Validation of High Reynolds Number, Unsteady Multi-Phase CFD Modeling for Naval Applications

Transcription

1 Validation of High Reynolds Number, Unsteady Multi-Phase CFD Modeling for Naval Applications Jules W. Lindau Robert F. Kunz David A. Boger David R. Stinebring Howard J. 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, highorder 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 stateof-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 numerical and measured results, the reliability of the unsteady capabilities of the code may be understood. 1

2 Nomenclature Symbols: C 1, C 2 turbulence model constants est, C prod mass transfer model constants C P pressure coefficient drag coefficient D body diameter d m bubble diameter f cycling frequency (Hz) g i gravity vector k turbulent kinetic energy L bubble length - + m, m mass transfer rates P turbulent kinetic energy production Pr tk,pr tε turbulent Prandtl numbers for k and ε p pressure Re D Reynolds number based on body diameter Str Strouhal frequency ( fd) U s arc length along configuration (also seconds) tt,, t physical time, mean flow time scale, time step U velocity magnitude u i Cartesian velocity components Cartesian coordinates x i y + dimensionless wall distance ( ρ m yu t ) µ m α volume fraction, angle of attack β preconditioning parameter τ pseudo-time ε turbulence dissipation rate µ molecular viscosity ρ density p σ cavitation number ( p v ) 2 1/2ρ l U Subscripts, Superscripts: D l m ng t v Physical Model body diameter liquid mixture non-condensable gas turbulent condensable vapor free stream value 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): p u j m +m ρ m β 2 τ xj 1 1 = ρ l ρ v ( ρm u t i )+ ( ρm u τ i )+ ( ρm u xj i u ) = - j p u i + µm,t +ρm g x i xj x j i α l+ α l p α + l+ m + αl t ρ m β 2 u τ τ m - ( ) = xj j ρ l ρ l α ng+ α ng p α ng+ + t ρ m β 2 ( αng u ) = 0 τ τ xj j (1) Where mixture density and turbulent viscosity have been defined in Equation (2). ρ m = ρ l α l + ρ v α v + ρ ng α ng µ mt ρ m C µ k 2, = ε In the present work, the density of each constituent is taken as constant. Equation (1) represents the conservation of 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 ( / τ terms), defined by parameter β, 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 (2) 2

3 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 m. - C ρ dest v α MIN[ 0, p p l v ] m = /2ρ l U t 2 + C prod ρ v α l ( 1 αl ) m = t (3) est and C prod are empirical constants. For all work presented here, est = C prod = 100. Both mass transfer rates are non-dimensionalized with respect to a mean flow time scale. In this work, a high Reynolds number twoequation turbulence model with standard wall functions has been implemented to provide turbulence closure. Either the k-ε or RNG k-ε (Orszag et al. 1993) model are represented in Equation (4): ( ρm ε) t ( ρm k) t (4) As with velocity, the turbulence scalars are interpreted as being mixture quantities. Numerical Method µ m t + ( ρm ku ) , k = + P ρε x j j x Pr j tk x j µ m, t ε + ( ρm εu ) ε = + [ C x j j x Pr j tε x j 1 P C 2 ρε] - k 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 difference splitting is used for spatial discretization. An implicit procedure is adopted with inviscid and viscous flux Jacobians approximated numerically. A blocksymmetric 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 45 o (22.5 o 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 the results of Rouse and McNown (1948) indicated 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 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/m 3 ), and the vapor density was 1 (kg/m 3 ). For most computations, the liquid viscosity was then set equal to 10-3 (Pa-s), and that of the gas phases was set to 10-5 (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 1.36x10 7, the liquid kinematic viscosity was then set equal to

