On the Turbulence Modelling for an Air Cavity Interface

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1 Introduction On the Turbulence Modelling for an Air Cavity Interface Gem Rotte, Maarten Kerkvliet, and Tom van Terwisga TU Delft, Delft/The Netherlands MARIN, Wageningen/The Netherlands G.M.Rotte@tudelft.

1 1 Introduction On the Turbulence Modelling for an Air Cavity Interface Gem Rotte, Maarten Kerkvliet, and Tom van Terwisga TU Delft, Delft/The Netherlands MARIN, Wageningen/The Netherlands The use of air lubrication techniques can significantly reduce a ship s fuel consumption. One of the most promising techniques applicable to ships is the external air cavity technique. An external cavity is created beneath the ship s hull with the help of a cavitator, which is located directly upstream of an air injection point (figure 1a). The cavitator is extended in the span-wise direction and typically has a rectangular or triangular cross section. It is used to separate the mean water flow from the hull, thereby providing a stable air layer. Air cavity applications are claimed to lead to propulsive power reduction percentages of 10-20% by reducing the ship s frictional drag (Gorbachev and Amromin (2014), Zverkhovskyi (2014)). However, a complete understanding of the influence that the ship s hull form has on the relevant twophase flow physics and thereby also on the length and stability of the air cavities is still lacking. This inability to predict the air cavity characteristics hampers the application of air cavity techniques in the shipping industry. Multiphase CFD methods can help us to gain a better understanding of the relevant physics. Fig. 1: The external air cavity technique (a) and two possible mechanisms for air discharge: re-entrant jet (b) and wave pinch-off (c) The largest challenge in predicting the air cavity characteristics lies in correctly modelling their closure region (Zverkhovskyi et al. (2015), Shiri et al. (2012)). Here, the closure region is defined as the region where the separated air-water flow transforms into a more dispersed flow. Both the re-entrant jet mechanism and the wave pinch-off mechanism are cited as possible mechanisms for air discharge from the cavity in the closure region (e.g. Shiri et al. (2012), Zverkhovskyi (2014), Mäkiharju (2012)). The reentrant jet mechanism is provisionally assumed to be similar to the re-entrant jet mechanism responsible for the break-up of natural sheet cavities. Callenaere et al. (2001) describe the mechanism as follows. In the closure region of a sheet cavity, a region with high adverse pressure gradient is formed. This increase in local pressure forces a thin stream of liquid into the cavity. This thin stream is called the re-entrant jet and it is illustrated in figure 1b. Impingement of this jet with the gas-liquid interface results in a disturbance leading to a portion of the attached cavity being pinched off and transported by the mean flow. The wave pinch-off mechanism, as illustrated in figure 1c, is governed by waves on the air-water interface. These waves are believed to be formed by turbulence structures interfering with the interface. If the resulting wave amplitudes are of the same magnitude as the cavity thickness, the free surface interface will hit the ship s bottom and a pocket of air will be shed from the cavity. This article aims to link the physical modelling of the relevant phenomena to their numerical modelling, with an emphasis on the modelling of the re-entrant jet mechanism with a RaNS turbulence model. The article is based on the available literature in the public domain and on knowledge gained from research projects carried out at Delft University of Technology and at the Maritime Research Institute Netherlands (MARIN).

$It is complemented with turbulence models, cavitation models and volume-fraction transport equations for different phases. The equations are discretised using a finite-volume approach.$

