Numerical model to simulate cavitating flow

5th International Conference on Advances in Mechanical Engineering and Mechanics ICAMEM2010 18-20 December, 2010, Hammamet, Tunisia

Kanfoudi, Lamloumi and Zgolli 1

Numerical model to simulate cavitating flow

H. Kanfoudi,H. Lamloumi and R. Zgolli

National Engineering School of Tunis, B.P.37, Tunis 1002, Tunisia

[email protected]; [email protected]

Abstract

For numerical simulation of cavitating flows, many numerical models currently proposed use some assumptions or/and

empirical formulations that must limit their performance. We present here a new model based on the void fraction transport

equation solved with the source term evaluating vaporization and condensation processes. The model is coupled with a CFD

code solving the Reynolds-averaged Navier-Stokes equations for the mixture (liquid and/or vapor) to approach the cavitating

flow. To test the validation of the numerical simulation, we present results obtained with a 2D and 3D approach for the flow

around a NACA0009 and NACA4412 hydrofoil.

Nomenclature

α volume fraction of the vapor phase

𝑚 + mass transfer of vaporization

𝑚 − mass transfer of condensation

𝜌𝑚 mixture density

𝜌𝑙 liquid density

𝜌𝑣 vapor density

𝜇𝑚 mixture viscosity

𝜇𝑙 liquid viscosity

𝜇𝑣 vapor viscosity

𝑝𝑣 saturation pressure

n0 density of bubbles

R0 initial radius

u∞ reference velocity

c chord

p local pressure

R radius of the bubble

B bubble

i attack angle in degrees

surface tension

Introduction

The phenomenon of cavitation that occurs within the flow of a liquid can be searched for specific

industrial applications as it should be avoided in order not to suffer adverse consequences in other

applications. In all cases we must learn to predict, and in this regard there is more research work. We

contribute here with the presentation of our development with the aim to develop a numerical method

to simulate the cavitating flow. The model presented here is developed in an attempt to predict the

onset of cavitation as a result of pressure drop and also the changes in the flow. The model is based on

the source term of the transport equation computing the vapor volume fraction which has the special

permit to reflect the quality of the liquid and also its tension surface. To validate the method we

consider the flow around hydrofoils that have been the subject of experimental measurements and also

other numerical methods.

Mathematical formulation

Governingequations

Many existing cavitation models in literature are categorized in VOF method known as well as two-

fluid model. The governing equations consist of the conservative from the Reynolds averaged Navier-

Stockes equation and a volume fraction transport equation. These equations are in cartesian

coordinates, presented below:

The Continuity Equation:

0

m jm

j

u

t x

(1)

mailto:[email protected]

mailto:[email protected]



The Momentum Equations:

m j i jm i im t

j i j j i

u u uu up

t x x x x x

(2)

Transport equation :

v jv v c

j

uS m m

t x

(3)

The mixture density, viscosity and the turbulent viscosity are defined respectively, as follows :

1m v l

(4)

1m v l

(5)

Our proposed cavitation model;

For numerical simulation of cavitating flows, many numerical models currently proposed use some

assumptions or/and empirical formulations that must limit their performance. We present here a new

model based on the void fraction transport equation solved with the source term evaluating

vaporization and condensation processes. It is able to take account of the effect of pressure forces and

surface tension. Fistly, we compare sensibility of pressure distribution with results obtained by

various source terms proposed by different researchers.

, ( , )pdS C f g p sign g p

(6)

with :

13

03.3 v lpd

m

C n

(7)

232

01f n h

(8)

12

1 23 3

3 2

0 04 2.52 3, 1 1

3

v

l l

p p R Rg p

h h h

(9)

0 1

hn

(10)

Comparison of model

We present a comparative study between the differnet vaporization and condensation terms propsed for the the void fraction transport equation. Most of the terms depend mainly on the diffence between the local pressure and the vapour pressure p-pv. Thus, the following comparaison between the models is based on the expression of the sources terms as a function of p-pv. However, the void fraction α usually also appears in theexpression of the source terms. To expres them as a function of p-pvonly, the barotropic state law of Delannoy is used (Fig. 1).



