Universit`a degli Studi di Udine
Dottorato di Ricerca in
Tecnologie Chimiche ed Energetiche
PREDICTIONS OF NON-CAVITATING AND
CAVITATING FLOW ON HYDROFOILS AND
MARINE PROPELLERS BY CFD AND
ADVANCED MODEL CALIBRATION
Dott. Mitja MORGUT
COMMISSIONE
Dr. Francesco SALVATORE Revisore
Prof. Leopold ˇSKERGET Revisore
Prof. Tommaso ASTARITA Commissario
Author’s e-mail: mmorgut@units.it
Author’s address:
DICAR, Dipartimento di Ingegneria Civile e Architettura Universit`a degli Studi di Trieste
Abstract
Sommario
Acknowledgments
Contents
Introduction 1I
Thesis background
3
1 Cavitation 5 1.1 Introduction . . . 5 1.1.1 Definition of cavitation . . . 5 1.1.2 Types of cavitation . . . 71.1.3 The cavitation number . . . 9
1.1.4 Cavitation effects and related history . . . 9
1.2 Sheet cavitation . . . 11
1.2.1 Two-dimensional flow . . . 11
1.2.2 Three dimensional effects . . . 13
1.2.3 Nuclei content and roughness . . . 14
1.3 CFD methods for cavitating flow . . . 15
1.3.1 Interface-Tracking methods . . . 15
1.3.2 Continuum modelling methods . . . 15
2 Mathematical model 19 2.1 Inhomogeneous model . . . 19
2.2 Homogeneous model . . . 20
2.2.1 Incompressible cavitating flow . . . 21
2.3 Incompressible single phase flow . . . 22
2.4 Turbulence modelling . . . 22
2.4.1 RANS approach . . . 23
2.5 Mass transfer models . . . 24
2.5.1 Zwart model . . . 24
2.5.2 Full Cavitation Model . . . 25
2.5.3 Kunz model . . . 25
3 Numerical Method 27 3.1 Domain discretization . . . 27
3.2 Equation discretization . . . 27
ii Contents
II
Two-dimensional hydrofoils
31
4 Calibration of mass
transfer models 33
4.1 Introduction . . . 33
4.2 Description of the calibration strategy . . . 34
4.2.1 The idea and the logic of the calibration process . . . 34
4.2.2 Reference test case . . . 35
4.2.3 Solution strategy . . . 36
4.3 Preliminary study (fully wetted flow) . . . 37
4.4 Application of the calibration strategy . . . 39
4.5 Closure . . . 43
5 Assessment of the calibrated mass transfer models 45 5.1 Evaluation of the calibrated mass transfer models . . . 45
5.1.1 NACA66(MOD) at AoA=6° . . . 45
5.1.2 NACA0009 . . . 47
5.2 Influence of the turbulence model . . . 51
5.2.1 NACA66(MOD) at AoA=4° . . . 51
5.2.2 NACA0009 . . . 51
5.3 Closure . . . 56
III
Model scale propellers
57
6 CFD strategy for the prediction of the flow around marine model scale propellers 59 6.1 Description of the CFD strategy . . . 596.2 Nomenclature and conventions . . . 62
7 P5168 propeller 65 7.1 Introduction . . . 65
7.2 Numerical setup and meshing . . . 66
7.3 Fully wetted flow . . . 69
7.3.1 Propeller performances . . . 69
7.3.2 Flow Field . . . 71
7.4 Closure . . . 76
8 E779A propeller 77 8.1 Introduction . . . 77
8.1.1 Numerical setup and meshing . . . 78
8.2 Fully wetted flow . . . 81
8.3 Cavitating flow . . . 86
Contents iii
9 PPTC propeller 95
9.1 Introduction . . . 95
9.2 Numerical setup and meshing . . . 97
9.3 Fully Wetted Flow . . . 99
9.3.1 Propeller performances . . . 99 9.3.2 Flow field . . . 100 9.4 Cavitating Flow . . . 105 9.5 Closure . . . 109 Conclusions 111 A Turbulence modelling 115 A.1 Introduction . . . 115
A.2 Eddy viscosity models . . . 115
A.2.1 Classification . . . 116
A.2.2 Two equation models . . . 117
A.3 Reynolds stress turbulence models . . . 121
Introduction
Cavitation is a complex multiphase phenomenon that consists in the formation and ac-tivity of bubbles inside a liquid medium. It happens when the liquid pressure is decreased below a certain threshold value. In flowing liquids it can appear due to the variations in local flow values (pressure, velocity) induced by the geometry of the system. Depending on the quality of the liquid, geometry configuration and roughness of the solid boundaries it can take different forms.
In the specific case of the marine propellers, cavitation is, in general, an undesirable phe-nomenon since it is usually associated with negative design implications such as noise, vibration, surface erosion and performance loss (thrust breakdown).
In this Thesis a stable and accurate CFD (Computational Fluid Dynamics) strategy for the prediction of non-cavitating and cavitating flow around marine model scale pro-pellers working in uniform inflow was developed, and validated considering three different model scale propellers.
As a matter of fact, owing to the continuous improvements of CFD technologies and steady increase of computer performances, numerical simulations are nowadays exten-sively used for design purposes, allowing the experimental tests to be performed only at the final stages of the project. However, a successful CFD procedure is influenced by many factors, such as: CAD geometry, topology and dimension of the computational domain, meshing strategy, and physical modelling.
In this work special attention was paid to meshing and physical modelling.