4 (Pa-s), and that of the gas phases was set to 10-7 (Pa-s). The model body diameter for this case was thus, (m). Prior to initiating unsteady computations, for purposes of computational expediency, a steady state, t =, 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. t= t= t= t= t= t= t= t= Figure 1: Zero caliber ogive in water tunnel at Re(D)=2.9x10 5, σ=0.35 (approximate) (Stinebring, 1976). t= t= t= t= 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 1.46x10 5 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, t =, initial condition. Thus it is expected that there was some startup 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 U = seconds, and the physical integration step size, t = 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= and t= 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. Even after significant integration effort, a clearly periodic result t= t= t= t= Figure 2: Modeled flow over a 0-caliber ogive. Liquid volume fraction contours and corresponding drag history. UNCLE-M result. σ=0.3. Re D =1.46x

5 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= to 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. σ=0.3. Re D =1.46x10 5. The low frequency mode apparent in most of the experimental 0-caliber results appears to have been captured at the lowest cavitation number (σ=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 4: Model time record of drag coefficient for flow over a 0-caliber ogive at Re D =1.46x10 5 and σ=0.3. In model units, D U = (s), physical time step, t = (s). 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 startup 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 1.46x10 5 and cavitation number of The Strouhal frequency based on this result is Here it is apparent that the computational modeling was incapable of reproducing 5

6 Amplitude approximate model cycle. In this case the model Strouhal frequency is 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 f (Hz) Figure 5: UNCLE-M result. 0-caliber ogive at Re D =1.46x10 5 and σ=0.3. Power spectral density function with 50% confidence intervals shown. t=0.1 t=1.1 t=0.6 t=1.6 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. t=2.6 t=3.6 t=3.1 t= t=4.6 t= Figure 6: UNCLE-M result. Time record of drag coefficient for flow over a 0-caliber ogive at Re D =1.46x10 5 and σ=0.35. In model units, D U = (s), physical time step, t = (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 1.36x10 5 and a cavitation number of 0.2. This result is presented over a period slightly longer than the Figure 7: Liquid volume fraction contours. Modeled flow over a hemispherical forebody and cylinder. UNCLE-M result. σ=0.2, Re(D)=1.36x10 5. 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. 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 1.36x10 5. 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 6

7 Figure 8 Unsteady drag coefficient. Flow over a hemispherical forebody and cylinder. UNCLE-M result. σ=0.2, Re(D)=1.36x10 5. In model units, = (s), physical time step, t = (s). D U time Figure 9: UNCLE-M result. Time record of drag coefficient for flow over a hemispherical forebody and cylinder at Re D =1.36x10 5 and σ=0.25. In model units, D U = (s), physical time step, t = (s). cylinder at a Reynolds number of 1.36x10 5 and cavitation number of The Strouhal frequency based on this result is Figure 10 contains a similar time record of drag coefficient during modeled flow over a hemispherical forebody and cylinder at a Reynolds number of 1.36x10 5 and cavitation number of The Strouhal frequency based on this result is Figure 11 contains a time record of drag coefficient during modeled flow over a hemispherical forebody and cylinder at a Reynolds number of 1.36x10 5 and cavitation number of The Strouhal frequency based on this result is In addition, the higher σ, 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 12 contains a time record of drag coefficient during modeled flow over a hemispherical forebody and cylinder at a Reynolds number of 1.36x10 6 and cavitation number of 0.3. The Strouhal frequency based on this result is Figure 13 contains a time record of drag coefficient during modeled flow over a hemispherical forebody and cylinder at a Reynolds number of 1.36x10 7 and cavitation number of 0.3. The Strouhal frequency based on this result is 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 1.36x10 5 and cavitation number of 0.2. The Strouhal frequency based on this result is As anticipated, due to the expected stability of cavities about this Figure 10: UNCLE-M result. Time record of drag coefficient for flow over a hemispherical forebody and cylinder at Re D =1.36x10 5 and σ=0.3. In model units, D U = (s), physical time step, t = (s). shape, this model flow exhibited very regular cycling with little additional strong components from secondary modes. Figure 15 contains a time record of drag coefficient during modeled flow over a hemispherical forebody and cylinder at a Reynolds number of 1.36x10 5 and cavitation number of This is another UNCLE-M result; however, rather than the standard k ε model, the RNG k-ε turbulence model has been applied. For the hemispherical forebody with cylindrical afterbody, when using the standard k-ε 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, t= seconds. However, with the RNG k-ε model, to obtain the same reduction in the unsteady residual, it 7

