance. This includes supercavitating and cavitating flows, bubbly flows, and water entry flows. Thus far the tool has been applied to a variety of configurations. Axisymmetric sheet cavity flow-fields have been modeled. In particular, an attempt to validate the unsteady reliability of UNCLE-M with consideration of the effect of cavitation number, Reynolds number and turbulence model has been made. Analysis of the modeled unsteady flow-field is also made and conclusions regarding the causes of success and shortcomings in the computational results are drawn. INTRODUCTION The ability to properly model unsteady multiphase flows is of great importance, particularly in naval applications. Cavitation may occur in submerged high speed vehicles as well as rotating machinery, nozzles, and numerous other venues. Traditionally, cavitation has had negative implications associated with damage and/or noise. However, for high speed submerged vehicles, the reduction in drag associated with a natural or ventilated cavity has great potential benefit. Yet, cavitation modeling remains a difficult task, and only recently have full Reynolds-averaged, three-dimensional, multi- phase, Navier-Stokes tools reached the level of utility that they might be applied for engineering purposes. UNCLE-M (Kunz, 1999(I, II)) is a fully implicit, pre-conditioned, multi-phase, 3-D, fully generalized multi-block, parallel, Reynolds-averaged Navier-Stokes solver. The code was initially evolved from a version of the single-phase UNCLE code developed at Mississippi State University (Taylor, 1995), and has undergone significant further development. UNCLE-M incorporates mixture volume and constituent volume fraction transport/generation for liquid, condensable vapor and non-condensable gas fields. Mixture momentum and turbulence scalar equations are also solved. Flux limiting has been applied to the inviscid flux terms based on the local slope of the solution volume fraction. As a result, high-order accurate solutions containing crisp, physically reasonable interfaces at the cavity boundary may be obtained with minimal nonphysical oscillations. Non-equilibrium mass transfer modeling is employed to capture liquid and vapor phasic exchange. The code can handle buoyancy effects and the presence/ interaction of condensable and non-condensable fields. This level of modeling complexity represents the state-of-the-art in CFD analysis of cavitation. The restrictions in range of applicability associated with inviscid flow, slender body theory and other simplifying assumptions are not present. In particular, the code can plausibly address the physics associated with high-speed maneuvers, body-cavity interactions and viscous effects such as flow separation. The principal interest here is in modeling high Reynolds number, unsteady flow about bodies with running cavities. These cavities are presumed to be sheet cavities amenable to a homogeneous approach. In other words, it is presumed that the nonequilibrium dynamic forces of bubbles are of negligible magnitude. In the present work, the effect of surface tension is not incorporated, since interface curvatures are very small for the configurations considered. This assumption is supported by model results of sheet cavitation with a full two-fluid approach (Grogger and Alajbegovic, 1998). In previous work (Kunz, 1999(I)), the fidelity of UNCLE-M has been demonstrated for steady state fluid flows. However, due to the reentrant jet, cavity pinching, and other effects of turbulent separated flow, multi-phase flows of naval importance are generally unsteady. In the work presented here, UNCLE-M will be applied to several configurations of naval relevance. Each of these configurations presents an experimentally documented, unsteady fluid dynamic test case. Model results will be presented for several ballistic, cavitator geometries. Both the steady (averaged) and unsteady (time domain and spectral) behavior of the flow will be presented and compared with data. In addition, interesting unsteady numerical results will be presented in a field form for comparison with photographic data. By comparison of the the authoritative version for attribution. numerical and measured results, the reliability of the unsteady capabilities of the code maybe understood.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 424 Nomenclature Symbols: C1, C2 turbulence model constants Cdest, Cprod mass transfer model constants Cp pressure coefficient Cd drag coefficient D body diameter dm bubble diameter cycling frequency (Hz) f gi gravity vector k turbulent kinetic energy L bubble length mass transfer rates P turbulent kinetic energy production Prtk, PrtÎµ turbulent Prandtl numbers for k and Îµ p pressure ReD Reynolds number based on body diameter Str Strouhal frequency (fD)/Uâ s arc length along configuration (also seconds) t, tâ, ât physical time, mean flow time scale, time step U velocity magnitude Cartesian velocity components ui Cartesian coordinates xi y+ dimensionless wall distance (ÏmyUt)/Âµm Î± volume fraction, angle of attack Î² preconditioning parameter Ï pseudo-time Îµ turbulence dissipation rate Âµ molecular viscosity Ï density Ï Subscripts, Superscripts: body diameter D liquid l mixture m non-condensable gas ng turbulent t condensable vapor v â free stream value Physical Model The physical model equations solved here have been described previously (Kunz 1999 (I, II)). The basis of the model is the incompressible multiphase Reynolds Averaged Navier Stokes Equations in a homogeneous form. Each phase is treated as a new species and requires the inclusion of a separate continuity equation. Three species, representing a liquid, a condensable vapor, and a noncondensable gas, are included. Mass transfer between the liquid and vapor phases is achieved through a differential model. Other researchers have applied similar models with a single species approach. However, the multiple species model of multiphase flow is presented as a more flexible physical approach. A high Reynolds number form of two-equation models with standard wall functions provides turbulence closure. The governing differential equations, cast in Cartesian tensor form are given as Equation (1): (1) Where mixture density and turbulent viscosity have been defined in Equation (2). (2) In the present work, the density of each constituent is taken as constant. Equation (1) represents the conservation of the authoritative version for attribution. mixture volume, mixture momentum, liquid phase volume fraction and noncondensable gas volume fraction, respectively. Physical time derivatives are included for unsteady computations. The formulation incorporates preconditioned pseudo- time-derivatives (â/âÏ terms), defined by parameter Î², which provide favorable convergence characteristics for steady state and unsteady computations, as discussed further below. The formation and collapse of a cavity is modeled as a phase transformation. Detailed modeling of this process requires knowledge of the thermodynamic behavior of the fluid near a phase transition point and the formation of interfaces. Simplified models are presented here, resulting in the use of empirical factors. Given as Equation (3), two separate models are used to describe the transformation of liquid to vapor and the transformation of vapor back to liquid. For transformation of liquid to vapor, m is modeled as being proportional to the product of the liquid volume fraction and the difference between the

