Numerical Evaluation of Cavitating Flows Using

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Vito Salvatore and Ciro Pascarella CIRA, Italian Aerospace Research Center Capua, CE, Italy 81043 3

Luca d’Agostino University of Pisa Pisa, PI, Italy 56126

$EVWUDFW This paper focuses on the modeling of cavitating flows by means of a barotropic law of state. Within a robust 2D Navier-Stokes flow solver, capable to handle all the complex aspects of cavitating flow, different cavitation models are implemented. In particular two different classes of barotropic cavitation models have been used: a sinusoidal model and a quasi-homogeneous isenthalpic cavitation model suitably modified to account for thermal effects. The paper deals with the differences between these models and hence with the differences in the resulting flow fields. The analysis has been made for the flow over an ogival headform.

,QWURGXFWLRQ In the last decades, the study of cavitation has been performed mainly through experimentation, while modeling activities have been based essentially on analytical approaches. In recent years, numerical computations of cavitating flow fields, especially in turbomachinery equipment, have been increasing due to the development of more and more efficient numerical algorithms and to the availability of increasingly more powerful computers. However, the modeling of cavitation is still hampered by the remarkable difficulties in describing the complex flow phenomena occurring in a cavitating medium. In addition, a cavitating fluid often corresponds to a complicated situation from a numerical point of view; in fact, both incompressible zones (pure phases) and regions where the flow may become highly supersonic (liquid-vapor mixtures) are present in the flow field and need to be resolved. The numerical stiffness of the phenomenon is further increased both by the high 5 density ratio between the two phases (for water at 293 K the density ratio is equal to 1.7 10 ) and by the strong shock discontinuity occurring at re-condensation. This singular behavior is due to the huge variation of the speed of sound as the flow transitions from a pure phase to a two-phase mixture. For instance, in the above hypotheses, the speed of sound is equal to about 1500 m/s for the liquid phase, 400 m/s for vapor phase and 3.2 m/s for a liquid-vapor mixture corresponding to a value of void fraction equal to 0.51. Moreover, the latter value of speed of sound is calculated for thermal equilibrium conditions and neglecting both the exchange of mass between the two phases and surface tension effects. Indeed, the calculation of the minimum speed of sound can vary largely depending on the hypotheses that one assumes. For example, using the steam tables (Rivkin, 1988) and assuming cavitation as an isenthalpic transformation (Avva and Singhai, 1995), the minimum speed of sound at 293 K is approximately equal to 0.004 m/s. In addition, the speed of sound is a parameter that can be easily defined only when the thermodynamics of phase change is very fast or very slow (frozen - equilibrium) with respect to the characteristic time of the

Aerospace Eng neer,v.sa vatore@c ra. t Mechan ca Eng neer,c.pascare a@c ra. t 3 Professor, Department of Aerospace Eng neer ng 2

phenomenon (Brennen, 1995), and is strictly linked to the cavitation model. For all the other cases a speed of sound cannot be simply defined. This paper will present the analysis of different cavitation models and will not focus on the physical uncertainty in the definition of the correct speed of sound. The work presented herein represents a step of a larger project aimed at the development of an advanced reliable tool for the numerical simulation of cavitating flows, and capable to provide useful indications on cavitation in devices such as turbopump inducers (Pascarella et al., 2000).

&DYLWDWLQJIORZPRGHO All the past main efforts to numerically model cavitating flows can be roughly divided into two classes: nonhomogeneous methods and homogeneous or continuum methods. Non homogeneous methods can be sub-divided into interface tracking and bubbly flows methods. Typically, there are separate balance equations for each phase (Brewe, 1986, Vijayaraghavan and Keith, 1990, Deshande and Merkle, 1979, and Chen and Heister, 1994(b)). These methods present two main disadvantages: they are computationally expensive, due to the double number of balance equations, and, in the case of interface tracking, they cannot be easily extended to 3-D flows because of the obvious difficulties involved in tracking complex interfaces in space. Continuum methods treat the flow as a homogeneous mixture employing a void fraction variable to quantify the intensity of cavitation; this variable is defined as the ratio between the vapor volume and the whole volume (liquid plus vapor):

a=

 - r r/  - r9 r/

(1)

Continuum methods can be sub-divided into three categories:

