Towards for Cavitation
6.2 Governing equations
model is developed to solve RANS equations based on local turbulent length scales, such as de-tached eddy simulation (DES) model [132] and the scale-adaptive simulation (SAS) model [101].
Moreover, it has been observed that in the presence of cavitation, the resolved eddy viscosity with current RANS methods will be overestimated and damp the re-entrant jet, which leads to that the unsteady cavitation shedding cannot be simulated. To this end, Reboud et al. [31, 123]
proposed an artificial modification (called hereafter the Reboud correction) on the eddy viscosity for unsteady cavitation. It can dramatically reduce the effective viscosity in the mixture and thus capture the shedding behaviours, but there is still no quantitative validation due to lack of experimental measurements on the turbulent quantities over the past two decades. The RANS methods coupling with Reboud correction have been investigated extensively for cavitation simulations. For instance, Coutier-Delgosha et al. [27] compared the standard and modifiedk–
RNG model, and k–ω model with and without compressibility effects. They demonstrated that the Reboud correction could significantly improve the numerical simulation results and the compressibility effects have to be taken into account. Decaix and Goncalves [34] provided the comparison of a class of RANS simulation based onk–ltransport equation model, including the standardk–l,k–l SST andk–l SAS model. They suggested that the SAS model with Reboud correction could have a good agreement with experiments. However, in the above work, only one component of the bubble velocities is considered in these comparisons; hence the comparison with the velocity of the homogeneous mixture is questionable. Moreover, they mainly focus on the time-averaged velocity and void fraction. The statistical turbulent quantities in cavitating flows, such as the turbulent kinetic energy and Reynolds shear stress, are not well analyzed due to the difficulty of experimental measurements on these quantities in cavitating regime. It is well known that the Reynolds stress is the main uncertain source in RANS simulations. Hence, to understand the discrepancy between RANS simulation and experimental measurements, it is necessary to compare and analyze RANS model-form uncertainty associated with Reynolds stress.
The recent development of fast X-ray imaging in experiments [82] provides a set of reliable data for cavitating flows, including turbulent kinetic energy and Reynolds shear stress. That makes it possible to gain some insights of the cavitation/turbulence interactions and further improve current RANS models. This chapter will dedicate to compare the performance of different RANS models on the cavitating flows and investigate the effects of the Reboud correction. As a result, we propose a modified Reboud correction and validate its performance based on experimental data and numerical tests.
6.2 Governing equations
To simulate cavitating flows, we assume that the two phases of liquid and vapor are strongly coupled and the slip in the phases interface is neglected. Based on that, the two-phase flow is
CHAPTER 6. TOWARDS FOR CAVITATION
governed by one group of RANS equations as:
(6.3)
The mixture density ρ as defined in (6.1) is resolved with the cavitation model associated with local pressure. The Reynolds stress−ρu0iu0j in the momentum equation is estimated by turbulence model. The turbulence models and cavitation model used in the present work will be presented in the following subsections.
6.2.1 Turbulence models
Diverse turbulence models have been proposed over decades. Here, we mainly focus on two advanced turbulence models, namely k–ω SST and SST–SAS. In this subsection, we give a brief introduction on these models.
6.2.1.1 k–ω SST model
Thek–ω SST is proposed by Menter [102] which combine the k–model andk–ω model. In the sub-layer of the boundary layer, the model adopts thek–ω model, while in the free shear layer away from the wall, it transforms to the k–model. Thus, this blend method can keep the respective merits of the two models where they perform the best. Moreover, the model leverages the Bradshaw assumption to ensure that the Reynolds shear stress varies as the turbulent kinetic energy, which can avoid the overestimations of the Reynolds shear stress in the adverse pressure gradient regions. The eddy viscosity is constructed by:
(6.4) µt= a1ρk
max (a1ω, SF2) The transport equation forkand ω is formulated as follows:
∂(ρk)
6.2. GOVERNING EQUATIONS
The limiter 10β∗kω on the turbulent kinetic energy (TKE) production is recommended by Menter [103]. The blend functionF1 is defined as:
(6.7) F1= tanh
The parameters in the model are blended from the k–ω model and k–as:
(6.9) φ=φ1F1+φ2(1−F1),
where φ1 stands for the constant with subscript 1 whileφ2 is the constant with subscript 2.