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 425 computational cell pressure and the vapor pressure. This model is similar to the one used by Merkle et. al. (1998) for both evaporation and condensation. For transformation of vapor to liquid, a simplified form of the Ginzburg-Landau potential is used for the mass transfer rate (3) Cdest and Cprod are empirical constants. For all work presented here, Cdest=Cprod=105. Both mass transfer rates are non- dimensionalized with respect to a mean flow time scale. In this work, a high Reynolds number two-equation turbulence model with standard wall functions has been implemented to provide turbulence closure. Either the k-Îµ or RNG k-Îµ (Orszag et al. 1993) model are represented in Equation (4): (4) As with velocity, the turbulence scalars are interpreted as being mixture quantities. NUMERICAL METHOD The baseline numerical method has been evolved from the work of Taylor and his coworkers at Mississippi State University (Taylor et. al. (1995), for example). Primitive variable interpolant type Roe flux implicit procedure is adopted with inviscid and viscous flux Jocobians approximated numerically. A block-symmetric Gauss-Seidel iteration is employed to solve the approximate Newton system at each timestep. The multi-phase extension of the code retains these underlying numerics but incorporates two additional volume fraction constituent transport equations. During flux formulation, a Jameson style (Jameson 1981) flux limiter based on liquid volume fraction is applied to the primitive interpolants. A nondiagonal pseudo-time-derivative preconditioning matrix is also employed. While the time derivative term vanishes from the mixture continuity equation as the limit of incompressible constituent phases is approached, the effect of preconditioning is to reduce the associated stiffness. This preconditioner gives rise to a system with well-conditioned eigenvalues which are independent of density ratio and local volume fraction. This system is well suited to high density ratio, phase-separated two-phase flows, such as the cavitating systems of interest here. A temporally second-order accurate dual-time scheme was implemented for physical time integration. At each time step, the turbulence transport equations are solved subsequent to solution of the mean flow equations. The multiblock code is instrumented with MPI for parallel execution based on domain decomposition. During unsteady time integration, to obtain results presented here message passing was applied after each symmetric Gauss-Seidel sweep. Each inner iterate involved twenty symmetric Gauss-Seidel sweeps, and each time step involved fifteen inner iterations. This procedure was sufficient to reliably reduce the unsteady residual by at least two orders of magnitude. However, a case by case examination likely could have reduced the expended computational effort yielding results similar in solution fidelity. Further details on the numerical method and code are available in Kunz et. al. (1999(II)). RESULTS Axisymmetric sheet cavity flow-fields have been modeled. In particular, an attempt to validate the unsteady reliability of a multiphase, computational fluid dynamics tool with consideration of the affects Reynolds number and turbulence model has been made. Steady, average, measurements of relevant cavitation parameters for the shapes chosen have been documented by Rouse and McNown (1949). Stinebring et al. (1983) documented the unsteady cycling behavior of several axisymmetric cavitators. Their report included results for both ventilated and natural cavitation. The unsteady performance of a 45Â° (22.5Â° in profile from centerline to outer edge) conical, hemispherical, and 0-caliber ogival cavitators at a range of cavitation numbers were documented. Although UNCLE-M has the capability to model ventilated cavitation (Kunz 1999(I)), only natural cavitation results have been included here. It should be noted that for the cavitator types and flows at or above the range of experimental Reynolds numbers reported and investigated here, the flow should be turbulent over a significant portion of the forebody. Therefore, for single phase flow, particularly for geometrically the authoritative version for attribution. smooth shapes, this should serve to avoid the well known chaotic, critical laminar separation and transition regime. The numerical results employ a fully turbulent model. Results presented here are given in the model computational system (SI) units. For all computations, the free stream velocity was set to 1 (m/s), the liquid density was 1000 (kg/m3), and the vapor density was 1 (kg/m3). For most computations, the liquid viscosity was then set equal to 10â3 (Pa-s), and that of the gas phases was set to 10â5 (Pa-s). Then the body diameter was chosen to achieve the desired model Reynolds number. In the case of the hemispherical forebody run at a body diameter based Reynolds number of 1.36x107, the liquid kinematic viscosity was then set equal to 10â5

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 426 (Pa-s), and that of the gas phases was set to 10â7 (Pa-s). The model body diameter for this case was thus, 0.136 (m). Prior to initiating unsteady computations, for purposes of computational expediency, a steady state, ât=â, integration was carried out. At the completion of this integration, it was possible to determine if the model solution was physically unsteady. In general, physically unsteady conditions were indicated by marginally convergent, flat-lined steady-state residual histories, themselves containing large amounts of unsteadiness. Figure 1: Zero caliber ogive in water tunnel at Re(D) =2.9x105, a=0.35 (approximate) (Stinebring, 1976). Figure 2: Modeled flow over a 0-caliber ogive. Liquid volume fraction contours and corresponding drag history. UNCLE-M result. Ï=0.3. ReD=1.46x105. A photograph of a 0-caliber axisymmetric cavitator operating at conditions similar to those modeled here is given in Figure 1 (Stinebring 1976) Figure 2 contains a series of snapshots of the volume fraction field from an unsteady model computation of flow over a blunt cavitator. Here the Reynolds number (based on diameter) was 1.46x105 and the cavitation number was 0.3. The time history for this case is given in model seconds, and at t=0, unsteady integration was initiated after obtaining a steady-state, ât=â, initial condition. Thus it is expected that there was some start-up transient associated with initialization from an artificially maintained set of conditions. For the volume fraction contours, dark blue indicates vapor, a liquid volume fraction of less than 0.005, and bright red indicates liquid, a volume fraction of one. Some significant numerical integration time parameters for this case are the body diameter to free stream velocity ratio, D/ Uâ=0.146 seconds, and the physical integration step size, ât=0.001 seconds. This result is presented over an approximate model cycle. The figure also includes the corresponding time history of drag coefficient. Note that the spikes in drag near t=37.725 and t=38.925 seconds correspond to reductions in the relative amount of vapor near the sharp leading edge. This marks the progress of a bulk volume of liquid from the closure region to the forward end of the cavity as part of the reentrant jet process. Although far from regular, these spikes also delineate the approximate model cycle. This picture serves to illustrate the basic phenomenon of natural sheet cavitation as it is best captured by UNCLE-M. This result is notable for the spatial and temporally irregular nature of the computed flow field. the authoritative version for attribution. Even after significant integration effort, a clearly periodic result