á analyses introducing a vapor source production term, á analyses including the bubble dynamics and á analyses assuming a barotropic state law. The model that employs a vapor production equation (Song and He, 1998) resolves, together with the momentum and continuity equations for the mixture, a phasic continuity equation when p> dT, Plesset-Zwick hypothesis, then it can be written

d7 = 5

a/

(13)

b

where

b=

S9 - S  r & Q FS  /

(14)

()

with

S=

/ r 9

(15)

r/ F 3/ 7Š a /

a purely thermodynamic quantity that is a function only of the liquid temperature 7Š . C(n) is a constant value, defined as 

] Q + Ë Q +Û &(Q ) = Q Ì  G]Q ³ () Ü Í p Ý   - ]  Q +

×

(16)

of order unity for most values of n of practical interest (Brennen, 1995). The previous analysis has been developed for an insulated cavitation bubble. Considering that the presence of multiple bubbles reduces their growth rate and so alter the fluid dynamic response of the field, in the aim of simulating more realistic cavitation occurring, a correction term c is needed, (Rapposelli, 2000).

In this hypothesis, hence,

d7 / r 9 =(7Š ) = = 5 r/ F 3/ 7Š &(Q)F (S 9 - S ) (S 9 - S )

(17)

with =(7Š ) a thermodynamic parameter depending only on the liquid temperature. The appeal of the above model consists in its independency from empirical assumptions. In fact, the relationship (17) is fully defined when the working fluid reference temperature has been assigned. The pressure appearing in (17) is the local pressure, depending on temporal and/or spatial coordinates, but, in the calculations performed here, has been assumed equal to its exit value. Considering that eL and eV must be in the range defined by homogeneous frozen model (eL = eV =0) and homogeneous equilibrium model (eL = eV =1), the following limiter has been introduced:  Ñ Ër Þ áÔ Û ÎË d Û Ô e / (S r) = PLQÒ ÌÌ / - ÜÜ ÏÌ + 7 Ü - ß â 5Ý ßà Ô Ý ÏÐÍ ÔÓ Í r ã

(18)

By integrating the Equation (5), density as a function of pressure as been obtained. For example, curves corresponding to this model are reported in Figure 4 for water at different values of the reference temperature and s=2.0, and in Figure 5 for different values of the cavitation parameter, and, hence, dT/R, at 7Š =323K

Figure 4 Isenthalpic curves for water at different values of

7Š

Figure 5 Isenthalpic curves for water/vapor mixture at different values of s

1XPHULFDODSSURDFK In the homogeneous approach and in the hypothesis of thermodynamic equilibrium the governing equations to solve are the Navier Stokes equations written in the unsteady, compressible form for one phase without the energy conservation equation; the barotropic state law completes the set of equations. The model may be written as three partial differential equations and one algebraic equation of state:

›r + ¶ ¼ r9 =  ›W

(19)

›rX ›S + ¶ ¼ rX9 = + ¶ ¼ m¶X ›W ›[

(20)

›rY ›S + ¶ ¼ rY 9 = + ¶ ¼ m ¶ Y ›W ›\

(21)

r = rS

(22)

where r and m are, respectively, the density and the viscosity of the mixture. The barotropic law of state, Eq. (22), defines the value of the density; a simple and stable (even if not very accurate) way to calculate m is the following (Kubota et al., 1992):

m = m Y a + m O  - a

(23)