The constants in the model are given as:
(6.10)
The scale-adaptive simulation (SAS) model is proposed by Menter and Egorov [101]. They introduced the von Karman length-scale into the turbulent scale equation. Thus, the SAS model can dynamically self-adjust to resolve the turbulent structure with unsteady Reynolds averaged Navier–Stokes equations based on the von Karman length scale. The approach can provide a LES-like behavior in the detached flow regions as DES model, but without explicit grid dependence in the RANS regime. The formulation of the eddy viscosity and the transport equations are same as thek–ω SST model, except that a termQSAS is added in right hand of the special dissipation ω transport equation (6.5b). TheQSAS is defined as:
(6.11) QSAS=ρmax
In the formula above, Lis the modelled length scale and the LvK is the von–Karman length scale which is defined as:
L=√
CHAPTER 6. TOWARDS FOR CAVITATION
The model constants are given as:
(6.13) C= 2, α= 0.8, ζ2 = 3.51, σφ= 2/3.
The added SAS term can be considered as a source term in the ω equation. It can detect the regions of the instabilities based on the local turbulent length scale and adjust the turbulent dissipation to reduce the turbulent viscosity.
6.2.2 Reboud correction
Cavitating flows are usually highly unstable with fluctuations at various scales. The unsteady cavitation typically have periodic behaviours with four different stages in each cycle, as shown in Fig. 6.4. The cavitation first expands along the wall and form the cloud cavitation. Then the downstream cavity detaches from the wall and is driven by the main stream to the wake, eventually breaking up in the zone of high pressure.
sheet expansion
cloud collapse cloud convection
cloud formation
Figure 6.1: a cycle of cavitation behaviour
Cloud cavitation is characterized by a primary large scale instability based on the periodical shedding of the rear part of the cavity. However, the conventional RANS models cannot capture the shedding behaviour, because the eddy viscosity in the cavitation region is overestimated and that will block the re-entrant jet which is main factor to arise the bubble separation. In order to capture the shedding behaviour, Reboud et al. [123] imposed an artificial modification f(ρ) on the original eddy viscosity as a multiplicative correction. The modification can reduce dramatically the eddy viscosity in the cavitation regime based on the vaporization extent. The formulation can be expressed as:
(6.14) f(ρ) =ρv+ (1−β)n(ρl−ρv).
The plot of f(ρ) is shown in Fig. 6.2. When the void fraction is equal to zero,f(ρ) equals the liquid density, as in the original model. Oppositely, when the void fraction decreases down to 1, f(ρ) will be the vapor density. For intermediate values of void fraction, the function reduces significantly the original modelled eddy viscosity once there occurs the cavitation.
Different investigations have been carried out and demonstrate the success of this modification for different geometries [31, 75, 155]. With this modification, the periodical behavior can be correctly reproduced, usually with the right frequency.
114
6.2. GOVERNING EQUATIONS
0 200 400 600 800 1000
0 200 400 600 800 1000
f()
baseline Reboud correction (n=10) Reboud correction (n=20)
Figure 6.2: Plot of the Reboud correction function
6.2.3 Phase model
Regarding the phase model, we take the homogeneous assumption to treat the cavitation region as a mixture. Further, we apply the barotropic state law to describe the phase change due to its robustness. However, the barotropic model has some drawbacks. For instance, it is assumed that the cavitation phenomenon is isothermal, and thermodynamic effects are neglected. On the other hand, the barotropic law relates the pressure to the void fraction directly, and the void fraction varies immediately with change of pressure, which is not physical since the time of phase change is not took into account. Despite the limitations of barotropic model, a systematical good agreement between the barotropic model and transport equation based model is obtained. [52]
Hence, basically there is no major improvement in using the transport equation model.
In the barotropic state model, the Tait equation and the perfect gas law are utilized for the pure liquid and vapor, respectively, to estimate the relationship between the pressure and the density. The formulations are shown as:
(6.15)
ρl ρref = 7
s P+P0 PrefT +P0, P
ρv =C,
where PrefT andρref is the reference pressure and density, P0 = 3×108. In the mixture interval, the state law is characterized with a sinusoidal transition. The maximum slope is defined by 1/A2min, whereAmin=∂P/∂ρ. The plots are shown in Fig. 6.3 with two different Amin.