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 427 had not emerged. Thus, to deduce the dominant frequency with some confidence, it was necessary to apply ensemble averaging. An examination of the flow pattern captured suggests qualitative validity. Note, in Figure 2. that over a significant portion of the sequence, the leading, or formative, edge of the cavity sits slightly downstream from and not attached to the sharp corner. In their experiments, Rouse and McNown (1948) observed this phenomenon. They suggested that this delay in cavity formation was due to the tight separation eddy which forms immediately downstream of the corner and, hence, locally increases the pressure. The corresponding evolution of cavitation further downstream, at the separation interface, was proposed to be due to tiny vortices. These vortices, after some time, subsequently initiate the cavity. Figure 3 shows a single frame at t=37.8 seconds from the same model calculation (as shown in Figure 2). Here, to clarify what is captured, the volume fraction contours have been enhanced with illustrative streamlines. Note that these are streamlines drawn from a frozen time slice. Nonetheless, if all of the details envisioned by Rouse and McNown were present, the streamlines should indicate smaller/tighter vortical flows. The current level of modeling was unable to capture small vortical structures in the flow. However, the overall computation was apparently able to capture the gross affects of these phenomena and reproduce a delayed cavity. In fact from examination of the cavity cycle evolution shown in Figure 2, and the streamlines shown in the snapshot, it appears that gross unsteadiness is driven by a combination of a reentrant jet and some type of cavity pinching (Brennan 1992). The pinching process is particularly well demonstrated in Figure 2 from t=38.125 to 38.325 seconds. However, rather than complete division and convection into the free stream, it should be noted that, in later frames of Figure 2, the pinched portion of the cavity appears to rejoin the main cavity region. Figure 3: Snapshot of modeled flow over a 0-caliber ogive. Liquid volume fraction contours and selected streamlines. UNCLE-M result. Ï=0.3. ReD=1.46x105. Figure 4: Model time record of drag coefficient for flow over a 0-caliber ogive at ReD=1.46x105 and Ï=0.3. In model units, D/Uâ=0.146 (s), physical time step, ât=0.001 (s). The low frequency mode apparent in most of the experimental 0-caliber results appears to have been captured at the lowest cavitation number (Ï=0.3), as shown in Figure 2, and is evidenced in the test photograph (Figure 1). In Figure 4, the drag coefficient history for a 40 model second interval from the same computation as in Figure 2 is shown. Here, a clear picture of the persistence, over a long integration time, of the irregular flow behavior is documented. At higher cavitation numbers, the current set of 0-caliber cavitator results indicate a more regular periodic motion. This is contrary to the experimental data. However, as Figure 3 indicates, the ability to capture this motion at any cavitation number may not necessarily require the explicit capture of the finer flow details of the vortical flow structure. This is encouraging and suggests that with increased computational effort, without altering the current physical model, the representation of this phenomenon could be improved over a greater range of cavitation numbers. Figure 5 presents the spectral content of the result given in Figure 4. This power spectral density plot is based on four averaged Hanning windowed data blocks of the time domain result. To eliminate the start-up transient effect, the record was truncated, starting at t=10 seconds and, to tighten the resulting confidence intervals, more time domain results, after t=40 seconds were included. As is typical of highly nonlinear sequences, the experience of this unsteady time integration demonstrated that, additional time records merely enrich the power spectral density function. However, the additional records do serve to improve the confidence intervals, and, therefore, add reliability to the numerical convergence process. The model result used, was, as indicated by the confidence intervals, sufficient for a comparison to experimental, unsteady results. Figure 6 contains a time record of drag coefficient during modeled flow over a 0-caliber ogive at a Reynolds number of 1.46x105 and cavitation number of 0.35. The Strouhal frequency based on this result is 0.0909. Here it is apparent that the authoritative version for attribution. the computational modeling was incapable of reproducing

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 428 what should have been a lower frequency result with flow around the forebody dominated by a more irregular cavity. It is supposed that the correct result, in comparison with the experimental data in the Strouhal frequency plot (Figure 22) would have been similar in nature to the results presented for a cavitation number of 0.30 in Figure 4. In addition to lacking the rich frequency content of the result for lower cavitation numbers, it appears that the amplitude of the unsteadiness present is an order of magnitude lower. Figure 6: UNCLE-M result. Time record of drag Figure 5: UNCLE-M result. 0-caliber ogive at coefficient for flow over a 0-caliber ogive at ReD=1.46x105 and Ï=0.3. Power spectral density ReD=1.46x105 and Ï=0.35. In model units, D/Uâ=0.146 function with 50% confidence intervals shown. (s), physical time step, ât=0.001 (s). Figure 7 contains a series of snapshots from the unsteady model computation of a hemispherical cavitator at a Reynolds number (based on diameter) of 1.36x105 and a cavitation number of 0.2. This result is presented over a period slightly longer than the approximate model cycle. In this case the model Strouhal frequency is 0.0326. There are ten frames presented, and the first (or last) nine of those ten constitute an approximate model cycle. The drag history trace in Figure 8 demonstrates how, relative to the modeled flow over the blunt forebody, the pattern of flow over the hemispherical forebody is regular and periodic. This is consistent with experimental observations made (for example) by Rouse and Mcnown (1948). Note the evolution of flow shown in Figure 7 as it compares to the drag history shown in Figure 8. As would be expected, the large spike in drag corresponds to the minimum in vapor shown near the modeled t=1.6 seconds. Figure 7: Liquid volume fraction contours. Modeled flow over a hemispherical forebody and cylinder. UNCLE-M result. Ï=0.2, Re(D)=1.36x105. The next three figures demonstrate the expected and captured dependence of Strouhal frequency on cavitation number. Here the trend of increasing cycling frequency with cavitation number during flow over a hemispherical forebody is reproduced. The result has been demonstrated at a Reynolds number of 1.36x105. This Reynolds number was intended to scale the problem properly with the data available. Here, the magnitude of the drag and the amplitude of the unsteadiness may be examined. Figure 9 contains a time record of drag coefficient during modeled flow over a hemispherical forebody and the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 429 cylinder at a Reynolds number of 1.36x105 and cavitation number of 0.25. The Strouhal frequency based on this result is 0.0484. Figure 10 contains a similar time record of drag coefficient during modeled flow over a hemispherical forebody and cylinder at a Reynolds number of 1.36x105 and cavitation number of 0.30. The Strouhal frequency based on this result is 0.0622. Figure 11 contains a time record of drag coefficient during modeled flow over a hemispherical forebody and cylinder at a Reynolds number of 1.36x105 and cavitation number of 0.35. The Strouhal frequency based on this result is 0.0933. In addition, the higher Ï, higher frequency results contain smaller cavities. In these situations, cavities drive the overall