where mG and mL are the vapor viscosity and the liquid viscosity respectively, and a is the void fraction. The choice of the numerical method to approach cavitation is very critical. In fact the flow field goes from incompressible in the one-phase regions to highly supersonic where a liquid-vapor mixture is present. Thus the flow solver must be capable to work well in a wide Mach number spectrum, from zero to Mach>>1. Furthermore, a cavitating flow is characterized by huge gradients at the cavitation interface and in correspondence of recondensation shocks. Hence the code must have shock-capturing or interfacecapturing properties in order to correctly solve the rapid transitions to cavitating conditions. The most suitable methodology to solve the governing equations seems to be a pressure-based method with a finite volume discretization written in conservative form, based on the SIMPLE (Semi Implicit Method for Pressure Linked Equations) algorithm (Patankar, 1980), with Karki (1986) compressibility correction. The code developed in the present work, named ALL2D, can be applied to both plane and axi-symmetric configurations under steady or unsteady conditions, and can be used to simulate both inviscid and viscous nd s laminar flows. In the developed code a non oscillatory TVD 2 / 1 order scheme, derived from SMART (Gaskel, and Lau, 1988), is adopted. ALL2D works with a body-fitted grid; the governing equations (19)-(21) have been reported here only in a Cartesian coordinate system for simplicity; in any case, applying the tensor rules, the system of equations (19)-(21) can be transformed in an equivalent system written in a curvilinear reference frame. The actual form of the governing is more complex and can be found in Kadja and Leschziner (1987). As in all the methods derived from the SIMPLE algorithm, also in ALL2D it is necessary to use the underrelaxation at the end of each iteration to promote stability (Patankar, 1980). Particular attention must be paid when the flow is cavitating; in this case, to avoid abrupt oscillations of density, the under-relaxation of pressure must be limited by imposing values to the pressure correction always less than (pL-pV). A way to achieve this is to introduce a varying under-relaxation factor which depends on the pressure correction itself as follows:

a3 =

S/ - S 9  D3PD[ x

(24)

where DPmax is the maximum value of pressure correction and c is an arbitrary number greater than one.

9DOLGDWLRQ The numerical method has been validated using the experimental data available from Rouse and McNown (1948), consisting in pressure measurements on the surface of some head forms at zero angle of yaw in water under steady state conditions. Although these experiments are very old the data are quite reliable. Among the several tests reported in Rouse and McNown (1948) two different head form shapes have been chosen. Both head forms belong to the so-called "Rounded Series"; in particular, they are called " 2’’ Caliber Ogival" and "Hemispherical". The comparison between numerical and experimental data has been made for each geometry at two different flow conditions corresponding to incipient cavitation and fully developed cavitation, see Table 1).

2”-Caliber Ogival Headform

Hemispherical Headform

K=0.3 (incipient cavitation)

K=0.7 (incipient cavitation)

K=0.2 (fully developed cav.)

K=0.2(fully developed cav.)

Table 1 Validation test cases

Following the terminology adopted in Rouse and McNown (1948) a cavitation index K has been defined as:

S - S9 .= Š S7 - SŠ

(25)

The simulations have been performed neglecting viscous effects (Euler approximation) and assuming a minimum speed of sound of 0.12 m/s. The calculations have been carried out assuming a temperature of 320 K; the corresponding density ratio is equal to 13305 (989/0.072) and the vapor pressure is equal to 10530 Pa. Both the very high density ratio and the very low speed of sound represent quite severe conditions: in most of the works in the literature on numerical modeling of cavitation using a barotropic law of state, a maximum value of density ratio of 1000 and a minimum speed of sound of 0.5-1 m/s is reported, for example, in Delannoy and Keuny (1998) and Hoeijmakers et al. (1998). For all simulations the total pressure at the inlet and the static pressure at the outlet have been imposed as boundary conditions. The comparison between calculated and measured CP distributions along the surface of the ogival head form is shown in Figure 6, for the case of incipient cavitation, as shown by the small zone where the CP remains almost constant. A good agreement between computed and experimental values is attained. This agreement is still good for the fully developed cavitation case, as shown Figure 7: the length of the cavitation region has been captured rather well. The discrepancy in the re-condensation zone, where the computation shows a shock, is probably due to thermal non-equilibrium effects and to the inevitable averaging errors in the measurement of a highly unsteady region of the flow. Figure 8, shows the CP distribution on the hemispherical head form corresponding to the onset of cavitation; the comparison of numerical and experimental values is again quite satisfactory.

Figure 6 Ogival Headform: Incipient Cavitation, K=0.3

Figure 7 Ogival Headform: Developed Cavitation, K=0.2

The most severe case is represented the fully developed cavitation on the hemispherical head form. Also for this case, agreement is excellent up to the end of the cavity where some discrepancy can be noted at recondensation, Figure 9. More details can be found in Pascarella et al. (2000, 2001).

