Application of the Full Cavitation Model to Pumps and Inducers

A new ``full cavitation model'' has been recently developed for performance predictions of engineering equipment under cavitating ̄ow conditions. A vapor transport equation is used for the vapor phase and it is coupled with the turbulent N ±S equations. The reduced Rayleigh±Plesset equations are used to account for bubble formation and to derive the time-mean phase-change rates utilizing the local pressures and characteristic velocities. Eects of turbulent ̄uctuations and noncondensable gases are also included to make the model complete. The model has been incorporated into an advanced ®nite-volume, pressure-based, commercial CFD code (CFD± ACE1) that uses unstructured/hybrid grids to integrate the N ±S equations. Full model details are being published separately. Presented here are simulations of cavitating ̄ows in three types of machines: water jet propulsion axial pump, a centrifugal water pump, and an inducer from a LOX turbo pump. The results show cavitation zones on the leading edgesuction side of each of the machines as expected. Simulations at dierent suction speci®c speeds were performed for the waterjet pump and the inducer and showed the proper trends of changes in cavity strength and sizes. All the test cases with cavitation show plausible results (no negative pressures, and good convergence characteristics). Computations on the waterjet pump for dierent noncondensible gas concentrations showed sizeable changes in the pump head developed.

Cavitation is usually found in ¯uid ¯ows where the local pressure drops below the saturation vapor pressure of the liquid, causing liquid evaporation and generation of vapor bubbles in the low pressure region.Fluid machinery is a common application where low pressures are routinely generated by the machine action, e.g., on blade surfaces, with a consequent possibility of cavitation.Other applications where cavitation is present include hydrofoils (Shen, 1989), submerged moving bodies (Kunz, 1999), and automotive fuel injectors, and draft tubes in turbines (Song, 1999).
Existence of cavitation is often undesired, because it can degrade the device performance, produce undesirable noise, lead to physical damage to the device and aect the structural integrity.Details of the existence, extent and eects of cavitation can be of signi®cant help during the design stages of ¯uid machinery, in order to minimize cavitation or to account for its eects and optimize the designs.Use of computational ¯uid dynamics (CFD) codes to analyze the viscous, turbulent ¯ows in ¯uid machinery has become fairly common, and advanced CFD code, together with a robust, validated cavitation model will be of signi®cant use for performance optimization of ¯uid machinery and devices in presence of cavitation.
The current eort is based on the application of the recently developed full cavitation model that utilizes the modi®ed Rayleigh±Plesset equations for bubble dynamics and includes the eects of turbulent pressure ¯uctuations and non-condensible gases (ventilated cavitation) to rotating cavitation in dierent types of ¯uid turbomachines.

DESCRIPTION OF THE NUMERICAL MODEL
The cavitation model accounts for all ®rst-order eects, i.e., phase change, bubble dynamics, turbulent pressure ¯uctuations, and non-condensible gases present in the liquid.The phase change rate expressions are derived from the Rayleigh±Plesset equations, and limiting bubble size (interface surface area per unit volume of vapor) considerations.Full derivation, along with model validations, are being presented in a separate paper.Here only the ®nal expressions and pertinent assumptions are summarized.
The vapor transport equation governing the vapor mass fraction, f, is given by: @ @t f r 1 Vf r 1 ÿrf R e ÿ R c 1 where V is the velocity vector, ÿ is the eective exchange coecient, and R e and R c are the vapor generation and condensation rate terms.The evaporation and condensation rates are functions of the instantaneous, local static pressure and are given by: where suxes l and v denote the liquid and vapor phases, V ch is a characteristic velocity, is the surface tension of the liquid, P sat is the saturation vapor pressure of the liquid for the given temperature, and C e and C c are empirical constants (in the present form these constants have appropriate dimensions to balance Eqs. [2] and [3]).0 The above relations are based on the following assumptions: 1.In the bubble ¯ow regime, the interface area per unit volume is equal to 3/R b , and the bubble size (R b ) is limited by the balance between the aerodynamic shear force and the surface tension force.This implies that the phase change rate should be proportional to V 2 rel =.However, in most practical two-phase ¯ow conditions with large number of bubbles, or bubble cloud type of conditions, the dependence on velocity is somewhat weaker and is assumed to be linear rather than quadratic.2. In the bubble ¯ow regime, the relative velocity between the vapor and liquid phases is of the order of 1 to 10 percent of the mean velocity.In most turbulent ¯ows the local turbulent velocity ¯uctuations are also of this order.Therefore, as a ®rst pragmatic approximation, square root of local turbulent kinetic energy k p is used for V ch in Eqs.[2] and [3]. 3. The preliminary values of the coecients C e and C c are set to 0.02 and 0.01, respectively.These values were arrived at by parametric studies over a wide range of ¯ow conditions in ori®ce and hydrofoil problems.4. A constant surface tension value of 0.075 N/m was used for calculations with water.0 Further calibration and validation is in progress and the ®nal values will be reported in a separate paper with the full description of the model.