unsteadiness of the flow, and the problem of sufficient grid points to define an unsteady cavity becomes apparent. Thus, by pushing the limits of reasonable discretization, the limits of effective modeling also are tested. Figure 8 Unsteady drag coefficient. Flow over a hemispherical forebody and cylinder. UNCLE-M result. Ï=0.2, Re(D) =1.36x105. In model units, D/Uâ=0.136 (s), physical time step, ât=0.001 (s). Figure 12 contains a time record of drag coefficient during modeled flow over a hemispherical forebody and cylinder at a Reynolds number of 1.36x106 and cavitation number of 0.3. The Strouhal frequency based on this result is 0.0614. Figure 13 contains a time record of drag coefficient during modeled flow over a hemispherical forebody and cylinder at a Reynolds number of 1.36x107 and cavitation number of 0.3. The Strouhal frequency based on this result is 0.133. Here, the standard trend of increased turbulent cycle frequency with increased Reynolds number appears to have been presented. Figure 14 contains a time record of drag coefficient during modeled flow over a conical forebody and cylinder at a Reynolds number of 1.36x105 and cavitation number of 0.2. The Strouhal frequency based on this result is 0.0383. As anticipated, due to the expected stability of cavities about this shape, this model flow exhibited very regular cycling with little additional strong components from secondary modes. Figure 9: UNCLE-M result. Time record of drag Figure 10: UNCLE-M result. Time record of drag coefficient for flow over a hemispherical forebody and coefficient for flow over a hemispherical forebody and cylinder at ReD=1.36x105 and Ï=0.25. In model units, D/ cylinder at ReD=1.36x105 and Ï=0.3. In model units, D/ Uâ=0.136 (s), physical time step, ât=0.0025 (s). Uâ=0.136 (s), physical time step, ât=0.0025 (s). Figure 15 contains a time record of drag coefficient during modeled flow over a hemispherical forebody and cylinder at a Reynolds number of 1.36x105 and cavitation number of 0.25. This is another UNCLE-M result; however, rather than the standard k-Îµ model, the RNG k-Îµ turbulence model has been applied. For the hemispherical forebody with cylindrical afterbody, when using the standard k-Îµ model, to obtain, during a complete dual time cycle, a reduction in the unsteady residual of two orders of magnitude, it was sufficient to apply a time step, ât=0.0025 seconds. However, with the RNG k- the authoritative version for attribution. Îµ model, to obtain the same reduction in the unsteady residual, it

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 430 was necessary to run a physical time step of 0.001 seconds. Note that in this time history trace, there is a great deal of unsteadiness. The result appears far less coherent than the standard k-Îµ result given in Figure 9. The Strouhal frequency based on this result is 0.1855. Based on the measured data (Stinebring 1983), this frequency is far too high. When applied for a higher cavitation number, Ï=0.30, the RNG k-e based model again required a smaller time step (0.001 units) and predicted a Strouhal frequency of 0.068. Here the value is nearly the same as that predicted by the model using the k-Îµ turbulence model. Clearly, the trend based on these results is incorrect. It appears that the current implementation of the RNG model has yielded results consistent with the k-Îµ model at one cavitation number, Ï=0.30, but at a lower value, the cycle frequency is far greater than the standard k-Îµ modeled result or the measured data. It seems probable that with finer time and space discretization, the current RNG k-Îµ model implementation would achieve results comparable with the k-Îµ model at all cavitation numbers. As expected, the RNG model increased the overall unsteadiness of the results. However, the computational cost of the current results is already significant, and based on the UNCLE-M solutions obtained thus far, and comparison to experimental data, little benefit appears to be had from the current application of the RNG k-Îµ model. Figure 11: UNCLE-M result. Time record of drag Figure 12: UNCLE-M result. Time record of drag coefficient for flow over a hemispherical forebody and coefficient for flow over a hemispherical forebody and cylinder at ReD=1.36x105 and Ï=0.35. In model units, D/ cylinder at ReD=1.36x106 and Ï=0.3. In model units, D/ Uâ=0.136 (s), physical time step, ât=0.0025 (s). Uâ=1.36 (s), physical time step, ât=0.025 (s). Figure 14: UNCLE-M result. Time record of drag Figure 13: UNCLE-M result. Time record of drag coefficient for flow over a conical forebody and cylinder coefficient for flow over a hemispherical forebody and at ReD=1.36x105 and Ï=0.3. In model units, D/Uâ=0.136 cylinder at ReD=1.36x107 and Ï=0.3. In model units, D/ (s), physical time step, ât=0.0025 (s). Uâ=0.136 (s), physical time step, ât=0.0025 (s). Where applicable, for the unsteady results presented here, the arithmetically averaged results have been compared to the results of Rouse and McNown (1948). Figure 16 contains a comparison for flow over the 0-caliber cavitator, Figure 17 contains a similar comparison for flow over a hemispherical cavitator, and Figure 18 contains a similar comparison for flow over a the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 431 conical cavitator. In each of these figures, the overall performance of the code seems to generally agree with the data. Clearly as the cavitation number is reduced, the UNCLE-M result tends further from the data. For both the numerical and experimental results, the average initiation and termination point of the cavity may be deduced from this figure. Accordingly, the ability of the code to properly model the average cavity is well represented in these figures. The averaged performance over the 0-caliber cavitator appears better than that of either of the others. The performance over the conical shape is the worst. It is clear that the formation point of the average cavity should be well defined in the axisymmetric shapes with discontinuous profile slopes. Thus it is not clear why the prediction of termination of the cavity should, on average, be worst for the conical shape. Figure 15: UNCLE-M/RNG k-Îµ turbulence model result. Figure 16: Flow over a 0-caliber cavitator (s/d=arc length Time record of drag coefficient for flow over a over diameter). Averaged unsteady pressure hemispherical forebody and cylinder at ReD=1.36x105 computations and measured data (Rouse and McNown and Ï=0.25. In model units, D/Uâ=0.136 (s), physical 1948). time step, ât=0.001 (s). Figure 18: Flow over a conical cavitator (s/d=arc length Figure 17: Flow over a hemispherical cavitator (s/d=arc over diameter). Averaged unsteady pressure length over diameter). Averaged unsteady pressure computations and measured data (Rouse and McNown computations and measured data (Rouse and McNown 1948). 