Figure 1 : Comparison between the sources terms, the left the vaporization process and the right the condensation

process.

In literature, empirical factors are determined through numerical/experimental results and are adjusted for different geometries and different flow conditions.To make this comparison possible, the empirical factors (production/destruction coefficients) are adjusted to obtain the same maximum value for the source terms.The empirical factors have the following values:

Cp=10. , Cd=0.7 for the Kunz model,Cp=410-4

. , Cd=1.4for the Singhal model,Cp=50. , Cd=0.005for

the Schnerr model and n0=1010

B/m3 for the Yuan and New models.

It is remarkable that the new model has the same form as the Kunz el al. model. It is sensitivity to pressure change for the two processes.

Result and discussion

To compute the flow close to the wall, standard wall-function approach was used, and then the

enhanced wall functions approach has been used to model the near-wall region. For this model, the

used numerical scheme of the flow equations was the segregated implicit solver. For the model

discretization, the HIGH resolution scheme was employed for pressure-velocity coupling, for the

momentum equations, and first-order up-wind for other transport equations (e.g. vapor transport and

turbulence modeling equations).

The domain is 9 blocks C-type grid of 88000 mesh cells. The steady state RANS simulations with the

k-/SST in this case. The boundary conditions are set using a velocity inlet (u∞=20 m/s) and an

average static pressure at the outlet (the parameter which fixes the cavitation number). The turbulence

is set to 1% of intensity and 0.001m of eddy length scale. Both upper and lower section walls and the

hydrofoil are modelled using non-slip conditions with classical log-law functions (y+ =1). Both lateral

sides of the domain are modelled as symmetrical planes. Numerical convergence is set to a maximum

of 10−4

for all the simulations.

Figure 2:NACA0009 domain grid (c=100 mm)

Inlet Outlet

Zoom In



Validation

The numerical approach of cavitating flow use the CFD code with introduction of our proposed model

will be tested in first time into a 2-D flow around a hydrofoil that we have an experimental result. It

concerns a NACA0009 hydrofoil, truncated at 90% of the original chord length. It has the final

dimensions of 100mm of chord length and 150mm of span. The hydrofoil is placed in the test section

of the EPFL high-speed cavitation tunnel (Ait bouziad, 2006).

The performance of our proposed model is based on the critical values of computed volume

fractionused to locate the interface between pure liquid and pure vapor of the mixture flow. The same

assumptions of spherical bubbles used to establish the expression of source term of the proposed

model will be considered to calculate the tow critical values αvap and αliq . Where αvap is the limit value

of volume fraction to the passage from liquid to vapor and corresponding to the engagement of

evaporation, and αliq is the limit value of α to the passage from vapor to liquid and corresponding to

the beginning of condensation.

If we consider equally sized spherical particles forming a rhomboid array, the average distance

between the centers of two adjacent particles with diameter D (D=2R) and volumetric fraction α is:[7]

𝑙 = 𝐷 𝜋

6

2

𝛼 1 3

(17)

The non-oscillating particles will touch each other if l=D, this happens for a volume fraction value;

𝛼 =𝜋

6 2 ≈ 0.74 (18)

Whichis sometimes called in the literature the maximum packing density volume concentration. This

consideration leads to the conclusion that bubblesfill the whole field of flow for;

𝛼 > 0.74 = 𝛼𝑣𝑎𝑝 (19)

And inversely the mixture can be considered pure liquid for;

𝛼 < 0.26 = 𝛼𝑙𝑖𝑞 (20)

Then the range of volume fraction values (figure 3) is used by the proposed numerical model to

characterize the shape of the possible presence of vapor cavity.

Figure 3: Range of vapor volume fraction ()

In order to assess the capabilities of the proposed model of cavitation to evaluate the shape of cavity

attached on the walls of the hydrofoil, we consider two cases of cavitating flow (figure 4-a for = 0.8

and figure 4-b for =0.85) around the same profile for which we have experimental (Ait bouziad,

2006). We can remark the good concordance of the pressure coefficient computed with the

experimental result. We can also note that the shape of the cavity can be correctly evaluated using this

model.