As far as mesh generation is concerned, hexa-structured and hybrid-unstructured (tetra-hedral + prisms on solid surfaces) meshing approaches were compared, for the following reasons. Generally speaking, the simulations carried out on hybrid-unstructured meshes are less accurate than those performed with hexa-structured ones. Nevertheless, the hybrid-unstructured meshing approach is automatic while the hexa-structured approach in not automatic and in the case of complex geometries (for instance marine propellers) can require a significant amount of work in order to generate a superior quality grid. Thus, we verified, mainly for the prediction of the propulsive performances, if hexa-structured meshes can guarantee such better performances (in terms of accuracy and convergence) than hybrid-unstructured ones, in order to justify the greater effort re-quired for their generation. The study revealed that for the prediction of the propeller performances hybrid-unstructured and hexa-structured meshes can guarantee similar lev-els of accuracy. However, hybrid-unstructured meshes, being more diffusive, show to be less suited for the detailed investigations of the flow field.
cavitat-2 Contents
ing flow regimes the homogeneous (one-fluid) model was used. In the homogeneous model three widespread mass transfer models were employed in order to take into account the mass transfer rate due to cavitation. The considered models, commonly to some others available in literature, share the common feature of employing empirical coefficients to adjust the mass transfer rate from liquid to vapour and vice versa. Unfortunately, these empirical coefficients do not have a general character, and their values can affect both the accuracy and stability of the numerical simulations. Thus, in order to compare the differ-ent mass transfer models fairly and congrudiffer-ently, the empirical coefficidiffer-ents of the differdiffer-ent models were properly tuned by means of an advanced calibration strategy. The models were calibrated on the basis of the two-dimensional stable sheet cavity flow around a hydrofoil, and the calibration process was driven by the modeFRONTIER optimization system. The calibrated mass transfer models were then further evaluated for the predic-tion of stable-sheet cavity flow around hydrofoils. The numerical results obtained with the different calibrated mass transfer models were in line with each other and, in general, compared well with the available experimental data, even though the cavity extent was under-estimated in the case of more severe cavitating flow regimes.
The calibrated mass transfer models were applied to the numerical predictions of cavi-tating flow around marine model scale propellers. Also in this case the three different calibrated mass transfer models ensured similar levels of accuracy. The numerical results compared well with the experimental data, but considering more severe operational con-ditions, the propeller thrust breakdown was not properly predicted.
For both hydrofoils and marine model scale propellers, the influence of the turbulence model was evaluated, revealing in particular that, for the prediction of the propeller performances the two-equation SST (Shear Stress Transport) turbulence model and the computationally more expensive BSL-RSM (Baseline - Reynolds Stress Model) turbu-lence model, capable of taking into account the anisotropic turbuturbu-lence field, can ensure similar levels of accuracy.
In the following, the Thesis is organized in three parts.
In the first one, the cavitation phenomenon is presented with focus on sheet cavity flow. A brief overview of different CFD methods for cavitating flow predictions is given.The mathematical models and the numerical methods used in this work to simulate non-cavitating and non-cavitating flow are presented.
In the second part, the calibration strategy developed for the tuning of the mass transfer models is discussed, first, and then the calibrated mass transfer models are evaluated for the prediction of stable sheet cavity flow around two-dimensional hydrofoils.
In the third part, the numerical simulations carried out considering three different model scale propellers are discussed.
I
1
Cavitation
An introduction to the cavitation phenomenon is provided first. Then a detailed de-scription of a particular type of cavitation frequently observed on hydrofoils and marine propellers is given. At the end a brief overview of the CFD cavitation modelling is provided.
1.1
Introduction
In this section the main physical aspects of cavitation are discussed. Usual types of cav-itation in flowing liquids are shown and briefly described. The non-dimensional number used to identify different cavitating flow conditions is formulated. The consequences of cavitation and the interesting historical aspects related to the cavitation appearance are briefly discussed.
1.1.1
Definition of cavitation
Cavitation is a complex multiphase phenomenon and as pointed out by Knapp et al. [48] it is not easy to give a concise definition of cavitation and at the same time convey much significant information about it. For this reason three different definition of cavitation are provided in the following. As a matter of fact cavitation can be defined as, for instance: • the vaporization of a liquid when the static pressure decreases below its vapour
pressure [19].
• the formation1 and activity of bubbles (or cavities) in a liquid [97]. • the breakdown of a liquid medium under very low pressures [37].
Considering the above definitions it is possible to state firstly that, cavitation implies a phase change from liquid to vapour. As a matter of fact, cavitation appears, similar to boiling with the difference that the (phase change) driving mechanism is not a tempera-ture change but a pressure change.
6 1. Cavitation
In this respect Fig. 1.1 shows qualitatively how in the case of boiling starting from point A, evaporation can be induced by crossing the vapour pressure saturation curve, PV(T ), at constant pressure. On the other hand in the case of cavitation, cool evaporation can be induced by crossing the saturation curve at almost constant temperature.
In fact, in most cases (and especially for cold water) the thermal effects can be neglected and the cool evaporation can be assumed practically isothermal as showed in Fig 1.1. However in some cases the thermal effects can not be neglected because the thermal ex-change needed to the phase ex-change is such that the phase ex-change occurs at temperature lower than the ambient liquid temperature. This temperature difference called cavitation thermal delay can be significant for instance in cryogenic fluids such as oxigen, nitrogen, hydrogen which are popular fuels for launch vehicles [8, 37, 90].
Figure 1.1: Different driving phase change mechanisms (paths) of boiling and cavitation. The precise value of the pressure threshold at which cavitation initiates depends strongly on the purity of the liquid. More precisely it depends on the amount of the cavitation nuclei present in the liquid and the time that nuclei are exposed to the low pressure. Cavitation nuclei are microbubbles of vapour or gas. They can be suspended in the liquid or may be trapped in tiny cracks either in the liquid boundary surfaces or in the solid particles travelling in the liquid.
These impurities interfere with the uniform attraction that might otherwise exist be-tween liquid molecules and can thus operate as a starting points for a liquid breakdown that leads to cavitation.
A liquid free or with a very low content of nuclei (such is pure water) can even withstand high negative pressure (tension) without undergoing cavitation. Otherwise the tap water (rich of nuclei), used in most of industrial devices, initiates to cavitate at a pressure close to its vapour pressure2.
Finally it is important to clarify that cavitation, (also following the second definition presented hereafter) embraces the entire sequence of events from bubble formation to
1.1. Introduction 7
bubble collapse. In fact, the phase change from liquid back to vapour (condensation) is a very important part of the overall phenomenon because when bubbles collapse, pressure waves of high intensity, which may damage the nearby surfaces, can be emitted.
1.1.2
Types of cavitation
Cavitation can appear in both flowing and static or nearly static liquids.
In the latter case cavitation can occur when an oscillating pressure field disposing of a sufficiently large oscillating amplitude is applied on the liquid. In this case cavitation is termed acoustic and the cavitation pattern is generally composed of one or more bubbles that grow and contract in the sound field. The bubbles can be stable and oscillate for many periods of the sound field, or transient and exist for less than one cycle.