8 Figure 11: UNCLE-M result. Time record of drag coefficient for flow over a hemispherical forebody and cylinder at Re D =1.36x10 5 and σ=0.35. In model units, D U = (s), physical time step, t = (s) Figure 13: UNCLE-M result. Time record of drag coefficient for flow over a hemispherical forebody and cylinder at Re D =1.36x10 7 and σ=0.3. In model units, D U = (s), physical time step, t = (s) Figure 12: UNCLE-M result. Time record of drag coefficient for flow over a hemispherical forebody and cylinder at Re D =1.36x10 6 and σ=0.3. In model units, D U = 1.36 (s), physical time step, t = (s). was necessary to run a physical time step of 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-ε result given in Figure 9. The Strouhal frequency based on this result is Based on the measured data (Stinebring 1983), this frequency is far too high. When applied for a higher cavitation number, σ=0.30, the RNG k-e based model again required a smaller time step (0.001 units) and predicted a Strouhal frequency of Here the value is nearly the same as that predicted by the model using the k-ε 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-ε model at one cavitation number, σ=0.30, but at a lower value, the cycle frequency is far greater than the standard k-ε modeled result or the measured data. It seems probable that with finer time Figure 14: UNCLE-M result. Time record of drag coefficient for flow over a conical forebody and cylinder at Re D =1.36x10 5 and σ=0.3. In model units, D U = (s), physical time step, t = (s). and space discretization, the current RNG k-ε model implementation would achieve results comparable with the k-ε 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-ε model. 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 8

9 Figure 15: UNCLE-M/RNG k-ε turbulence model result. Time record of drag coefficient for flow over a hemispherical forebody and cylinder at Re D =1.36x10 5 and σ=0.25. In model units, D U = (s), physical time step, t = (s). C p computation, σ=0.3 data, σ=0.3 computation, σ=0.4 data, σ= s/d Figure 16: Flow over a 0-caliber cavitator (s/d=arc length over diameter). Averaged unsteady pressure computations and measured data (Rouse and McNown 1948). 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. C p computation, σ=0.2 data, σ=0.2 computation, σ=0.3 data, σ= s/d Figure 17: Flow over a hemispherical cavitator (s/d=arc length over diameter). Averaged unsteady pressure computations and measured data (Rouse and McNown 1948). C p computation, σ=0.3 data, σ=0.3 computation, σ=0.4 data, σ= s/d Figure 18: Flow over a conical cavitator (s/d=arc length over diameter). Averaged unsteady pressure computations and measured data (Rouse and McNown 1948). Several parameters of relevance in the characterization of cavitation bubbles include body diameter, D, bubble length, L, bubble diameter, d m, and form drag coefficient associated with the cavitator,. 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 afterbody diameter (which can range from 0 [isolated cavitator] to the cavitator diameter). Accordingly, bubble length is often, and here, taken as twice the 9

10 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 associated with the back of the cavitator is assumed equal to the cavity pressure ( p v ). For the model computations, d m is determined by examining the α l = 0.5 contour and determining its maximum radial location. 1/2 In Figure 20, the quantity L/ ( D ) 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 1/2 correlation between L/ ( D ) and σ has been long established (see Reichardt (1946), Garabedian (1958), for example). Despite the significant uncertainties associated with experimental and computational evaluation of L and C D, the data and simulations do correlate well, close to independently of cavitator shape. Another parameter that has been established to be well correlated with cavitation number is the fineness ratio, L/d m. 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/d m, 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 d m L/2 D Figure 19: Definition used to determine bubble length, L, and diameter, d m. L/d m UNCLE-M hemisphere UNCLE-M 0-caliber UNCLE-M cone data cone data stagnation cup data sphere data disk σ L/(D 1/2 ) 10 3 UNCLE-M hemisphere UNCLE-M 0-caliber UNCLE-M cone data sphere 10 2 data stagnation cup data cone σ 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). 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 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 10