Eects of Turbulent Pressure Fluctuations
Several experimental investigations have shown signi®cant eect of turbulence on cavitating ¯ows (e.g., Keller, 1997).A few years ago, present authors reported a numerical model (Singhal, 1997), using a probability density function (PDF) approach for accounting the eects of turbulent pressure ¯uctuations.This approach required: (a) estimation of the local values of the turbulent pressure ¯uctuations as: P 0 turb 0:39k 4 and (b) computations of time-averaged phase-change rates ( R e and R c ) by integration of instantaneous rates (R e and R c ) in conjunction with assumed PDF for pressure variation with time.In the present model, this treatment has been simpli®ed by simply raising the phase-change threshold pressure value from P sat to P sat P 0 turb =2.

Eects of Noncondensible Gases
In most engineering equipment, the operating liquid has small ®nite amount of non-condensible gases present.These can be in dissolved state, or due to leakage, or by aeration.However, even the small amount of (e.g., 10 ppm) of non-condensible gases can have signi®cant eects on the performance of the machinery.The primary eect is due to the expansion of gas at low pressures.This can lead to signi-®cant values of local gas volume fraction, and thus have considerable impact on density, velocity and pressure distributions.The working ¯uid is assumed to be a mixture of the liquid phase and the gaseous phase, with the gaseous phase comprising of the liquid vapor and the noncondensible gas.The density of the mixture, , is variable, and is calculated as where l is the liquid density, and v and g are the vapor and noncondensible gas densities, respectively, and f l , f v and f g are the mass fractions of the liquid phase, vapor phase and noncondensible gas.The volume fractions of the individual components can be calculated from the mass fractions as, e.g., v f v v : 6 The secondary eect of the non-condensible gases can be on the inception of cavitation by increasing the number of nucleation sites for bubble formation.This can be included as a re®nement in the empirical coecients (C e and C c ), perhaps as a functional dependence on f g .This is planned to be investigated later.

AND DISCUSSION
The full cavitation model has been implemented in an advanced, general purpose, commercial CFD code, CFD ± ACE .It was applied to a number of rotating cavitation problems encountered in turbomachines to assess its robustness and capability of predicting the occurrence, extent and strength of cavitation zones.Cavitation, when present, typically alters the pressure ®eld on the working surfaces of the turbomachine, with two obvious eects.Changes in blade pressure will alter the amount of work being put into the ¯uid, which changes the machine performance.Cavitation also will change the blade loading and have impact on the mechanical design of the blades.
Three dierent applications of turbomachines are presented here: an axial pump, a centrifugal pump and a rocket inducer.Each of the applications presents a dierent set of conditions and serves to illustrate the usability of the cavitation model.Given below are details of the ¯ow simulations.

Axial Pump
The pump considered operates in a waterjet propulsion system used on marine ships (e.g., see Allison, 1993).It has two stages: an inducer with four blades and an impeller (kicker) with eight blades mounted on a single shaft.The pump operates at a near-constant ¯ow rate and speed.The static pressure at the inlet depends on the ship speed, and can vary between 0.1 to 1 bar.This can lead to cavitation on the inducer blades over a large portion of the operating envelope.In the extreme cavitation case the entire inducer blade could be covered with cavitation with minimal zones on the impeller blades.
The computational model used one blade-to-blade passage of the inducer and 2 passages of the impeller to maintain periodicity.A single-domain structured grid was generated with approx.65000 cells.Both the inducer and impeller blades have tip gaps and 3 cells were used in the gap region.Working ¯uid was water at 290 K, with a liquid-to-vapor density ratio of about 70000.Turbulence in the ¯ow was handled using standard k±" model with wall functions.Appropriate stationary or rotating wall conditions were imposed on solid surfaces.Inlet boundary was a speci®ed ¯ow rate with a static pressure condition at the outlet.The exit static pressure was adjusted during computations to produce the desired inlet static pressure levels.A very small noncondensible air fraction of 0.5 ppm was used.Figure 1 shows the computational grid and boundary conditions.