In flowing liquids, examined in the present work, the cavitation appearance is caused by variations in local flow values (pressure, velocity) induced by the geometry of the system (in this work hydrofoil or propeller). In this case cavitation is termed hydrodynamic. Depending on the quality of the liquid (presence of nuclei and solid particles), the pressure field in the cavitation zone, as well as on the geometry configuration and roughness of solid-boundaries, it can take different forms. The different possible cavitation patters can be classified according to [48], [37], [97] in the categories described below, however, it has to be kept in mind that some cavitation patterns can be a combination of several types. Examples of hydrodynamic cavitation over a hydrofoil and propeller are given in Fig. 1.2 and Fig. 1.3, respectively.
• Fixed or Attached or Sheet Cavitation appears as a region of detached flow filled with vapour. Its shape is usually under steady/quasi-steady conditions or changes relatively slowly and/or periodically. If the cavity extents well beyond the body, it is called supercavitation. Fixed cavities typically form downstream steep decreasing pressure gradients such is the leading edge of a body.
• Travelling (Bubble) Cavitation is characterized by cavities or bubbles which form in the liquid and travel with the liquid as they expand and collapse. The bubbles appear in low-pressure regions of the liquid as a result of fast growing of nuclei present in the liquid and then subsequently collapse when they are convected to higher pressure regions. This type of cavitation most commonly occurs on hy-drofoils with a small angle of attack and is less likely to be present in hydraulic machines.
• Cloud Cavitation appears as a mist or cloud of very small bubbles. Frequently it is formed by the periodic break up of the sheet cavity. This type of cavitation usually causes engineering problems such as erosion, noise and vibrations, especially when bubbles collapse near the surfaces.
8 1. Cavitation
Figure 1.2: Examples of cavitation over a hydrofoil: (a) partial leading edge cavitation, (b) supercavitation, (c) travelling bubble cavitation, (d) sheet-cloud cavitation. Figure (a) taken from [74], (b)-(d) taken from [20]
1.1. Introduction 9
1.1.3
The cavitation number
The susceptibility of a flow to cavitate is commonly identified by the non-dimensional governing parameter introduced by Thoma in 1920 known as cavitation number or Thoma number defined as:
σ = PREF − PV(T )
∆P (1.1.1)
where PREF is the reference pressure, PV is the vapour pressure at the operating tem-perature of the fluid T , ∆P is the pressure difference characterizing the system.
As a matter of fact the lower is the value of σ, the larger is the risk of the appearance of a severe cavitation phenomenon.
In general, the Thoma number can be expressed for the hydrofoil case as: σ = PREF − PV(T )
0.5ρLV2
(1.1.2) where ρL and V are the density and free-stream velocity of the given working liquid, respectively.
For the propeller case as:
σn=
PREF − PV(T ) 0.5ρL(nD)2
(1.1.3)
where n and D are the propeller rotational speed and diameter, respectively.
However, the cavitation number itself is not sufficient to uniquely identify a certain cav-itation phenomenon. For the comparison of experiments and/or numerical predictions it is fundamental to guarantee the same geometric arrangement and fluid properties. As a matter of fact, for instance, both surface roughness as well as liquid gas content can significantly affect the cavitation appearance.
1.1.4
Cavitation effects and related history
The interest in cavitation is strictly related to its effects which may be desirable or un-desirable.
Even though cavitation was already recorded by Newton in 1704 and later, in 1754, the Swiss mathematician Euler guessed the conditions that might cause the appearance of cavitation in rotating flow machinery and its effects on their performances, the intensive study of cavitation phenomena did not start earlier than at the end of the nineteenth century.
10 1. Cavitation
developed on the blades which caused the so called thrust breakdown.
In the case of Turbinia however, Parson was able to overcome the problem of cavitation by enlarging the blade area, increasing the number of propellers per shaft from one (orig-inal design) to three mounted in tandem. With the new arrangement of propellers the vessel was able to absorb all the power at the correct shaft speed and was also able to reach the remarkable trial speed of 32.75 knots.3
In order to fully understand the reasons of his correct decision, Parson in 1895 con-structed an enclosed circulating channel and investigated the nature of cavitation on propeller models. It was the first cavitation tunnel ever made which enabled Parson to test 2-in diameter propellers.
In the same period, the term cavitation derived by the Latin word “cavus,-a,-um” (En-glish. hollow), suggested by Froud to identify the phenomenon of appearance of voids and clouds of bubbles (or cavities) appeared in literature in the work of Barnaby & Thornycroft [4], for the first time.
Since Parson a lot of cavitation facilities have been constructed in various locations of the world in order to investigate the nature of different cavitation phenomena as well as of their related effects. Some of the best know facilities are for instance at: MARIN (Wa-geningen, the Netherlands), INSEAN (Rome, Italy), HSVA ( Hamburg, Germany), SSPA (Gothenburg, Sweden), NSWCCD(a.k.a. the David Taylor Model basin, W.-Bethesda, Memphis, Tennessee, USA), etc.
The negative effects of the cavitation such as the thrust breakdown in marine propellers or the decrease in both power output and efficiency in hydraulic turbines can be also accompanied by excessive levels of noise, vibration and surface erosion.
The surface erosion and production of noise are related to the bubble collapse because during collapse a large shock wave (thousands of Pascal) can be emitted, see Fig. 1.4 (left). Moreover, if the bubble collapses near a solid surface a micro-jet perpendicular to the surface can form, see Fig. 1.4 (right).
Figure 1.4: Erosive mechanisms formed during bubble collapse. (left) pressure waves from bubble collapse, (right) microjet formation close to surface. Adapted from [11]. The microjet reaches very high velocities (order of 100 m/s) in the final stages of the
1.2. Sheet cavitation 11
collapse and can thus induce an overpressure on the surface having the same order of magnitude as that produced by the shock wave. For this reason, shock waves and micro-jet are both possible hydrodynamic mechanisms of cavitation erosion.
If one of the fluctuating forces induced by the phenomenon matches a natural frequency of a portion of equipment vibrations can occur, as for instance on turbine and propeller blades.
For the sake of completeness it is important to point out that cavitation, can have also positive effects. As a matter of fact it can be used to concentrate energy on small sur-faces or to reduce the viscous drag. The energy concentration can be used for ultrasonic materials cleaning or rock-cutting. The drag reduction can be applied, for instance, to super-cavitating projectiles, torpedos, etc.