11 of 1.36x10 5. For the 0-caliber ogive, computations were made at a Reynolds number of 1.46x10 5. In addition, for the hemisphere, results are included for Reynolds numbers of 1.36x10 6 and 1.36x10 7. The experimental results included in the figure were obtained at Reynolds numbers ranging from 3.5x10 5 to 1.55x10 6. Str hemisphere Re D =( )x cal Re D =( )x10 5 cone Re D =3.9x10 5 model 1/4 cal Re D =1.36x10 5 model 0 caliber Re D =1.46x10 5 model hemisphere Re D =1.36x10 5 model hemisphere Re D =1.36x10 6 model hemisphere Re D =1.36x10 7 model cone Re D =1.36x σ 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 1.36x10 7 (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 σ 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 Re D =1.36x10 5, the maximum cavitation number yielding an unsteady result was σ 0.35, at Re D =1.36x10 6, that number was σ 0.45, and at Re D =1.36x10 7, 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, It is worth considering that the Reynolds number of the experimental flow was 3.9x10 5 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 Re D 10 5 ) 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. σ=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 σ. 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 Re D =1.36x10 6. 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 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 Re D =1.36x10 5 and σ=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, t = 0.005, Str= with a physical time step, t = seconds, and Str= with a physical time step, t = seconds. As demonstrated in the figure, for the smaller two integration step sizes, over the range of relevant 11

12 (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. Amplitude C P 10 2 t=0.005 t= t= f Figure 23: Spectral comparison of effect of physical integration time step size on history. UNCLE-M result. Flow over a hemispherical forebody with cylindrical afterbody. Re D =1.36x10 5. σ= Fine Grid Medium Grid Coarse Grid s/d Figure 24: Comparison of predicted surface pressure distributions for naturally cavitating axisymmetric flow over a conical cavitator with cylindrical afterbody, σ = 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 t = ) using UNCLE-M have been found to be quite consistent with arithmetically averaged timedependent results. This result is expected to be useful in expediting the future interpretation of complex threedimensional 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 flowfield, 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 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 12

13 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 #N , with Dr. Kam Ng as contract monitor. References Brennan, C.E., Cavitation and Bubble Dynamics, Oxford University Press, New York, Garabedian, P.R., Calculation of Axially Symmetric Cavities and Jets, Pac. J. of Math 6, Grogger, H.A. & Alajbegovic, A., Calculation of the Cavitating Flow in Venturi Geometries Using Two Fluid Model, ASME Paper FEDSM Jameson, A., Schmidt, W., & E. Turkel, Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes, AIAA Paper , Kunz, Robert F., et al., Multi-Phase CFD Analysis of Natural and Ventilated Cavitation about Submerged Bodies, ASME Paper FEDSM , 1999(I). Kunz, Robert F., et al., A Preconditioned Navier- Stokes Method for Two-Phase Flows with Application to Cavitation Predication, AIAA Paper , 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 75-2, Merkle, C.L., Feng, J., & Buelow, P.E.O., Computational Modeling of the Dynamics of Sheet Cavitation, 3rd International Symposium on Cavitation, Grenoble, France, Orszag, S.A. et al., Renormalization Group Modeling and Turbulence Simulations, Near Wall Turbulent Flows, Elsevier Science Publishers B.V., Amsterdam, The Netherlands, 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, 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, Schlichting, H., Boundary-Layer Theory, McGraw- Hill, New York, Stinebring, D.R., Billet, M.L., & Holl, J.W., An Investigation of Cavity Cycling for Ventilated and Natural Cavities, TM 83-13, The Pennsylvania State University Applied Research Laboratory, Stinebring, D.R., Scaling of Cavitation Damage, M.S. Thesis, The Pennsylvania State University, University Park, Pennsylvania, August 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 ,