Esra Sorguven, Gunter H. Schnerr

Modified k − ω Model for Simulation of Cavitating Flows

Although cavitating flows are generally turbulent, the interaction of turbulence and cavitation dynamics is still notwell understood. In general, two-equation models are employed, which were originally developed for single-phaseflows. Therefore they fail by handling cavitation based flow phenomena with very high density variations (dependenton operating condition up to 40000:1). This sudden change of the density causes strong pressure gradients, secondaryflows and local compressibility. The aim of this study is to enhance the Wilcox’s k-ω model with empirical correlationsin order to simulate turbulent cavitating flows more precisely and efficiently.

1. Introduction

In order to demonstrate the effect of the correlations, developed cavitating flow over a NACA 0015 hydrofoil withan angle of attack of 60 is simulated. The results are compared with experiments [1]. The simulations are carriedout with our 2-D Finite-Volume Code CAVKA. Cavitation is modeled with the Rayleigh equation and the Volume-of-Fluid method, and turbulence with the k − ω model of Wilcox. In this study the original single-phase k − ωmodel is modified for two-phase flows. There are several correlations listed in the literature to enhance two-equationturbulence models. Here the following three models are investigated.

The turbulent viscosity correlation [2]: A well known shortcoming of two-equation turbulence models is thatturbulent viscosity tends to be overestimated even in single-phase flows. In cavitating flows this can influence theflow dynamics strongly, since the overestimated viscosity enforces steadiness of the flow. In order to overcome thisphenomena, the linear dependence of µt and ρ is replaced by an exponential relation with the model parameter N:

µturb = f(ρ) · k

ω(1)

f(ρ) = ρvap + (ρvap − ρ

ρvap − ρliq)N · (ρliq − ρvap)· (2)

The local compressibility correlation [2]: Although the speed of sound is very large in pure liquids and inthe vapor phases, it decreases rapidly in gas-liquid mixtures. In the single-phase regime the Mach number is low,generally less than 0.3, but it becomes considerably high (over 1) in the two-phase regime. Thus local compressibilityhas been taken into account by introducing the turbulent Mach number Mt =

√2k/a into the equations for the

turbulent kinetic energy and the turbulent dissipation. Here Sarkar’s and Wilcox’s correlations are tested.

k, ω = f(β, β∗) (3)

β =9

100(1 + ξ∗F (Mt)) (4)

β∗ =9

100− 9

125ξ∗F (Mt) (5)

Sarkar’s correlation:ξ∗ = 1 (6)F (Mt) = M2

t (7)

Wilcox’s correlation:ξ∗ = 3/2 (8)F (Mt) = [M2

t −M2t0]H(Mt−Mt0)(9)

Mt0 = 1/4· (10)

The turbulent pressure fluctuations [3]: Usually the onset of cavitation is defined by a critical pressure.Generally, this threshold value is kept constant and equal to the thermodynamic equilibrium saturation pressure.But in turbulent flows, all flow variables become dependent on turbulence and exhibit variations from the meanvalues. Therefore, the critical pressure can be defined as follows:

pturb = 0.39ρk (11) pcrit = psat + pturb/2· (12)

2. Results

The experiments show that the cavitating flow around the NACA 0015 hydrofoil is periodic unsteady with a frequencyof 16 Hz. First, the cavitating flow is simulated with the original single-phase k − ω model of Wilcox. This leadsto steady flow, mainly because turbulent viscosity is overestimated. The re-entrant jet is damped and cavitation

PAMM · Proc. Appl. Math. Mech. 2, 386–387 (2003) / DOI 10.1002/pamm.200310177

cloud separation is not observed. The turbulent viscosity modification results in lower values of k,ω and µt, e.g.the original single-phase k-ω model predicts the maximum turbulent kinetic energy as kmax = 25 m2/s2, whereasthe turbulent viscosity modification with N=1000 predicts it as kmax = 10 m2/s2. Now the cavitating flow isperiodic unsteady. As N is increased the periodicity of the flow becomes apparent and the frequency increases(Fig. 1). Simultaneously the mean vapor volume production Vvap decreases. Although the total cavity lengthremains unchanged, the area of liquid-vapor transition is larger compared with the results obtained by applyingthe original single-phase k-ω model. Euler simulations predict periodic unsteady cavitating flow with instantaneoustotal disappearance of the vapor production. This is not in agreement with experimental observations. The modifiedturbulence models capture this phenomenon more correctly. Two versions of local compressibility correlations aretested, 1. Sarkar’s and 2. Wilcox’s. Both models estimate the turbulent variables lower than the original single-phasek-ω model (e.g. kmax(Sarkar′s) = 8 m2/s2), which leads to more unstable flow dynamics. At the interface the speedof sound decreases. Corresponding to that the flow becomes supersonic and the maximum turbulent Mach numbersare of the order of 0.6. In the last modification cavitation is allowed to begin at pressures higher than the saturationpressure. The maximum increase in the critical pressure due to fluctuations is 30%. This modification initiateshigher transient fluctuations after initiation of the calculation, but finally it leads to a steady state cavity. Thecomparison of the modifications shows that the turbulent viscosity affects the flow dynamics of this test case muchmore than local compressibility or pressure fluctuations. One of the main important aspects of cavitation modelingand of appropriate flow simulations is the interaction of cavitation and turbulence. Understanding of this stronglycoupled phenomenon is necessary to control the periodic unsteady cavitation dynamics, which is most critical withrespect to erosion.

α: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Steady void fraction distribution Instantaneous void fraction distribution Steady void fraction distribution

original single-phase k-ω model turbulent viscosity correlation N=1000 Sarkar’s local compressibility correlation

t [s]

Vva

p[d

m3 ]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

Unsteady vapor volume distribution, modified k-ω model N=1000, periodic, f= 11.62 Hz, Vvap = 0.0768dm3)

Figure 1: Cavitating NACA0015 hydrofoil, αA = 60, turbulent Tuinlet = 0.01, poutlet = 74175Pa, uinlet = 12m/s,σcomp = 1.0, n0 = 108nuc./m3, R0 = 1.62x10−7m, water at T = 293.15K, Grid=134x41

3. References

1 Keller, A., Arndt, R.A.: Hochgeschwindigkeits- Videoaufzeichnungen von Kavitation an einem NACA 0015 Profil.Versuchsanstalt fur Wasserbau und Wasserwirtschaft, TU Munchen, Private Communication (1999).

2 Coutier-Delgosha, O., Fortes-Patella, R., Reboud, J.L.: Evaluation of the turbulence model influence on thesimulations of unsteady cavitation. Proc. of ASME FEDSM’01, New Orleans, Lousiana (2001).

3 Athavale, M.M., Li, H.Y., Jiang, Y., Singhal, A.K.: Application of the full cavitation model to pumps and inducers.Proc. of ISROMAC-8, Honolulu (2000).

Prof. Dr.-Ing. Habil. Schnerr G.H., Fachgebiet Stromungsmaschinen University of KarlsruheKaiserstr. 12 D-76128 Karlsruhe, Germany (guenter.schnerr@mach.uni-karlsruhe.de)

Section 12: Multiphase Flows, Flows of Reactive Fluids 387