1948). Several parameters of relevance in the characterization of cavitation bubbles include body diameter, D, bubble length, L, bubble diameter, dm, and form drag coefficient associated with the cavitator, Cd. Some ambiguity is inherent in both the experimental and computational definition of the latter three of these parameters. Bubble closure location is difficult to define due to unsteadiness and its dependence on after-body diameter (which can range from 0 [isolated cavitator] to the cavitator diameter). Accordingly, bubble length is often, and here, taken as twice the the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 432 distance from cavity leading edge to the location of maximum bubble diameter (see Figure 19). The form drag coefficient is taken as the pressure drag on an isolated cavitator shape. For cavitators with afterbodies, such as here, the pressure contribution to Cd associated with the back of the cavitator is assumed equal to the cavity pressure For the model computations, dm is determined by examining the Î±l=0.5 contour and determining its maximum radial location. In Figure 20, the quantity is plotted against cavitation number for experimental data sets assembled by May (1975). Arithmetically averaged UNCLE-M results are included for ten unsteady computations made with three cavitator shapes. The correlation between and Ï has been long established (see Reichardt (1946), Garabedian (1958), for example). Despite the significant uncertainties associated with experimental and computational evaluation of L and CD, the data and simulations do correlate well, close to independently of cavitator shape. Figure 19: Definition used to determine bubble length, L, and diameter, dm. Figure 20: Dimensionless drag to bubble length parameter and cavitation number. Flow over axisymmetric cavitators. Arithmetically averaged, unsteady UNCLE-M results and data (May 1975). Another parameter that has been established to be well correlated with cavitation number is the fineness ratio, L/dm. May (1975) noted that this parameter is particularly independent of ambient pressure, vapor pressure, free stream velocity, and whether the cavity was filled with vapor or a mixture of vapor and air. Once again, May assembled a large quantity of experimental results for this parameter. Figure 21 contains a comparison of the fineness ratio, L/dm, for averaged unsteady UNCLE-M computations and data. As a blanket observation, the spread of data between the experiments and computations in Figure 22 appears to be rather large. However, there are several encouraging items to be reviewed. It is clear that (for a given cavitation number) the computational results are bounded by the experimental data, and the proper trends (rate of change of Strouhal frequency with cavitation number) are well captured. More insight into the physical relevance of the data requires examination of specific results. Figure 21: Cavity fineness ratio and cavitation index. Flow over axisymmetric cavitators. Arithmetically averaged, unsteady, UNCLE-M results and data (May 1975). When run at similar cavitation numbers, the extremely low frequencies observed in the 0-caliber ogive testing was not captured by the model. However, considering only model results at a cavitation number of 0.3 (see Figure 4), it the authoritative version for attribution. appears that the observed of behavior was captured. Figure 22 contains a large survey of unsteady computational and experimentally obtained data (Stinebring 1983). The numerical results in this figure summarize this validation effort. Here, Strouhal frequency is shown over a range of cavitation numbers. Computational results are given for hemispherical, 1/4-caliber, conical, and 0-caliber forebodies. Unsteady experimental data is included for the hemispherical, conical and 0-caliber shapes. Computational results for the hemisphere, 1/4-caliber and conical forebodies, were obtained at a Reynolds number based on diameter

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 433 of 1.36x105. For the 0-caliber ogive, computations were made at a Reynolds number of 1.46x105. In addition, for the hemisphere, results are included for Reynolds numbers of 1.36x106 and 1.36x107. The experimental results included in the figure were obtained at Reynolds numbers ranging from 3.5x105 to 1.55x106. Figure 22: Axisymmetric running cavitators with cylindrical afterbodies. Strouhal frequency and cavitation number. UNCLE- M results (open symbols) and data reported in Stinebring (1983). For the hemispherical forebody results, as may be seen in Figure 22, there is a significant but almost constant offset between the measured unsteady data and the modeled results both of which appear to follow a linear trend over the range presented. An interesting result occurs in the model data for the hemispherical forebody with a Reynolds number of 1.36x107 (pentagrams in Figure 22). Here the numerical results appear to agree quite well with the experimental data for hemispherical forebodies. The experiments were taken at an order of magnitude lower Reynolds number, but the agreement is apparent in both cases where model results have been obtained. For design purposes, this may suggest an avenue towards model calibration. Another result found in the Str versus Ï plot (Figure 22) is the tendency of the modeled flows to become steady at higher cavitation numbers. For the 0-caliber or the conical cavitators, this is the reason model results are not included for cavitation numbers greater than 0.4. For the modeled hemisphere, the upper limit of cavitation number to yield unsteady model results was found to be Reynolds number dependent. At a ReD=1.36x105, the maximum cavitation number yielding an unsteady result was Ïâ0.35, at ReD=1.36x106, that number was Ïâ0.45, and at ReD=1.36x107, the maximum cavitation number for unsteady computations was not determined. This result may indicate a limit of the computational grid applied to the problems rather than a limit of the level of physical modeling. In addition, physically in the mode of unsteadiness present, a transition does occur from cavity driven to separated, turbulent, but single phase driven flow. For the conical forebody, the datum shown in Figure 22 suggests that the cycling frequency should be higher, 0.123. It is worth considering that the Reynolds number of the experimental flow was 3.9x105 and that the general trend with increasing Reynolds numbers is to increased frequency. However, based on the standard level of dependence of Strouhal frequency (see Schlicting 1979 for example) on Reynolds number for bluff body flows, it would seem unlikely that the rate of change in frequency with Reynolds number (at ReDâ105) would be as high as three to two. In addition, compared to shapes with geometrically smooth surfaces, the nature of unsteady flow over a conical shape is not expected to be nearly so dependent on Reynolds number. In the case of a cone, at low values of cavitation number (i.e. Ï=0.3), the separation location, and, hence, the likely forward location of the cavity, is rarely in question. A trend that is captured in the model results but not represented in the experimental data included here, is the tendency for the Strouhal frequency of a given cavitator shape to exhibit two distinct flow regimes. The first regime exists at moderate cavitation numbers and is indicated by a low Strouhal frequency where the value of Str will have an apparent linear dependence on Ï. The second regime tends toward much higher cycling frequencies. Here the dependent Strouhal frequency appears to asymptotically approach a vertical line with higher cavitation number, just prior to the complete elimination of the cavity. This is documented in Stinebring (1983) and demonstrated in Figure 22 for the modeled hemisphere at ReD=1.36x106. Based on the model results, it appears that this is characteristic of a change from a flow mode dominated by a large unsteady cavity to one dominated by other, single-phase, turbulent, sources of unsteadiness. During this investigation, some effort towards the establishment of temporal and spatial discretization independence was made. As a requirement of the model, to accommodate the use of wall functions, for regions of attached liquid flow, fine-grid near-wall points were established at locations yielding 10<y+<100. Temporal convergence was established by the authoritative version for attribution. the successive reduction of time integration step for a selected few cases. Figure 23 contains a comparison of the spectral content of results for flow over a hemispherical forebody and cylindrical afterbody, with ReD=1.36x105 and Ï=0.3, for three, successively smaller, integration step sizes. Here, the computed flow resulted in a Strouhal frequency, Str=0.600 with a physical time step, ât=0.005, Str=0.0622 with a physical time step, ât=0.0025 seconds, and Str=0.0680 with a physical time step, ât=0.001 seconds. As demonstrated in the figure, for the smaller two integration step sizes, over the range of relevant