1.2
Sheet cavitation
In this section the sheet cavitation is further described. Considering the two-dimensional flow the main physical mechanism related to the leading edge and closure cavity region are discussed first. Then the three dimensional effects as well as the effects of the nuclei content and roughness are introduced.
1.2.1
Two-dimensional flow
Leading edge regionIn the past, several experimental investigation have been carried out in order to under-stand the mechanism allowing for the cavity attachment and consequent appearance of attached (sheet) cavitation.
In particular, Arakeri [1], Arakeri & Acosta [2] verified, and later Franc & Michel [36] and Tassin & Ceccio [88] confirmed that attached cavitation can occur if the laminar boundary layer separation is present in the non-cavitating (fully-wetted) flow conditions. In such a flow condition, the attached cavitation can be generated by the small bubbles starting to grow in the re-attachment region.
Later, Farhat & Avellan [33] and Farhat et al. [34] observed that the presence of the laminar boundary layer separation in the non cavitating flow condition is not strictly necessary for the appearance of the attached cavity. They observed that in the absence of the laminar separation (in fully wetted flow conditions) attached cavity can appear consequently to a complex interaction between the travelling bubbles and the boundary layer. They observed that the attached cavity can forms as soon as the flow separates upstream to the bubbles forming from the nuclei exploding in the low pressure region at the leading edge.
In fact, for the appearance of the attached cavitation the boundary layer has to separate just upstream to the cavity. The separation may be already present in the fully-wetted flow conditions or induced by the changes in pressure distribution generated by the de-veloping bubbles. As a matter of fact, if the boundary layer does not separate, the cavity is not sheltered from the incoming flow and can not attach to the wall.
12 1. Cavitation
Figure 1.5: Sketch of sheet cavitation (right) and detail of the leading edge region (left).
quasi stable) bubble, as qualitatively depicted in Fig. 1.5.
Considering the flow situation sketched in Fig. 1.5 it is worth to clarify that in the absence of the diffusion of dissolved gasses, the pressure in the sheet cavity is constant and very close to the liquid vapour pressure, while the pressure upstream of the cavity detachment (point C in Fig. 1.5) is lower. This adverse pressure gradient forces the boundary layer to separate, in point S, just upstream of the cavity detachment. Due to the fluid viscosity circulation can occur in the small region between the flow separation and cavity detachment.
Cavity closure region
Since the minimum pressure occurs inside the cavity itself, at the cavity closure region the surrounding streamlines tend to be directed towards the cavity. In fact in most cases, as depicted in Fig. 1.6, in the cavity closure region the flow splits into two parts, i.e. the re-entrant jet flow (red line) and the outer flow (green line). Both parts are separated by a streamline which, ideally ends to a stagnation point.
Figure 1.6: Characterization of the cavity closure region
The re-entrant jet is a thin stream of liquid that is forced into the cavity by the adverse pressure gradient present in the cavity closure region. If the jet has enough momentum it can travel, with the same velocity as the free-stream flow velocity, along the surface underneath the vapour sheet, and impinge the liquid/vapour interface near the cavity detachment. As a consequence a large portion of the cavity is swept (or separated or shed) and transported by the main flow in downstream direction. The advected structure is termed cloud cavitation as it quickly turns into a frothy structure that moves at a velocity smaller than the mean free stream velocity and has a concentrated vorticity region at its centre [51].
1.2. Sheet cavitation 13
vapour cloud shedding a new cavity develops and a new re-entrant jet is formed. The cloud cavitation generally occurs in attached (partial) cavities of medium length. This because the shedding of the large vapour structures may occur when the thickness of the cavity is significantly larger than the re-entrant jet thickness and the adverse pressure gradient at the cavity closure region is sufficiently high to force the re-entrant jet to travel towards the leading edge of the cavity. (See [37], [10].) As a matter of fact the large vortex structures are not shed neither by long cavities, having small pressure gradient in cavity closure region nor by short cavities with small thickness.
1.2.2
Three dimensional effects
In the previous discussion the flow was assumed to have a two dimensional character. However, even on two dimensional bodies the cavitating flow was often observed to be three-dimensional.
The three dimensionality of the flow is related to the behaviour of the re-entrant jet which is reflected at the cavity closure line. This statement was put forward by de Lange & de Bruin [24] (See also [23]) who assuming that the pressure gradient is zero along the closure line, predicted that the tangential component along the closure line should remain constant. Thus, as shown in 1.7, when the closure line is not perpendicular to the flow the re-entrant jet is directed sideways and thus contribute to the formation of the three-dimensional flow field.
Figure 1.7: Behaviour of the re-entrant jet. (a) cavity closure line perpendicular to the free-stream velocity, (b) cavity closure line inclined. V is the incoming flow velocity, Vjet is the re-entrant jet velocity. Figure adapted from [37]
Later Dang & Kuiper [22] and Dang [21] using a twisted foil with a span wise varying angle of attack found that the direction of the re-entrant jet is strongly influenced by the cavity topology. Laberteaux & Ceccio [55] show the strong influence of the leading edge sweep of a hydrofoil on both the topology of the cavity and the direction of the re-entrant jet. Dular et al. [28] confirmed the hypothesis for the asymmetric cloud separation proposed by de Lange & de Bruin [24] and Duttweiler & Brennen [30] but pointing out that the flow deviation is not as pronounced as suggested.
14 1. Cavitation
1.2.3
Nuclei content and roughness
The fluid nuclei content and the wall roughness of a given device are important because they can influence not only the extent but also the structure of the cavity.
Fig. 1.8-1.9 qualitatively show how the water nuclei content and the wall roughness can impact on the appearance of the sheet cavitation on a model scale propeller. In fact the fluid with a higher nuclei content is more susceptible to cavitate, and the wall roughness trigger cavitation. The application of a certain level of roughness can be very useful in experimental investigations of the model scale propeller. As a matter of fact in the case of the full-scale propeller the boundary layer is considered to be fully turbulent, except for a very small region close to the leading edge, [11]. On the other hand at the model scale the boundary layer can be laminar on a considerable region of the blade. Since the increase of the Reynolds number generally does not move transition to the leading edge the solution is to roughen the model scale leading edge. In this manner the transition is close to the leading edge and the cavitation pattern observed at the model scale represents the possible full scale phenomenon much better.
Figure 1.8: Effect of the nuclei content. Lower nuclei content (left), higher nuclei content (right). Figure reconstructed from [11]
.