We begin by testing two models of turbulence k-ε and SST and we present the results in figure 4. We

note that the SST turbulence model is most stable and reflect the effect of the presence of cavitation

pocket closest to the experimental results, as shown in this figure.We also note that the k- is the

turbulence model with the most attenuation of cavitation effects, which is due to a possible

overestimation of the turbulent viscosity.



Figure 4: Influence of turbulence model on the calculated pressure coefficient (i=2.5 °

,𝐮∞=30 m/s, =0.8 and =0.85 ).

A 3D approach of cavitating flow

To test the performance of our proposed model with 3D flow simulation, we consider a water flow

around hydrofoil NACA 4412 with chord equal 3 in and Span equal 10 in., for which we have

experimental results (Knap, 1944). The computed domain is represented by figure 6.

The numerical solution was very sensitive to liquid quality that is managed here by the number n0. For

water the number n0 can be fixed around 108 but physical explanations rest quite limited. To make our

model able to approach the real solution, we calibrate this number by comparison with experimental

results. Figure 7 shows the numerical results of lift force obtained for different values of n0, and we

can conclude that the best agreement with experimental measurements for <1 (favorable conditions

for the appearance of cavitation) is the case with n0 = 5.108.

Figure 5: Computed domain of water flow around a Hydrofoil NACA 4412.

Figure 6: Cavitating flow computed (with our model) and observed (Knap, 1944)

Inlet

700 mm

254 mm

300 mm

a - = 0.8 b - = 0.85



We present in this figure the numerical result obtained for cavitating water flow around the hydrofoil,

and we can conclude the good concordance with the experimental result.

Conclusion

This study present a numerical method approaching cavitating flows that uses a CFD code solving the

Navier-Stokes equations with homogeneous mixture consideration. This method is based on the

introduction of a model with a form of source term of transport equation coupling the pressure

calculation with the volume fraction distribution. The model presented here has shown the ease with

which one can calibrate to suit different qualities of the liquid considered. Previously we gave some

insights on how to validate and especially the opportunities to rely on such models to monitor changes

in length and also the shape of the cavity which occurs in cavitating flow.Finally for temporal

evolution of the flow structure from cavity formation to cavity growth towards the trailing edge, we

present the unsteady behavior of cavitating flow simulated by this numerical method.

References

R. F. Kunz, D.A. Boger, D. R. Stinebring, et al. (2000)., “A preconditioned Navier-Stokes method for two-phase flows with

application to cavitation prediction, “Computers & Fluids, vol. 29, no. 8, pp. 849-875.

R. F. Kunz, D.A. Boger, D. R. Stinebring, T.S. Chyczewski, J. W. Lindau and T.R. Govindan, (July 1999) “Multi-phase

CFD analusis of natural and ventilated cavitation about submerged bodies”, in Procceedings of 3rd ASME/JSME Joint

Fluids Engineering Conference (FEDSM’99), p. 1, San Francisco, Calif,USA.

A.K. Singhal, M. M. Athavale, H. Li, and Y.Jiang, (2002) “Mathematical basis and validation of the full cavitation model”,

Journal of Fluids Engineering, vol. 124, no. 3, pp.617-624.

G. H. Schnerr and J. Sauer, (May-June 2001) “Physical and numerical modeling of unsteady cavitation dynamics”

inProceedings of the 4th International Conference on Multiphase Flow (IMCF’01), New Orleans, La, USA.

Yuan, W. Sauer, J.,Schnerr, G.H (2001). Modeling and computation of unsteady cavitation flows in injection nozzles.

Mec.Ind., vol. 2, pp. 383-394.

AIT BOUZIAD Youcef (2006). Physicalmodelling of leading edge cavitation: computational methodologies and application

to hydraulic machinery, Thesis, French, pp. 70-80.

KNAP, Robert T. (1944). Force and cavitation characteristics of the NACA 4412 hydrofoil. USA : s.n.. pp. 20-21. 6.1-sr207-

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Numerical model to simulate cavitating flow