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 434 (shown) frequencies, there was very similar modal behavior. Unfortunately only the fine-grid models tended to provide unsteady results. Thus time and spatial fidelity were judged independently. A demonstration of the steady-state spatial convergence of the modeled conical forebody and cylindrical afterbody is given in Figure 24. Figure 23: Spectral comparison of effect of physical integration time step size on Cd history. UNCLE-M result. Flow over a hemispherical forebody with cylindrical afterbody. ReD=1.36x105. Ï=0.3. Figure 24: Comparison of predicted surface pressure distributions for naturally cavitating axisymmetric flow over a conical cavitator with cylindrical afterbody, Ï=0.3. Coarse (65x17), medium (129x33) and fine (257x65) mesh solutions are plotted. It should be noted that during this investigation, steady state results (time integrations based on ât=â) using UNCLE- M have been found to be quite consistent with arithmetically averaged time-dependent results. This result is expected to be useful in expediting the future interpretation of complex three-dimensional flows. In addition, real single phase flows, at the Reynolds numbers considered, over these axisymmetric bodies are in fact unsteady. However, with the grids and level of modeling applied here, the UNCLE-M solutions tended to be steady. It seems possible that increased resolution and incorporation of low Reynolds number turbulence modeling would resolve this issue. CONCLUSIONS The effect of Reynolds number on the results for the hemispherical cavitator was not anticipated. It was assumed that with the appropriate application of the high Reynolds number turbulence model at the wall, the inviscid external flow would dominate the flow-field, determining cavity shape and size (i.e. surface pressure). However, it appears that strong flow-field interactions due to the highly turbulent separated closure region are important to determining the unsteady mode. To some extent, based on the average results, these phenomena are being accurately captured. However, there are shortcomings in the currently employed level of single-phase turbulence modeling. The validation cases examined have demonstrated the capabilities of UNCLE-M over a range of important flow conditions. The most prominent result for validation is that the unsteady frequencies obtained in numerical results appear to be bounded by the experimental data of Stinebring (1983) for all the modeled cases. Other qualitative observations made regarding the modeled case of the 0-caliber cavitator at a cavitation number of 0.3 suggest that UNCLE-M has the ability to correctly represent the overall nature of unsteady, complex, multiphase flows without necessarily capturing some of the finer flow details. This in itself is a validation of the approach taken here. Validated modeling based on parameters related to profile drag, cavity length, cavity shape, and trends of Strouhal frequency with cavitation number has been accomplished. It seems clear that with higher fidelity turbulence and mass transfer modeling and subsequent improved modeling of the closure region, a benefit to the modeling of unsteady cavitating flows would be obtained. However, the current approach has allowed rendering of unsteady multiphase flows at Reynolds numbers relevant to engineering applications in a modeling method amenable to complex geometries and design applications. The authors continue to develop the capabilities of UNCLE-M. This includes the pursuit relevant validation cases for the authoritative version for attribution. complex three-dimensional flows. In addition, new levels of physical modeling, such as compressible phases via isothermal and full energy modeling, will be incorporated. These new capabilities, in addition to the already incorporated abilities to model buoyancy and ventilation, are critical

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 435 to a current research goal, the full configuration modeling of a high speed supercavitating vehicle undergoing maneuvers. ACKNOWLEDGMENTS This work is supported by the Office of Naval Research, contract #N00014â98â0143, with Dr. Kam Ng as contract monitor. REFERENCES Brennan, C.E., Cavitation and Bubble Dynamics, Oxford University Press, New York, 1995. Garabedian, P.R., Calculation of Axially Symmetric Cavities and Jets, Pac. J. of Math 6, 1958. Grogger, H.A. & Alajbegovic, A., Calculation of the Cavitating Flow in Venturi Geometries Using Two Fluid Model, ASME Paper FEDSM 98â5295 1998. Jameson, A., Schmidt, W., & E.Turkel, Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes, AIAA Paper 81â1259, 1981. Kunz, Robert F., et al., Multi-Phase CFD Analysis of Natural and Ventilated Cavitation about Submerged Bodies, ASME Paper FEDSM99â7364, 1999 (I). Kunz, Robert F., et al., A Preconditioned Navier-Stokes Method for Two-Phase Flows with Application to Cavitation Predication, AIAA Paper 99â 3329, 1999 (II) to be published in Computers and Fluids. May, A., Water Entry and the Cavity-Running Behaviour of Missles, Naval Sea Systems Command Hydroballistics Advisory Committee Technical Report 75â2, 1975. Merkle, C.L., Feng, J., & Buelow, P.E.O., Computational Modeling of the Dynamics of Sheet Cavitation, 3rd International Symposium on Cavitation, Grenoble, France, 1998. Orszag, S.A. et al., Renormalization Group Modeling and Turbulence Simulations, Near Wall Turbulent Flows, Elsevier Science Publishers B.V., Amsterdam, The Netherlands, 1993. Reichardt, H., The Laws of Cavitation Bubbles at Axially Symmetrical Bodies in a Flow, Ministry of Aircraft Production Volkenrode, MAP-VG, Reports and Translations 766, Office of Naval Research, 1946. Rouse, H. & McNown, J.S., Cavitation and Pressure Distribution, Head Forms at Zero Angle of Yaw, Studies in Engineering Bulletin 32, State University of Iowa, 1948. Schlichting, H., Boundary-Layer Theory, McGrawHill, New York, 1979. Stinebring, D.R., Billet, M.L., & Holl, J.W., An Investigation of Cavity Cycling for Ventilated and Natural Cavities, TM 83â13, The Pennsylvania State University Applied Research Laboratory, 1983. Stinebring, D.R., Scaling of Cavitation Damage, M.S. Thesis, The Pennsylvania State University, University Park, Pennsylvania, August 1976. Taylor, L.K., Arabshahi, A., & Whitfield, D.L., Unsteady Three-Dimensional Incompressible Navier-Stokes Computations for a Prolate Spheroid Undergoing Time-Dependent Maneuvers, AIAA Paper 95â0313, 1995. the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 436 DISCUSSION W.Shyy University of Florida, USA In this paper, the authors have summarized a vast amount of information resulting from their research in developing and refining a CFD tool for single- and multi-phase flows. The results have been impressive. In particular, it seems that unless the flow is massively separated and or cavitated, the present CFD tool can perform quite well, especially in terms of pressure coefficients and main time-dependent features. The authors should be congratulated for their accomplishments to date. With regard to the unresolved issues, there are several that one can name. First, the multi-phase, time dependent (for the ensemble averaged quantities) turbulent flows is obviously a major challenge. On the one hand, there are interactions