Figure 1.9: Effect of the leading edge surface roughness. Lower roughness (left), higher roughness (right). Figure reconstructed from [11]
1.3. CFD methods for cavitating flow 15
1.3
CFD methods for cavitating flow
In this section a brief overview of the CFD methods for the cavitating flow predictions is provided.
The computational fluid dynamic methods based on the Euler or Navier-Stokes equa-tions can be roughly classified in two main categories: the interface tracking (or fitting) methods and the continuum modelling methods (or interface capturing methods).
1.3.1
Interface-Tracking methods
In the Interface-Tracking methods only the physical equations of the liquid phase are resolved. The vapour phase is not considered and the pressure in the cavity is assumed constant and equal to the vapour pressure of the corresponding liquid.
The cavity interface can be described either by deforming the computational grid or by tracking the marker particles that define it. The cavity interface is by definition clear and distinct (sharp and without any thickness). For this reason it has to be tracked in conjuction with a cavitation closure model (wake model) in order to handle the cavity closure region where the distinction between liquid and vapour is not so clear.
First approaches were introduced by Deshpande et al. [26], and Chen & Heister [12] using Euler and Navier-Stokes equation, respectively, for the solution of two dimensional cavitating flow. Then, excellent works attempting to refine this modelling approach have been carried out amongst others by Deshpande et al. [27], Hirschi et al. [42] and more recently by Liu et al. [59].
This approach proved to be suitable only for the solution of simple problems where the cavity was described as a well defined volume of gas. In particular it proved to work well for the two-dimensional stable sheet cavitation.
1.3.2
Continuum modelling methods
The continuum modelling methods treat the flow as two phases with an average mixture density which continuously varies between the liquid and vapour extremes. They make no attempt to track the liquid/gas interface which is obtained directly from the flow calculations.
The continuum modelling methods, for cavitating flow, can be classified in two main categories: the inhomogeneous (or full two-fluid) models and homogeneous (or single fluid) models.
Inhomogeneous model
In the inhomogeneous model, phases coexist at every point in the flow field. Each phase is separately modelled and thus each phase is governed by its own set of conservation laws. (See [47, 68, 96]).
16 1. Cavitation
exchange, thermal transfer and even surface tension. However, some quantities such as exchange rates and viscous friction between phases have to be known a priori. Such quan-tities are usually very difficult to obtain whether experimentally or otherwise. Moreover this method (in combination with large meshes) can be computationally expensive. The total number of equations to be solved can be doubled compared to the single phase flow. For this reasons in the last decades many authors preferred to focus on the homogeneous model.
Homogeneous or one-fluid model
In the homogeneous model the cavitating flow can be modelled as a mixture of two or in some cases three species (liquid, vapour and in some cases non condensable gas) be-having as single one. The flow is modelled by solving a single set of mass, momentum and eventually energy equations and the variable density field can be generated using, for instance, one of the methods reviewed in the following.
Formulated equation of state: Some of the developed methods employ the complete set of governing equations and use a formulated equation of state to evaluate the mixture density. Such a type of methods have been developed amongst others by Ventikos & Tzabiras [92], Edwards & Franklin [31], Goncalves & Fortes Patella [39], Koop et al. [50], Koop [49], Schmidt et al. [76], Schmidt & Schnerr [77].
Barotropic law : Since most of cavitating flows can be assumed isothermal, arbitrary barotropic equations have been proposed to supplement the energy consideration. De-lannoy and Kueny [25] were the first to introduce a sinusoidal barotropic law relating pressure to density. Later, using this type of law several interesting works have been carried out amongst others by Rebound & Delannoy [72], Song & He [87], Delgosha et al . [16–19], Barre et al. [5]. The cavitation modelling based on the barotropic laws is attractive because it can be integrated into a single phase CFD code without a much effort. However it fails to capture some fundamental fluid physics such as the vorticity production which is an important aspect of cavitating flows, especially in the closure region as shown experimentally by Gopalan & Katz [40].
Transport equation: These models employ an additional transport equation where the mass transfer rate between phases is regulated with appropriate source terms.
The first attempt to model the phase change through an additional transport equation was made by Chen and Heister [13, 14]. They used density as dependent variable in transport equation. Density transport equation was added to the system of the Navier-Stokes equations. The cavity growth and collapse was governed by pressure differences introducing the pressure history concept.
Later authors mostly proposed models based on the volume or mass transport equation for one specie (water or vapour). The models proposed by the different authors differed principally in the formulations used for the source terms.
1.3. CFD methods for cavitating flow 17
phases. Senocak [80] and Senocak & Shyy [81, 82] proposed a model apparently free of empiricism by replacing the source terms proposed by Kunz [52] with the explicit calcu-lations for the interfacial velocity terms.
Also the source terms proposed by Sauer [74], Sauer and Schnerr [75], derived from the Rayleigh-Plesset equation for bubble dynamics do not rely on empirical support. How-ever uncertainty may exist on the values such as the bubble number density and initial bubble radius present in this type of source terms.
Finally, some author proposed models based on the Rayleigh-Plesset equation where empirical coefficients were additionally included in order to force different phase change rates of vaporisation and condensation, respectively. (See for instance Zwart et al. [99], Singhal et al. [84, 85]).
2
Mathematical model
The inhomogenous flow model is briefly described. The homogeneous model for laminar flow is derived, and the incompressible single phase flow Navier-Stokes equations are given. The Reynolds averaging is applied on both homogeneous model conservation equations and Navier-Stokes equations in order to derive suitable models for practical turbulent flow problems.
2.1
Inhomogeneous model
The inhomogeneous (or Euler-Euler or two-fluid) model is formulated considering each phase separately. Thus, the model is based on averaged equations of mass, momentum and energy expressed for each kth phase of the flow [47, 68, 96].
Averaged equations (omitting the energy equation, for convenience) can be obtained by volume (or ensemble) averaging the local instantaneous conservation equations yielding to:
∂(αkρk)
∂t + ∇ · (αkρkuk) = Γk (2.1.1)
∂(αkρkuk)
∂t + ∇ · (αkρkukuk) = −αk∇pk+ ∇ · (αkτk) + Γkuk+ Sk+ Fk (2.1.2) In the above mass and momentum conservation equations ρk, uk, pk represent the aver-aged density, velocity and pressure of kth phase. αkis the phasic volume fraction defined as:
αk= Vk
V (2.1.3)
where Vk is the volume filled by the kth phase within a certain region of volume V . Γk represents the volumetric mass transfer rate of the kth phase due to the interphase mass transfer process such as cavitation.