Figure 8 Hemispherical Headform: Incipient Cavitation, K=0.7

Figure 9 Hemispherical Headform: Developed Cavitation, K=0.3

5HVXOWV The dynamic response of the cavitating flow field around a 2" caliber ogival headform has been investigated, using different barotropic cavitation models. Water has been chosen as working fluid, at different temperature (see Table 2 for the thermodynamic properties). The domain has been discretized in 151x69 grid points; the computational grid is illustrated in Figure 10. A grid independence analysis has been initially performed using a finer grid and a coarser grid point. Some discrepancies were observed in the results obtained with the coarse and medium (reference) grids. Only one computation was carried out with the finer grid, which involved a longer computer time, with negligible differences in the flow when compared to the medium grid. Therefore, all calculations were performed with the medium grid (151x69). At inlet, a uniform value of the velocity has been assumed and an uniform pressure has been assigned at the exit plane. A no-penetration condition has been imposed at the headform surface. For the other boundaries, classical free-stream conditions have been adopted. Ten test cases have been performed by varying only the water temperature and the cavitation parameter and hence the exit pressure (see Table 3) The value of the minimum speed of sound of the sinusoidal barotropic model has been set equal to 0.12 m/s. This value is the best tuning value obtained to match the experimental data during validation activity. The cavitation number has been defined as follow: s =

S UHI - S /

   r UHI 9UHI

26

where reference values are specified at inlet of the computational domain The minimum speed of sound of two-phase mixture for the sinusoidal model has been set equal to 0.12 m/s, as in the validation activity.

Analyzing Figure 11, it can be noted that the differences between cavitating flow fields, obtained by using the two different cavitation models, become more evident at higher temperature. Differences are more remarkable in term of intensity, increasing with increase of temperature, as shown by Table 4 where the minimum value of density achieved in the flow field and the percentage error between the solutions are reported. The same Table 4 shows that the point of the flow field where rmin is achieved is quite independent by the model used. In terms of extension of the cavitation region, (see Figure 12) isenthalpic model give a longer sheet cavity, but, increasing temperature, difference vanishes, Figure 13. (The cavitation region in individuated by the horizontal segment in the pressure coefficient distribution) By increasing the cavitation parameter, differences are less evident, Figure 14, even if the trend is the absolutely same (see Table 4). Analyzing Figure 15,Figure 16 and Figure 17, it can be noted that the cavitating flow extension obtained by the two models is quite the same and some discrepancies are evident only in the recompression region. The available experimental data are not enough detailed and accurate to be used as support in the present analysis. In other words, the differences between the two numerical solutions are less than the differences between each numerical solution and experimental data.

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Table 2 Thermodynamic properties of water at different temperature



N   

T [K]

s



P Outlet [Pa]

V Inlet [m/s]



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Table 4 Main Results

&RQFOXVLRQV The development of a quasi-homogeneous isenthalpic cavitation flow model, suitably modified to account for thermal effects, and the results of calculations performed to simulate cavitating flows, have been presented and discussed. The flow solver, ALL2D, has shown excellent capabilities in view of the fact that all the peculiar aspects of cavitation flow have been caught. In fact, for the head form field used for validation, computed results are in good agreement with experimental data. Moreover, the simulation of a NACA0015 hydrofoil at different angles of attack has also verified the capability of the computer code to capture the vapor cavity and the relevant flow features (re-entrant jet). ALL2D has been applied to evaluate the different barotropic model of cavitation. Differences between the model based on a sinusoidal law of state and model based on the proposed isenthalpic approach have been underlined. A very encouraging message obtained by this study is the proof that the isenthalpic model with thermal effects gives excellent results.

Finally, further investigations must be needed especially for evaluating the influence of viscosity on cavitation occurring.

1RPHQFODWXUH a b CP cP h IMTE L K k p pT R SM s T t V

= speed of sound = mean separation distance among bubbles = pressure coefficient = specific heat = specific enthalpy = isenthalpic model with thermal effects = latent heat of evaporation = cavitation index (Rouse and McNown) = polytropic constant = pressure = total pressure = bubble radius = sinusoidal barotropic model = specific entropy = temperature = time = velocity vector

Greek a dT e f m c r s D Pmax S

= thermal diffusivity, void fraction = thermal boundary layer thickness = phase fraction = general variable = dynamic viscosity = arbitrary number > 1 = density = cavitation number = maximum value of pressure correction = thermal parameter