2 The numerical solver used for all simulations is ReFRESCO (MARIN (2017)). It is a viscous-flow CFD code that solves multiphase (unsteady) incompressible flows using the RaNS equations. It is complemented with turbulence models, cavitation models and volume-fraction transport equations for different phases. The equations are discretised using a finite-volume approach. Time integration is performed implicitly with first or second-order backward schemes. The implementation is face-based, which facilitates grids with elements consisting of an arbitrary number of faces (hexahedrals, tetrahedrals, prisms, pyramids, etc.) and, if needed, h-refinement (hanging nodes). For turbulence modelling, RaNS/URaNS, SAS (Scale Adaptive Simulation), DES (Detached Eddy Simulation), PANS (Partially Averaged Navier-Stokes) and LES (Large Eddy Simulation) approaches can be used. 2 Base case calculations The (2D) base case calculations are based on the measurements performed in the Delft Cavitation Tunnel by Zverkhovskyi (2014). The cavitation tunnel has a test section length of 2m and a m crosssectional area. The length of the computational domain upstream of the cavity is chosen such that the boundary layer profile in de simulations approaches the one from the measurements as close as possible. A schematic of the domain including the applied boundary conditions is shown in figure 2. The close-up shows the grid topology in the region including the cavity, with refinement zones in the region around the cavitator, the cavity closure and the expected cavity interface. Close to the wall grid clustering was applied to ensure y+ values well below 1. Unsteady simulations were carried out with a water tunnel inlet velocity of 1 m/s. Based on current experimental observations in the Delft Cavitation tunnel, this is the free-stream velocity at which the re-entrant jet mechanism is dominant over the wave pinch-off mechanism. All computations were carried out using the k kl (KSKL) turbulence model as described by Menter et al. (2006). A second order upwind scheme is used for the time discretisation and the time step used in the calculations was 0.001s to ensure Courant numbers in the order of 1. The discritisation of both the momentum and turbulence equations was done using the QUICK scheme and the volume fraction transport equation was discretised using a Fromm scheme with a SUPERBEE limiter. Fig. 2: Computational domain and boundary conditions Zverkhovskyi et al. (2015) found that prescribing the air flux at the air inflow did not yet give physical solutions. Figure 3 shows the instantaneous cavity profiles for simulations with an air inlet pressure of 40, 45, 50, 55 and 60 Pa respectively. The profiles are plotted using the contour where the air volume fraction equals 0.5. Based on linear theory and the experiments from Zverkhovskyi (2014) a cavity length 0.32 m and a cavity thickness of around 0.01 m is expected. The pressure at the air inlet for the base case calculation is chosen such that the the cavity profile from the simulations matches the one from the experiment as close as possible, which resulted in an air inlet pressure of 50 Pa. Fig. 3: Cavity profile sensitivity to pressure at the air inflow boundary

3 The base case simulation did not show correct cavity closure behaviour. No break-up of the air cavity and thus no shedding was observed. A thin layer of air is constantly present at the closure and downstream of the cavity, influencing the computed wall shear stress. Figure 4 shows that the overall computed velocity profiles compare reasonably well with the measured profiles. However, the computed upstream profile already differs from the experimental one. This could be the reason for the differences in the middle and downstream of the cavity. The simulated profile is somewhat thicker and seems more turbulent. As for the profile at the middle of the cavity, the simulations predict higher velocities outside the cavity. This seems a reasonable solution due to the narrowing of the tunnel as a result of the presence of the cavity. Unfortunately the results of the experimental measurements close to the wall are of insufficient resolution to quantitatively compare the viscous sub-layer of the boundary layer, which determines the wall shear stress. Fig. 4: Time-averaged boundary layer profiles at three different streamwise locations: before in the middle and after the cavity. The 2-phase base case simulations from the current project and the experiments from Zverkhovskyi (2014) are shown 3 Eddy-viscosity correction For the numerical modelling of the re-entrant jet in the air cavity cases, it was investigated to see whether RaNS methods can be used, since these are the most interesting option for application in practical ship design. For the simulation of unsteady natural sheet cavity dynamics, and in particular the re-entrant jet mechanism occurring in those cases, RaNS methods can already be applied. However, the commonly used two-equation turbulence models, such as the k-ɛ and k-ω models, were originally developed for RaNS simulations of single-phase flows. A frequently encountered problem with these turbulence models that are based on the Boussinesq assumption is the over-prediction of the eddy-viscosity around the liquid-gas interface and thus in and downstream of the cavity closure region. According to Reboud et al. (1998), this over-predicted eddy-viscosity reduces the development of the re-entrant jet and can thus prevent the occurrence of shedding. The above mentioned tendency is not just associated with one specific RaNS solver or cavitation model. There are several additional correction models that artificially lower the over-predicted eddy-viscosity, of which the Reboud-correction is just one. Here, the formulation for the turbulence viscosity is modified. The originally computed turbulence viscosity µ t is multiplied by a correction factor based on a function of the local density f (ρ). Computations were carried out using two different correction functions: the function as described in Coutier-Delgosha et al. (2001) and Reboud et al. (1998) and a correction based on a Gaussian function which is proposed here. The same numerical settings as given in section 2 are applied. The function as described by Coutier-Delgosha et al. (2001) is given in Eq. (1). Here, ρ g is the density of the gas, ρ l is the density of the liquid, ρ is the local cell density and n influences the amount of reduction. For natural cavitation problems, n usually has a value of 10. The eddy viscosity correction factor is obtained by dividing f (ρ) by the local cell density, resulting in a value between 0 and 1. ( ) ρg ρ n f (ρ) = ρ g + (ρ l ρ g ) (1) ρ g ρ l