Sk represents volumetric momentum sources acting on kth phase due to the interphase momentum transfer processes such as drag and lift.
Fk represents body forces acting on kth phase. τk is the viscous stress tensor which can be defined using the Stoke’s law as:
20 2. Mathematical model
The same pressure field p = pk is shared by all the phases [46, 96].
Since the governing equations are expressed for each kth phase of the flow this model is capable to predict the dynamic and non-equilibrium interactions between phases. As a matter of fact it can be very useful to carry out analysis of the transient phenomena, wave propagation and of the flow regime changes [47].
On the other hand if the phases are strongly coupled (such is often assumed in cavitation) and/or the interest is focussed on the effect of the mixture on the system rather than the local behaviour of each phase, simpler models can be successfully applied.
Note that, for turbulent flow the above equations required a second averaging procedure. This second averaging leads to the appearance of additional turbulent stresses in the momentum equation, which require a further modelling.
2.2
Homogeneous model
The homogeneous (or one-fluid) model is a limiting case of the inhomogeneous model where all the phases are assumed to share the same pressure as well as velocity field. This model is often suitable for cavitating flow predictions.
This model is based on the phasic continuity equation, here expressed as: ∂(αkρk)
∂t + ∇ · (αkρku) = Γk (2.2.1)
and the bulk momentum equation, obtained by adding all the phase conservation equa-tions together:
∂(ρmu)
∂t + ∇ · (ρmuu) = −∇p + ∇ · τm+ Fm (2.2.2)
Fmrepresents the mixture body forces, and the mixture stress tensor is evaluated as:
τm= µm[∇u − (∇u)T] (2.2.3)
The mixture density and dynamic viscosity are given by:
ρm= Nk X k=1 αkρk (2.2.4) µm= Nk X k=1 αkµk (2.2.5)
where Nk is the number of phases.
2.2. Homogeneous model 21
For implementation purposes it is recommended to replace one of the phasic continu-ity equations with the following volume continucontinu-ity equation
Nk X k=1 1 ρk ∂ ∂t(αkρk) + ∇ · (αkρku) = Nk X k=1 1 ρk Γk (2.2.6)
which is derived in that manner. Take Eq. 2.2.1 divide by phasic density and sum over all phases.
2.2.1
Incompressible cavitating flow
In the specific case of the cavitating flow, if both liquid ρL and vapour ρV densities are assumed constant the volume and liquid continuity equations simplified. Thus, the cavitating flow can be modelled according to:
∇ · u = ˙m 1 ρL − 1 ρV (2.2.7) ∂(ρmu) ∂t + ∇ · (ρmuu) = −∇p + ∇ · τm+ Fm (2.2.8) ∂γ ∂t + ∇ · (γu) = ˙ m ρL (2.2.9)
where ˙m depending on the pressure field, can be the mass transfer rate from vapour to liquid or from liquid to vapour. This source term, for instance, can be modelled using one of the mass transfer models described in section 2.5.
γ is the liquid volume fraction which is related to the vapour volume fraction α through the volume fraction constraint:
γ + α = 1 (2.2.10)
Thus considering Eq. 2.2.10, Eq. 2.2.4 and Eq. 2.2.5
ρm= γρL+ (1 − γ)ρV (2.2.11)
µm= γµL+ (1 − γ)µV (2.2.12)
22 2. Mathematical model
2.3
Incompressible single phase flow
The incompressible isothermal single phase flow can be modelled using the following set of simplified Navier Stokes equations.
∇ · u = 0 (2.3.1)
∂(ρu)
∂t + ∇ · (ρuu) = −∇p + ∇ · τ + F (2.3.2)
In the above equations F represents additional body forces and τ is the viscous stress tensor. The well known Reynolds Averaged Navier Stokes equation used in practical turbulent flow are given in section 2.4, for convenience.
2.4
Turbulence modelling
In this section a brief overview of the different approaches to turbulence is provided, first. Then the Reynolds Avereged Navier Equation for the incompressible single phase flow and incompressible homogeneous mixture flow are derived.
The most accurate approach to turbulence is that of solving directly the Navier-Stokes equations (or the multiphase flow equations, see. [68, 89]) without employing any sort of averaging. Such an approach termed Direct Numerical Simulation (DNS) is applicable only to simple geometries and flows at low Reynolds numbers (Re), because the required spatial resolution (number of grid points) scales as Re9/4and the corresponding comput-ing time as Re3[7, 96].
Thus, a certain level of turbulence approximation have to be adopted in order to simulate practical engineering turbulent flow problems.
Considering the level of approximation introduced, it is possible to distinguish between Large-Eddy Simulation (LES), Detached Eddy Simulation (DES), Scale Adaptive Simu-lation (SAS), Reynolds Averaged Navier Stokes (RANS) approaches.
The LES is an intermediate approach between DNS and RANS, where the large eddies structures are accurately resolved, while the effect of the small scales is approximate using relatively simple subgrid-scale models. The separation of large and small scales is obtained by filtering the time-dependent Navier Stokes equations in the physical space. Such an approach is justified by the observation that, while the large-medium vortex structures which transport the convective properties are flow dependent, the small scales of the turbulent motion posses a more universal character.
The LES is inherently unsteady and three-dimensional. Thus, even though, LES requires lower spatial resolution (number of nodes) than DNS, it remains computationally very expensive. Consequently, (at the time writing), it can not be routinely applied to com-plex engineering flow problems and/or optimization procedures.
2.4. Turbulence modelling 23
In the averaged equations additional terms appears which are approximate by means of turbulence models.
Compared to LES and DES the RANS approach requires coarser grids and the station-ary mean solution can be assumed (at least for attached or moderately separated flows [7]). It is important to remark that, due to the averaging procedure, the details of the turbulent structures can not be obtained.
The DES is based on hybrid RANS-LES formulation. The RANS approach is applied close to the slip surfaces, inside attached and mildly separated boundary layers, and the LES approach is applied further away from the surfaces, in massively separated regions. This method is often used in practical engineering problems in place of LES, because with a reduced computational requirements it is possible to simulate both high Reynolds number attached boundary layers and resolve large scale turbulent structures.