CHAPTER 6. TOWARDS FOR CAVITATION
Perfect gas law
(pure vapor) Mixture Tait equation (pure liquid)
p
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pv
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Cmin= 1.5m/s
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Pressure (bar) Density(kg/m3 )<latexit sha1_base64="nboRoLQa6XrAJSSEVET8JFKZxDk=">AAACAHicbVA9SwNBEN2LXzF+RS0sbBaDEJt4p4KWQS0sI5gPSGLY20ySJXt7x+6cGI40/hUbC0Vs/Rl2/hs3H4UmPhh4vDfDzDw/ksKg6347qYXFpeWV9GpmbX1jcyu7vVMxYaw5lHkoQ13zmQEpFJRRoIRapIEFvoSq378a+dUH0EaE6g4HETQD1lWiIzhDK7Wyew2ER0yuQRmBgyHN97vHwf3pUSubcwvuGHSeeFOSI1OUWtmvRjvkcQAKuWTG1D03wmbCNAouYZhpxAYixvusC3VLFQvANJPxA0N6aJU27YTalkI6Vn9PJCwwZhD4tjNg2DOz3kj8z6vH2LloJkJFMYLik0WdWFIM6SgN2hYaOMqBJYxrYW+lvMc042gzy9gQvNmX50nlpOC5Be/2LFe8nMaRJvvkgOSJR85JkdyQEikTTobkmbySN+fJeXHenY9Ja8qZzuySP3A+fwD7p5X6</latexit><latexit sha1_base64="nboRoLQa6XrAJSSEVET8JFKZxDk=">AAACAHicbVA9SwNBEN2LXzF+RS0sbBaDEJt4p4KWQS0sI5gPSGLY20ySJXt7x+6cGI40/hUbC0Vs/Rl2/hs3H4UmPhh4vDfDzDw/ksKg6347qYXFpeWV9GpmbX1jcyu7vVMxYaw5lHkoQ13zmQEpFJRRoIRapIEFvoSq378a+dUH0EaE6g4HETQD1lWiIzhDK7Wyew2ER0yuQRmBgyHN97vHwf3pUSubcwvuGHSeeFOSI1OUWtmvRjvkcQAKuWTG1D03wmbCNAouYZhpxAYixvusC3VLFQvANJPxA0N6aJU27YTalkI6Vn9PJCwwZhD4tjNg2DOz3kj8z6vH2LloJkJFMYLik0WdWFIM6SgN2hYaOMqBJYxrYW+lvMc042gzy9gQvNmX50nlpOC5Be/2LFe8nMaRJvvkgOSJR85JkdyQEikTTobkmbySN+fJeXHenY9Ja8qZzuySP3A+fwD7p5X6</latexit><latexit sha1_base64="nboRoLQa6XrAJSSEVET8JFKZxDk=">AAACAHicbVA9SwNBEN2LXzF+RS0sbBaDEJt4p4KWQS0sI5gPSGLY20ySJXt7x+6cGI40/hUbC0Vs/Rl2/hs3H4UmPhh4vDfDzDw/ksKg6347qYXFpeWV9GpmbX1jcyu7vVMxYaw5lHkoQ13zmQEpFJRRoIRapIEFvoSq378a+dUH0EaE6g4HETQD1lWiIzhDK7Wyew2ER0yuQRmBgyHN97vHwf3pUSubcwvuGHSeeFOSI1OUWtmvRjvkcQAKuWTG1D03wmbCNAouYZhpxAYixvusC3VLFQvANJPxA0N6aJU27YTalkI6Vn9PJCwwZhD4tjNg2DOz3kj8z6vH2LloJkJFMYLik0WdWFIM6SgN2hYaOMqBJYxrYW+lvMc042gzy9gQvNmX50nlpOC5Be/2LFe8nMaRJvvkgOSJR85JkdyQEikTTobkmbySN+fJeXHenY9Ja8qZzuySP3A+fwD7p5X6</latexit><latexit sha1_base64="nboRoLQa6XrAJSSEVET8JFKZxDk=">AAACAHicbVA9SwNBEN2LXzF+RS0sbBaDEJt4p4KWQS0sI5gPSGLY20ySJXt7x+6cGI40/hUbC0Vs/Rl2/hs3H4UmPhh4vDfDzDw/ksKg6347qYXFpeWV9GpmbX1jcyu7vVMxYaw5lHkoQ13zmQEpFJRRoIRapIEFvoSq378a+dUH0EaE6g4HETQD1lWiIzhDK7Wyew2ER0yuQRmBgyHN97vHwf3pUSubcwvuGHSeeFOSI1OUWtmvRjvkcQAKuWTG1D03wmbCNAouYZhpxAYixvusC3VLFQvANJPxA0N6aJU27YTalkI6Vn9PJCwwZhD4tjNg2DOz3kj8z6vH2LloJkJFMYLik0WdWFIM6SgN2hYaOMqBJYxrYW+lvMc042gzy9gQvNmX50nlpOC5Be/2LFe8nMaRJvvkgOSJR85JkdyQEikTTobkmbySN+fJeXHenY9Ja8qZzuySP3A+fwD7p5X6</latexit>
Figure 6.3: Plot of barotropic state law for the mixture