4 The correction based on a Gaussian curve is proposed in Eq. (2). Here, A and W are the amplitude and width of the peak and have a value between 0 and 1, and p controls the order and thus the width of the peak. Figure 5 shows the different correction functions for different values of n for cases of Reboud correction and different values of A, W and p for Gaussian correction cases. It can be observed that the Reboud correction function is asymmetrical and has a bias towards the lower density fluid. It also has large gradients when the mixture approaches pure air or pure water, which can possibly introduce difficulties when numerically solving the problem. For the proposed Gaussian functions the correction only is active when the cell is marked as an interface cell and the function is symmetric. ( ρ 1 2 (3ρ l ρg) ) 2 g(ρ) = Ae 2(W(ρ l ρg)/10) p (2) Fig. 5: Eddy-viscosity correction factor as a function of the local density ρ. The computed eddy viscosity is multiplied by this correction factor. Figure 6 shows the computed cavity profiles for three representative cases: the base case, one case with Reboud correction and one case with Gaussian correction. Most probably, the asymmetric nature of the Reboud correction function is the reason that the cavity profiles as computed including this function are very unstable. This instability is present for all n > 1, thus for all cases were the correction function is applied. For all simulations with n > 1, the cavity grows to its stable length which is similar to to profile of the base case simulation and then starts to get unstable. The air volume fraction inside the cavity is equal to or larger than 0.99, but due to the bias of the Reboud correction function towards the low density fluid, all eddy-viscosity ν t inside the cavity is reduced such that there is no eddy-viscosity at all. This results in large difference between the viscosity inside and outside the cavity, causing instabilities on the air-water interface. The total viscosity difference between inside and outside the cavity is in the order of ten times lower for the base case when compared to the the Reboud cases. The total eddy-viscosity outside the cavity is rather high, due to the developed boundary layer upstream of the cavity (Re x > 10 6 ). An exploratory sensitivity study confirmed that the onset of these instabilities is also the case for smaller time steps and with finer grids around the cavity. In contrast to the unstable behaviour which is found for the simulations including the Reboud correction function, all three simulations as carried out using the Gaussian correction produced very stable cavity profiles (figure 6). All three correction functions with an amplitude of 1 which are shown in figure 5 were applied and produced more or less the same result as found for the base case calculation, showing incorrect closure behaviour. This lead to the conclusion that the reduction of the eddy-viscosity on the air-water interface is probably not the only requirement for the re-entrant jet to form and the production of eddy viscosity at the cavity interface as well as inside the cavity have a large influence on the interface stability. Figure 7 shows the regions where the correction function was active

5 for both the Reboud function and Gaussian function. The Reboud function is active in a large region inside the cavity as well. whereas the Gaussian correction function only acts on the cavity interface. For the boundary layer profiles, similar profiles were found for all correction cases. Only the simulations including the Reboud correction did show some irregular behaviour behind the cavity closure, most probably due to passing pockets of air-water mixture bubbles. Fig. 6: Cavity profiles for different eddy viscosity correction functions Fig. 7: Regions where the eddy viscosity correction function was active; 1: no reduction, 0: maximum reduction 4 Discussion Most of this article focuses on the numerical simulation of the re-entrant jet mechanism for ventilated air cavity flows with a RaNS model. Solely correcting the eddy-viscosity on the air-water interface appears to be insufficient to initiate the re-entrant jet shedding mechanism. However, the computed eddy-viscosity at the cavity interface as well as inside the cavity have a large influence on the interface stability. In contrast to the RaNS modelling of natural cavitation problems, there is a developed boundary layer upstream of the cavity in the ventilated cavity cases. The eddy-viscosity produced in this region is transported with the flow, causing a thick band of viscous flow around the cavity. This could also hamper the formation of the re-entrant jet in the simulations. This hypothesis is to be strengthened by a simulation with a very short no-slip wall upstream of the cavity, thus almost no turbulent boundary layer formation. Maybe a re-entrant jet can then exist (simulation is running now). Another difference with respect to natural sheet cavitation is the lack of a condensation term. There is no destruction of gas at the cavity closure. Since the re-entrant jet mechanism is a 3D phenomenon, simulations also have to be carried out in 3D, this will most probably be added to the final version of this article. In cavitation problems you need very high convergence levels and small time steps. An exploratory study of the influence of the time step size did not show any differences for smaller timesteps. Also, the RaNS modelling of natural cavitation problems usually ask for a low-order and thus diffuse discretisation scheme for the volume fraction transport equation. Here a higher order scheme is used, since it is hypothesised that sharp interfaces are required in order to compute the cavity surface correctly, especially when one is also interested in the wave pinch-off mechanism and the total air flux leaving the cavity.