A similar capability is available in SAS approach where the large grid scale are resolved until the von Karman length scale reaches the grid limit [63, 98]
2.4.1
RANS approach
In order to derive the new sets of conservation equations, suitable for the turbulent flow, the flow variables are decomposed into a mean and fluctuating part according to
φ(x, t) = Φ(x, t) + φ0(x, t) (2.4.1)
where φ0(x, t) is the turbulent fluctuating component and Φ(x, t) is the mean value. Then, the governing equations employing the decomposed variables are time (or en-semble) averaged yielding to the following RANS (Reynolds Avereged Navier Stokes) equations for the incompressible single phase flow and homogeneous mixture flow. Capital letters represent mean quantities in the following equations.
Turbulent single phase flow
∇ · U = 0 (2.4.2)
∂(ρU)
∂t + ∇ · (ρUU) = −∇P + ∇ · τ + ∇ · τR+ SM (2.4.3) Turbulent homogeneous mixture flow
24 2. Mathematical model
As a consequence of the averaging procedure additional Reynolds Stress tensors τR, τRm appear in the single phase flow and mixture momentum equations, respectively. These tensors are composed by the so called Reynolds stresses, −ρu0u0, −ρ
mu0u0. Since these terms can not be represented uniquely in terms of the mean quantities a further modelling is required in order to close the systems of equations.
In order to explicitly resolve the Reynolds stress tensor or to approximate its effect on the mean flow, different turbulence models can be applied.
An exhaustive description of turbulence models used in the course of the present study is given in Appendix A. It is Important to clarify that in this study the same turbulence models developed for the single phase flow were applied to the homogeneous mixture flow.
2.5
Mass transfer models
The mass transfer models describe the inter-phase mass transfer rate due to cavitation. In the last two decades several authors proposed different mass transfer models.
Here only the the mass transfer models employed in the course of the present study are presented, for convenience. The following models, except for the Full Cavitation Model are mainly simplified versions of the original formulations where the contribution of the dissolved gasses was omitted.
2.5.1
Zwart model
The Zwart model is the native ANSYS-CFX mass transfer model. It is based on the the simplified Rayleigh-Plesset equation for bubble dynamics [9]:
˙ m = −Fe 3rnuc(1 − α)ρV RB r 2 3 PV − P ρL if P < PV Fc 3αρV RB r 2 3 P − PV ρL if P > PV (2.5.1)
In the above equations, PV is the vapour pressure, rnuc is the nucleation site volume fraction, RB is the radius of a nucleation site, Fe and Fc are two empirical calibration coefficients for the evaporation and condensation processes, respectively. In ANSYS-CFX the above mentioned coefficients, by default, are set as follows: rnuc= 5.0 × 10−4, RB = 2.0 × 10−6 m, Fe= 50, Fc= 0.01.
2.5. Mass transfer models 25
2.5.2
Full Cavitation Model
The mass transfer model proposed by Singhal et al. [84], originally known as Full Cavi-tation Model (for brevity FCM), is currently employed in some commercial CFD codes, i.e. FLUENT and PUMPLINX. This model is also based on the reduced form of the Rayleigh-Plesset equation for bubble-dynamics, and its formulation states as follows:
˙ m = −Ce √ k T ρLρV r 2 3 PV − P ρL (1 − fV) if P < PV Cc √ k T ρLρL r 2 3 P − PV ρL fv if P > PV (2.5.2)
where fV is the vapour mass fraction, k (m2/s2) is the turbulent kinetic energy, T (N/m) is the surface tension, Ce= 0.02 and Cc = 0.01 are two empirical calibration coefficients. It is important to note, that in this work, for convenience, we did not use the original formulation of the model, but the formulation derived by Huuva [45] in which the vapour mass fraction, fV, is replaced by the vapour volume fraction α.
2.5.3
Kunz model
The Kunz mass transfer model is based on the work by Merkle et al. [64] and currently is one of the mass transfer models implemented in OpenFOAM. In this model, unlike the above mentioned models, the mass transfer is based on two different strategies for creation ˙m+ and destruction ˙m− of liquid. The transformation of liquid to vapour is calculated as being proportional to the amount by which the pressure is below the vapour pressure. The transformation of vapour to liquid, otherwise, is based on a third order polynomial function of volume fraction, γ. The specific mass transfer rate is defined as
˙ m = ˙m++ ˙m−. ˙ m+= CprodρVγ 2(1 − γ) t∞ ˙ m−= CdestρVγ min[0, P − PV] (1/2ρLU∞2)t∞ (2.5.3)
3
Numerical Method
In this chapter the main features of the numerical method adopted by ANSYS-CFX 12 in order to solve the governing equations of a homogeneous multiphase flow model are presented.
3.1
Domain discretization
ANSYS-CFX 12 (for brevity CFX) uses an element-based finite volume method [71, 78]. Thus, the governing equations are discretized on control volumes constructed around each nodal point of the computational mesh (grid), which in the specific case of CFX can consist only of hexahedral, tetrahedral, prismatic and pyramid elements[46]. Fig. 3.1 shows in two dimensions, for convenience, a typical control volume (shaded area) constructed as union of element sectors sharing a particular node. The element sectors are defined by the lines joining the centres of the edges with the centres of the corresponding elements.
Figure 3.1: Example of a control volume constructed on a two-dimensional mesh
3.2
Equation discretization
The systems of the linear algebraic equations that can be solved using a numerical method are obtained discretizing the governing equations on each control volume.
28 3. Numerical Method
equation (2.2.9) are obtained as follows.
In the spirit of the finite volume approach the equations (2.2.7)-(2.2.9) are first expressed in integral form as:
Z V ∇ · u dV = Z V ˙ m 1 ρL − 1 ρV dV (3.2.1) Z V ∂(ρmu) ∂t dV + Z V ∇ · (ρmuu) dV = − Z V ∇p dV + Z V ∇ · τmdV + Z V FmdV (3.2.2) Z V ∂γ ∂t dV + Z V ∇ · (γu) dV = Z V ˙ m ρL dV (3.2.3)
where V represents the volume of a control volume constructed around a certain node. Then, some of the above volume integrals are converted to the surface integrals us-ing the Gauss’ divergence theorem leadus-ing to:
X ip Z Aip u · n dA = Z V ˙ m 1 ρL − 1 ρV dV (3.2.4) Z V ∂(ρmu) ∂t dV + X ip Z Aip (ρmuu)·n dA = − X ip Z Aip p·n dA+X ip Z Aip τm·n dA+ Z V FmdV (3.2.5) Z V ∂γ ∂t dV + X ip Z Aip (γu) · n dA = Z V ˙ m ρL dV (3.2.6)
where n is the outward vector and Aip are the subfaces delimiting the control volume. (In Fig. 3.1 the segments Lip is in analogy with surface Aip)
Finally, the volume and surface integrals are evaluated as follows.