between different physical mechanisms which produce, dissipate, convect, and diffuse the turbulent kinetic energy and the Reynolds stress components, on the other hand, there is substantial mass, momentum, and energy exchange between liquid and vapor phases. The resulting physical framework is extremely complicated beyond what we have been able to predict with adequate confidence. There is no quick, practical solution to handle this challenge. However, to the least, models capable of handling (i) substantial departure from equilibrium between production and dissipation of the turbulent kinetic energy, (ii) anisotropy between main Reynolds stress components, and (iii) turbulence-enhanced mass transfer across the phase interface, should be emphasized. The second issue is related to the need for resolving the liquid-vapor boundary with due accuracy. This issue is difficult to handle because the interface's location, shape, and velocity must be computed as part of the solution, resulting in a system that doesn't have either a predetermined configuration geometrically, or a fixed mass, momentum and energy budget within its domain. An accurate and robust interface tracking scheme can help improve the performance of the present CFD tool. The third issue is related to the numerical elements, including features such as dynamic adaptation of the grid to help maintain desirable resolution, satisfactory control of numerical dispersion and dissipation in view of the highly connective multi-phase flows, and ways to expedite and stabilize the computational procedures. Suffice it to say that the authors have already developed a highly impressive and effective CFD capabilities. In each of the issues discussed above, efforts are being made to help further improve its performance in various difficult and important application areas. To help develop these advanced capabilities, one must appreciate the need for acquiring experimental information with adequate resolution and comprehensiveness. For example, turbulent quantities, precise interface definition, and convection-diffusion ratios are some key information that to date, we have not been able to document based on first-hand experimental information. AUTHOR'S REPLY: Professor Shyy has made several valuable comments regarding the difficulties of resolving complex, multi-phase flows. With regard to his suggestions for improved turbulence modeling, the authors suspect that the incorporation of such models lies in the future of this and other Reynolds-Averaged Navier-Stokes based efforts. The authors plan to continue to incorporate improved turbulence modeling. In addition, the authors hope to incorporate some better form of turbulence enhanced mixing. Particularly of interest to the authors, in the context of the current modeling method, is the proper physics to apply in the presence of multiple gaseous species and a single liquid species. The authors consider the second and third issues raised by Professor Shyy to be necessarily related. It is believed that, for complex engineering configurations, with current computational limits, a reasonable way to capture sheet cavities is by application of a method similar to what has been applied here; i.e. the interfaces to be captured will be a solution to the homogeneous mixture flow equations, possibly with multiple species, mass transfer, and equations of state. Thus, the interface will be finite and sharpness will be grid dependent. This is analogous to the most popular methods of shock capturing for engineering purposes during modeling of compressible flows. As Professor Shyy has noted, when such a method is combined with grid adaptation, significant improvement in solution quality may be achieved. the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 437 DISCUSSION J.R.Edwards North Carolina State University, USA This paper details the validation of a sophisticated CFD approach for modeling incompressible, unsteady cavitating flows. Attention is correctly focused on resolving the time-dependent aspects of cavity formation and growth, as such processes are inherently unsteady and should be modeled as such. The approach is shown to predict time-averaged surface pressure distributions to good accord. Discrepancies evidenced may be a consequence of the modeling but it must be kept in mind that the Rouse and McNown database is over 50 years old, and measurement techniques have improved substantially over the years. The model also predicts bubble shapes that correlate well with more recent experimental data, giving confidence in its ability to resolve the bulk features of axisymmetric sheet cavity flowfields. The unsteady validation of the model is presented as plots of drag coefficient versus time, with dominant frequencies extracted from the signal by a spectral analysis. While the correct trend of an increase in the Strouhal number with increasing cavitation number is evidenced, the actual values are not in accord with experimental data. The authors conjecture that these deviations may result from several factors, including insufficient grid resolution (particularly for higher cavitation numbers) and the quality of the turbulence model. Factors that also could influence these comparisons include again the quality of the experimental data and three-dimensional effects. There is certainly no guarantee that the cavity motion will remain axisymmetric over time. Some questions that might be posed to the authors during the discussion section are as follows: 1. Are there any plans to repeat any of the calculations as three-dimensional runs? It would be interesting to see if the unsteady results change. 2. The authors employ an empirical rate equation to model the conversion of liquid to vapor and vice versa as the pressure drops below the vapor pressure. How sensitive are the results obtained to the rate coefficients, particularly as regards the time-averaged predictions? 3. The authors note that âsteady-stateâ results (using very large time steps) for surface pressure distributions are âquite consistent with arithmetically averaged time-dependent resultsâ (p. 12). Does this comment apply to the wake predictions as well? It would appear that capturing the large-scale motion of the re-entrant jet would be essential in predicting the correct delayed recovery of the pressure. I would think that the âsteadyâ calculations would predict a more abrupt recovery. In conclusion, I find the authors' work to be truly representative of the state-of-the-art in cavitation modeling. Only the extension to three dimensions and the validation thereof is required before a high-fidelity tool for unsteady cavitation prediction will emerge. AUTHOR'S REPLY: Professor Edwards makes insightful commentary regarding the three-dimensionality and transient nature of sheet cavitation. It is clear that he has spent a great deal of effort studying and modeling such flows. Responses to his three questions are listed below: 1. Subsequent to the comments of Professor Edwards, the authors have begun to undertake some three- dimensional modeling of the ogive cases that were originally measured experimentally by Rouse and McNown. Partially completed results are included here in Figure A. These results seem to indicate the presence of three-dimensional modes. However, a complete study, including sensitivity to small angles of attack, was not ready as of the time of the deadline for this reply. 