RESULTS
Flow simulations at three dierent suction speci®c speed (Nss) values (US standard) were considered which covered nearly the entire range of Nss likely to be encountered.The speci®c values were approximately 13400, 15800, and 19200.In each case the solution was considered converged after a minimum residual drop of 4 orders of magnitude.In all the cases this required less than 900 iterations.A sample convergence plot is shown in Figure 2. Figure 3A shows the single-phase pressure ®eld for Nss 19200 with the negative pressure areas on the suction side of the inducer blades.The cavitating ¯ow solutions of the pressure ®eld are shown in Figure 3B.Cavitation zone extent can be estimated using the void fraction plots and the surface distribution is shown in Figure 4A.(The value of the volume fraction that signi®es the apparent boundaries of the cavitation zone is rather arbitrary and the practice adopted here is to use a value that is slightly higher than the non-condensible gas fraction; the typical values used here are a volume fraction of 0.0001 or so).In the present case the cavitation zone extends over the entire suction side of the inducer blade.The calculations also predict a fair amount of cavitation on the kicker blade leading edge.Reduction in Nss in present case indicates a higher upstream pressure, and hence represents less favorable conditions for cavitation, Figure 4B shows the void fraction plot for Nss 15800.For this case a triangular-shaped cavitation zone, occupying perhaps 40% of the leading portion of blade suction side is seen.Eects of cavitation on blade pressure pro®les are of importance for mechanical design, and a representative pressure plot along the blade chord length near the suction side tip is shown in Figure 5 where the single-phase and cavitation solutions are plotted.The pressure in the cavitation zone is nearly constant and positive, compared to the negative pressure zone in the single-phase solution.
For a marine application the presence of non-condensing gas can be expected to aect the cavitation characteristics and the pump performance, and this was explored to some extent by conducting another simulation at Nss 19200, where the noncondensible air fraction was increased to 5 ppm.The corresponding pressure ®eld is plotted in Figure 6.The calculated pump head rise for this case showed a decrease of about 4.5% from the earlier calculation with 0.5 ppm air concentration.Further investigations of this eect are in progress.

Centrifugal Water Pump
This is a common design used for pumping water with a moderate head increase.For a given ¯ow rate, the inlet conditions depend on the height dierence between the inlet plane and the lower reservoir.If this dierence is large, the inlet static pressure can become low enough to produce cavitation on the leading portion of the blades.In the present simulation, the inlet pressure level corresponds to a Nss of approximately 6200.A 2-domain, structured grid with 53000 cells was generated for this 6 bladed pump, with a single blade-toblade passage as the working domain.A constant inlet axial velocity was speci®ed at the inlet plane.A vaneless diuser section was used after the impeller blade exit.Figure 7 shows the computational model of this pump.Working ¯uid was water at 290 K. Standard k±" turbulence model was used for the turbulent ¯ow.The non-condensible gas fraction was set to 0.5 ppm.

RESULTS
The computed static pressure ®eld for cavitating ¯ow is shown in Figure 8.The predicted cavitation zone is located in the leading portion of the suction side as shown by the volume fraction plotted on the solid surfaces, shown in Figure 9.The severity and extent of this case is relatively low and a only a small decrease in the head developed was seen for this particular case.This application is from a high-performance liquid oxygen pump used in a rocket engine.The pump consists of an induce followed by a centrifugal impeller.The inducer operates in a regime where cavitation is expected.The pressurization in the inducer is relatively high, and this eliminates the possibility of cavitation in the impeller.The inducer has 4 blades, with a blade angle of approx.76 degrees with the rotational axis.The rotational speed is  A single domain grid with approx.43000 cells was generated for this geometry, with blocked cells representing the blade (Figure 10).A mass ¯ux at the inlet boundary was speci®ed, with a speci®ed static pressure at the exit boundary.The standard k ÿ " turbulence model was used.
As in the previous cases, the single phase solutions yielded pressurization across the inducer, and then the exit static pressures were adjusted to yield the required inlet pressure levels.Solutions at three Nss values of 9000, 13000, and 15000 were obtained to evaluate the changes in the cavitation zone.