Subscripts

Š

L G re S V

= free stream condition = liquid; pure liquid in the barotropic two-phase representation = gas; pure vapor in the barotropic two-phase representation = reference = saturation = vapor

5HIHUHQFHV Avva, R. K. and Singhai, A. K., 1995, An Enthalpy Based Model of Cavitation, FED Vol. 226, Cavitation and Gas-Liquid Flow in Fluid Machinery Devices, ASME 1995 Brennen, C. E., 1995, Cavitation and Bubble Dynamics, Oxford Univ. Press. Brewe, D. E., 1986, Theoretical Modeling of the Vapor Cavitation in Dynamically Loaded Journal Bearings, Trans. Of ASME, J. of Tribology, Vol. 106, pp. 628-638 Chen, Y. and Heister, D. S., 1994(a), Two Phase Modeling of Cavitated Flows, FED-Vol 190, ASME Cavitation and GasLiquid Flow in Fluid-Machinery and Devices Forum Chen, Y. and Heister, D. S., 1994(b), A Numerical Treatment for Attached Cavitation, J. Of Fluid Eng., September, Vol.116

De Jong, F. J. and Sabnis, J. F., 1991, Simulation of Cryogenic Liquid Flows with Vapor Bubbles, AIAA-91-2257 Delannoy, Y. and Kueny, J. L., 1998 Two-Phase Approach in Unsteady Cavitation Modeling, FED Vol. 98, ASME Cavitation and Multiphase Flow Forum, Grenoble, France Deshande, M. and Merkle, C., 1979, Cavity Flow Predictions Based on Euler Equation, Trans. Of ASME, J. Fluid Eng., Vol. 116, pp. 36-44 Gaskell, P. H. and Lau, K. C. , 1988, Curvature Compensate Convective Transport: SMART, a New Boundeness Preserving Transport Algorithm, International Journal for Numerical Methods in Fluids, 8, pp. 617-641 rd

Hoeijmakers, H. W. M., Janssens, M. E. and Kwon, W., 1998, Numerical Simulation of Sheet Cavitation, 3 International Symposium On Cavitation, Grenoble, France Jakobsen, J. K., 1964, On The Mechanism of Head Breakdown in Cavitating Inducer, J. Of Basic Eng., Trans. ASME, June 1 964, pp. 291-305 Kadja, M. and Leschziner, M. A., 1987 The Team-Cog Computer Code, UMIST, Draft1 Karki, K. C., 1986, Ph.D. Thesis, Minnesota State University. Kubota, A., Kato, H. and Yamaguchi, H., 1992, A New Modeling of Cavitating Flows; a Numerical Study of Unsteady Cavitation on a Hydrofoil Section, Journal of Fluid Mechanics, Vol. 240, pp 59-96 Pascarella, C., Ciucci, A., Salvatore V. and d’Agostino L., 2000, A Numerical Tool for the Investigation of Cavitating th Flows in Turbopump Inducers, 36 AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, July 17-19 2000, Huntsville, AL, USA Pascarella, C. and Salvatore V., 2001 (a), Numerical Study of Unsteady Cavitation on a Hydrofoil Section Using a th Barotropic Model, 4 International Symposium On Cavitation, June 20-23 2001, Pasadena, CA, USA Pascarella, C., Ciucci, A. and Salvatore V., 2001 (b), Numerical Study on the Effects of Speed of Sound Variation on Unsteady Cavitating Flows, Submitted to Trans. of ASME, J. Fluid Eng. Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation. Rapposelli, E., 2000, Degree Thesis, University of Pisa, Italy Rivkin, S.L., 1988, Thermodynamic Properties of Gases, Hemisphere Publishing Corporation. Rouse, H. and McNown, J., S., 1948, Cavitation And Pressure Distribution:Head Form At Zero Angle Of Yaw, State University of IOWA, Eng. Bull.,N.32 rd

Song, C. and He, G., 1998, Numerical Simulation of Cavitating flow by Single-Phase Flow Approach, 3 International Symposium On Cavitation, Grenoble, France Vijayaraghavan, D. and Keith, T.,G., 1990, An Efficient, Robust, and Time Accurate Numerical Scheme Applied to a Cavitation Algorithm, Trans. Of ASME, J. Of Tribology, Vol 112, pp.44-51