6 5 Conclusions and further work To conclude, the simulated velocity profiles at different streamwise locations in the boundary layer around the cavity compare reasonably well with the experimental profiles. However, RaNS turbulence models show serious deficiencies for the modelling of the re-entrant jet mechanism for air cavities. Also not after including an ad-hoc eddy-viscosity correction model as proposed by Reboud and in this paper. Maybe a Reynolds Stress Model (RSM) could help, since the isotropic turbulence modelling as used in classical 2-eq. RaNS turbulence models is a violation of the anisotropic nature of the turbulence structures near the interface. The Reynolds stresses across the interface should be close to zero as no significant momentum flux should cross the interface. Next to the modelling of the re-entrant jet, RaNS methods are especially not suitable when one also wants to include the simulation of the wave pinch-off mechanism. This mechanism is hypothesised to be governed by waves formed by turbulence structures disturbing the interface. These structures are not solved using a RaNS method and one would need a scale resolving or hybrid model such LES, RaNS- LES or a PANS method. The use of DES requires caution, since the cavities are located inside a boundary layer. In these regions a regular DES is still in its RaNS-mode, but e.g. a iddes could be an option. From a practical point of view however, these scale resolving models are not believed to be viable for practical ship design purposes yet, due to the significant amount of added computational time. These models can however first be used to identify the physical conditions in the closure region and to study and identify scaling influences. Thereafter, one could develop a RaNS model for the numerical modelling of air cavitites. The dispersion (or diffusion) of the gas phase can then for example be modelled by an extra convection-diffusion term to break up the air layer. This extra term can then be activated when certain conditions characterised by the physics in the closure region are met. Further work will first focus on a performance study for different RaNS and hybrid RaNS-LES turbulence models around air-water interfaces using a benchmark case such as a Kelvin-Helmholtz stability analysis in the vicinity of a wall with a known disturbance. Thereafter hybrid RaNS-LES,LES or PANS simulations could be carried out to study the physical mechanisms responsible for air discharge in the closure region. Acknowledgements We thank the Netherlands Organisation for Scientific Research NWO, by whom this research is financially supported. References Callenaere, M., Franc, J.-P., Michel, J.-M. and Riondet, M. (2001). The cavitation instability induced by the development of a re-entrant jet. J. Fluid Mech 444, Coutier-Delgosha, O., Fortess-Patella, R. and Reboud, J.L. (2003). Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation. J. of Fluids Eng. 125(1), Gorbachev, Y. and Amromin, E. (2012). Ship Drag Reduction by Hull Ventilation from Laval to Near Future: Challenges and Successes. ATMA 2012 Mäkiharju, S.A. (2012). The Dynamics of Ventilated Partial Cavities over a Wide Range of Reynolds Numbers ans Quantitative 2D X-Ray Densitometry for Multiphase Flow. Ph.D. Thesis The University of Michigan ReFRESCO (2017) Online; accessed 23 March 2017 Menter, F.R., Egorov, Y. and Rusch, D. (2006). Steady and unsteady flow modelling using the k kl model. Ichmt Digital Library Online, Begel House Inc. Reboud, J., Stutz, B. and Coutier, O. (1998). Two-phase Flow Structure of Cavitation: Experiment and modelling of Unsteady Effects. Third International Symposium on Cavitation, Grenoble, France Shiri, A., Leer-Andersen, M.,Bensow, R.E. and Norrby, J. (2012). Hydrodynamics of a Displacement Air Cavity Ship. 29 th Symposium on Naval Hydrodynamics, Gothenburg, Sweden Zverkhovskyi, O. (2014). Ship Drag Reduction by Air Cavities. Ph.D. Thesis, TU Delft Zverkhovskyi, O., Kerkvliet, M., Vaz, G. and van Terwisga, T. (2015). Numerical Study on Air Cavity Flows. Proceedings of NuTTS 2015, Cortona, Italy

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