2. The rate equation used for mass transfer is a weak link in our model. However, it has been applied in a consistent manner. The rate coefficient was originally chosen empirically as one which produces approximately correct steady-state cavity size for a given ogive at a specific cavitation number. After this initial calibration, the rate coefficient has been left unchanged for all computations. This consistent application of the rate coefficient should allow results to be fairly assessed. No sensitivity study has been preformed on this value. the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 438 3. The original conclusions of the authors regarding the unsteady calculations reflect a comparison of the time- averaged unsteady computed pressure field to the steady state computed pressure field on the surface of the ogives. Thus it appears to the authors that the computed unsteady motion of the reentrant wake is captured in a manner that approximates the computed steady-state average and the average yielded by the data collected by May. DISCUSSION K.J.Farrell ARL, The Pennsylvania State University, USA First, I would like to congratulate my colleagues for an outstanding modeling effort in the area of time-dependent multi-phase flows, which are of particular interest in the hydrodynamics of underwater high speed vehicles. I applaud their persistence to push their analysis to the realm of engineering usefulness, where the true utility of the computational tool can be realized. The success of the simplified mass transfer models is notable. What is the nature of the higher fidelity mass transfer models that you are considering and their intended benefit? Please discuss the relative merits of advanced turbulence modeling versus mass transfer modeling in improving the prediction of the unsteady cavity dynamics. AUTHOR'S REPLY: The authors are pleased to receive such favorable commentary from Dr. Farrell. It was due to his expertise in the field of cavitation inception and cavitation modeling that it was suggested that he be an invited discussor. Regrettably, the authors have not been able to advance the physical quality of mass transfer modeling beyond the simple rate coefficient based model presented in the text. It is hoped that an inception model may be developed with physics based on the abundance of cavitation nuclei in the flow. This might be similar to work that has been done by Dr. Farrell [1]. The effect of turbulence modeling on the ability to capture a cavity flows is suspected to be strong. Some of the shortcomings of the wall-function based, two-equation approach when applied to single-phase flows have been commented on here by Professor Shyy and elsewhere by Wilcox [2], and others. A great deal of research has been devoted to turbulent flows and turbulent modeling. Thus, it is unlikely that, in the near future, a significant improvement in applicable turbulence models will be developed. In comparison, the mass transfer model employed is supported by far less research. Thus, it is suspected that, in the near term, improvements in applicable mass transfer modeling will be found and incorporated into the numerical model. It may then become clear whether improvement in quality of results is attainable by improvement in mass transfer modeling. DISCUSSION R.Arndt University of Minnesota, USA 1) How do you explain the discrepancy between frequency of pulsing cavities determined numerically and experimentally? 2) Have you carried out any analysis of partial cavitation on hydrofoils? AUTHOR'S REPLY: The authors suspect that turbulence modeling, mass transfer modeling, and three-dimensional effects all contributed to the lack of absolute agreement with experimental unsteady results. Of these possible avenues of improvement, the authors have included some three-dimensional results in Figure A. A preliminary examination of these results do indicate the presence of additional modes. However, at the deadline for submission of this reply, results are not yet complete. The authors do also intend to apply the current and future version of the computational model on other engineering configurations including partially cavitating hydrofoils. However none of this modeling is yet complete. DISCUSSION I.Celik West Virginia University, USA Strouhal number, i.e. the frequency of the primary vortex shedding, can be easily predicted by RANS codes if they are 2nd order in time. A the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 439 more appropriate validation is to compare the centerline velocity variation in the value of a bluff body. The code treats volume fraction of gas as a passive scalar and solves mixture momentum equations with variable density. In this regard it is not truly a multi-phase flow code. The slip between the two phases were not accounted for, and this can have significant consequences. AUTHOR'S REPLY: The authors thank Professor Celik for his h-sightful comments. Unfortunately the centerline velocity data for the experimental results used for comparison was not available. The authors readily acknowledge that the absence of a slip model renders the method less capable of capturing certain cavitation phenomena. This would probably manifest itself in flows dominated by bubbles. However, flows that the authors have concentrated on here are essentially phase separated, actually sheet cavities. For this type of flow, it is hypothesized that the inability to properly represent certain bubble physics is insignificant. A wide range of cavitating flows may be properly represented by homogeneous, equilibrium two- phase models. Successful researchers have modeled unsteady sheet and even cloud cavitation with homogeneous models: See, for example, Arndt, Song, et al. [3]. In the current work, the volume fraction is not thought to be a passive scalar. Here liquid volume fraction, is solved for in the liquid volume continuity equation as a dependent variable. It appears in the momentum equations via the formulation of rm and Mass transfer from liquid to vapor and from vapor to liquid takes place due to source terms in the continuity relations. Thus is fully coupled to and interdependent with solution of the flow field. If, for example, a different field is created due to alteration of some flow condition such as Reynolds Number or cavitation number, the rest of the solution flow field will be significantly altered as well. DISCUSSION H.Kato, Tokyo University, Japan Estimation of vaporization/condensation rate is important when we analyze cavitating flow because it decides the amount of vapor in the flow. I'd like to know how the authors decided the mass transfer rate at the interface between vapor and water, and how the authors verified it. AUTHOR'S REPLY: The reply to Professor Kato's question has been given in the earlier reply to Part 2 of the questions by Professor Edwards. Figure A: Three-dimensional, wall-function based, turbulent, unsteady, two-phase result. 1,245,184 cell grid. Modeled flow (from right to left) over blunt ogive (shown in gray) with an isosurface of volume fraction, colored by velocity magnitude on a field colored by velocity magnitude. s=0.30. ReD=1.46x105. Integration time step size Uâât/D=0.00685 a) Uât/D=6.85 b) Uât/D=10.27 Figure appears to show the capture of an unsteady and three-dimensional sheet-cavity flow. the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line VALIDATION OF HIGH REYNOLDS NUMBER, UNSTEADY MULTI-PHASE CFD MODELING FOR NAVAL APPLICATIONS 440 REFERENCES FOR DISCUSSION: 1. Farrell, K.J., An Eulerian/Lagrangian Computational Analysis for the Prediction of Cavitation Inception, Ph.D. Thesis, Department of Mechanical and Nuclear Engineering, Pennsylvania State University, August 2000. 2. Wilcox, D.C., Turbulence Modeling for CFD, DCW Industries, La Canada, California, USA, 1998. 3. Arndt, Song, et al., Instability of Partial Cavitation: A Numerical/Experimental Approach, ONR 23rd Symposium on Naval Hydrodynamics, val de Reuil, France, 17â22 Septermber, 2000. the authoritative version for attribution.