RESULTS
Results from the Nss 15000 case are presented in Figures 11 and 12.The surface pressures for the cavitating ¯ow are shown in Figure 11 and, as before, all positive.The predicted cavitation zones are approximately near the leading edge suction side of the blade, as shown by the cavitation bubble boundaries, shown in Figure 12A.As seen here, the LOX vapor zone extends out to nearly the mid passage away from the blade.When Nss was lowered to 13000, i.e., the upstream pressure levels were increased, the cavitation bubble became much smaller as shown in Figure 12B.Sample line plots of the blade pressure distributions on the suction side tip region are shown in Figure 13 for the single phase and cavitating ¯ow and clearly show the in¯uence of cavitation, where a fair   amount of pressure redistribution on the blade surface behind the cavitation zone is seen.

SUMMARY AND CONCLUSIONS
The full cavitation model has been applied to cavitating ¯ows in three dierent turbomachine elements, namely a two-stage axial pump, centrifugal water pump, and a highperformance rocket pump inducer.The three examples oered dierent challenges to the ¯ow solver in terms of either severity of operating conditions (rocket inducer) or very high density ratios (water) in the other two problems.In each case, the model predictions were physically plausible, and the numerical convergence behavior was very robust and stable.The simulation results predict several of the  anticipated trends correctly, namely variation of size, location and shape of cavitation zone with dierent Nss values.In all of the cases, the cavitating ¯ows showed positive pressure levels at all places in the cavitation zones.The eects of the noncondensible gases on cavitation and pump performance were explored for the waterjet pump.Initial tests show that the presence of noncondensible gases reduced the pump head as well as the extent of the cavitation zone.This aspect will be of importance in a variety of applications and needs further work.Further development is needed for better calibration of the coecients C e and C c .If necessary these can also be made functions of physical eects such as water quality ( f ), surface roughness, etc. Additional, more through validation work on the full cavitation model, and applications to other problems is in progress and will be reported separately.

FIGURE 1
FIGURE 1 Solids model of the axial waterjet pump with ¯ow and boundary condition.

FIGURE 2
FIGURE 2 Sample convergence plot for the cavitation run on the axial waterjet pump (See Colour Plate at back of issue.).

FIGURE 3
FIGURE 3 Normalized static pressure ®eld in the waterjet pump for Nss 19000 (See Colour Plate at back of issue.).

FIGURE 4
FIGURE 4 Vapor volume fraction on solid surfaces, indicating the extent of cavitation for two dierent Nss values.

FIGURE 5
FIGURE 5 Pressure pro®le on the suction side tip region of the inducer stage of waterjet pump.Cavitation solution at Nss 19200.

FIGURE 6
FIGURE 6 Static pressure distribution in the pump for an air concentration of 5 ppm.Compare with results from Figure 3B (See Colour Plate at back of issue.).

FIGURE 7
FIGURE 7 Computational grid and model of the centrifugal water pump.

FIGURE 8
FIGURE 8 Surface pressure ®eld for the centrifugal pump, cavitating ¯ow (See Colour Plate at back of issue.).

FIGURE 9
FIGURE 9 Void fraction distribution, showing the cavitation zone and extent, centrifugal pump, Nss 6200 (See Colour Plate at back of issue.).

FIGURE 10
FIGURE 10 Computational model of the rocket inducer.

FIGURE 11
FIGURE 11 Static pressures on solid surfaces for the inducer, cavitating ¯ow, Nss 15000 (See Colour Plate at back of issue.).

FIGURE 12
FIGURE 12 Extent of the cavitation zone in the inducer at dierent Nss values (See Colour Plate at back of issue.).