The volume integrals are discretized within each element sector and accumulated to the control volume to which the sector belongs [71, 78].
The surface integrals are evaluated using the mid-point approximation with the integral arguments evaluated at the integration points ip. Since the surface integrals are equal and opposite for control volumes adjacent to the integration points, the surface integrals are guaranteed to be locally conservative.
If the first order Euler scheme is used for the discretization of the transient terms the following set of the linear algebraic equations is obtained.
3.3. Solution strategy 29 ρmV ∆t (u − u o) +X ip (ρmu · A)ipuip = − X ip (p · A)ip+ X ip (τm· A)ip+ FmV (3.2.8) V ∆t(γ − γ o) +X ip (u · A)ipγip = ˙ m ρL V (3.2.9)
where A=nA. ∆t is the time step and the superscriptso represents the old time level. From the above set of algebraic equations it is possible to note that many quantities have to be evaluated at the integration points ip. Since all the solution variables and fluid properties are stored at the nodes (mesh vertices), the integration point quantities are approximated using node values. As a matter of fact the pressure and velocity gra-dients are computed from the nodal values using the finite element shape functions (see. [46]), while the advected variables are evaluated using an upwind-biased discretization. The advection schemes, available in CFX, for the approximation of the advected variable φ can be expressed as:
φip= φup+ β∇φ · r (3.2.10)
where φip is the integration point value, φupis the upwind node value and r is the vector from the upwind node to the integration point.
Depending on the choice of the β value between 0 and 1 it is possible to vary from the first-order bounded but highly diffusive upwind scheme (β=0) to the unbounded second-order upwind scheme (β = 1).
Moreover, a bounded High Resolution scheme is also available. It uses a special nonlinear recipe for β which makes the value of β as close to 1 as possible without introducing new extrema and preventing the occurring of overshoots and undershoots. This advection scheme was employed in all the simulations performed in the course of this study.
3.3
Solution strategy
The algebraic equations of the homogeneous model i.e. Eq. 3.2.7-3.2.9 and Eq. 2.2.10 (volume fraction constraint) form a coupled system of equations at each nodal point. However, the Eq. 3.2.9 and Eq. 2.2.10 can be decoupled from the pressure-velocity system and treated in a segregated manner.
Using such an approach (employed in this study) the solution strategy proceeds as follows: 1. Assemble and solve Eq. 3.2.7 and Eq. 3.2.8 for pressure and velocity
2. Assemble and solve Eq. 3.2.9 for the liquid volume fraction 3. Solve Eq. 2.2.10 for the vapour volume fraction
30 3. Numerical Method
In the case of the transient simulations this cycle is repeated within each time-step. In the case of the turbulent flow simulations, the turbulence model equations are addition-ally assembled and solved following the solution of the equation for the vapour volume fraction.
II
4
Calibration of mass
transfer models
Three widespread mass transfer models are calibrated using an advance calibration strat-egy developed in this study, for the prediction of the stable sheet cavity flow around a hydrofoil. The details of the calibration strategy are provided and the results of the calibration process are discussed.
4.1
Introduction
In this Thesis the homogeneous (one-fluid) model implemented in ANSYS-CFX (for brevity CFX) was employed in order to simulate the cavitating flow around both hydro-foils and marine propellers.
Referring to previous chapter 2 it is possible to remind that this model treats the cavi-tating flow as a mixture of two fluids behaving as a single one. As a matter of fact the set of the governing equations is composed by the (volume) continuity and momentum equations for mixture, plus a transport equation for the water volume fraction. The mass transfer rate due to cavitation is modelled by the same source term ( ˙m) appearing in the (volume) continuity and volume fraction equations.
In CFX this source term is modelled by the mass transfer model originally proposed by Zwart et al. [99], using the default settings. However, in literature several other mass transfer models can be found. ([38]). In this work we additionally implemented, in CFX, the model originally proposed by Kunz et. al. [53] (hereafter Kunz) and the model originally proposed by Singhal et al. [84] also known as Full Cavitation Model (for brevity FCM). These models share the common feature (see chapter 2) of employing empirical coefficients to adjust the mass transfer rate due to cavitation (evaporation and condensation), and their values can significantly affect both the stability and accuracy of the numerical predictions.
34 4. Calibration of masstransfer models
the best values of the empirical coefficients were determined by optimization algorithms. It is important to point out that the calibration strategy, hereafter described and applied to CFX, could be easily extended to other solvers.
In the following, the description of the calibration strategy is provided first. The re-sults of the calibration process are discussed after presenting the preliminary investiga-tions carried out for the non-cavitating flow regimes. At the end some conclusions are formulated.
4.2
Description of the calibration strategy
4.2.1
The idea and the logic of the calibration process
The calibration strategy was developed starting from the idea of searching the empiri-cal coefficients which minimized the differences between the numeriempiri-cal and experimental pressure distributions on the suction side of a NACA66(MOD) hydrofoil, for three dif-ferent cavitating flow regimes. This idea was applied as follows.
The sum of the differences between the numerical and experimental pressure distribu-tions, on the hydrofoil suction side, for the three different cavitating flow regimes were expressed as objective function f1according to:
f1= X σ N X i=1 |CPi− CPi,Exp| ; σ = 1.00, 0.91, 0.84 (4.2.1)
where CPi, and CPi,Exp were the numerical and experimental values of the pressure
coefficient, taken at N =12 locations on the suction side of the NACA66(MOD) hydrofoil, for three different cavitating flow regimes (σ=1.00, 0.91, 0.84), at an Angle of Attack, AoA=4o. In the preliminary stages, also a quadratic expression, Eq. 4.2.2 was derived.
f2= X σ v u u t N X i=1 (CPi− CPi,Exp) 2 ; σ = 1.00, 0.91, 0.84 (4.2.2)
No significant differences were observed with the results obtained using the linear ex-pression. Thus, in this study, all the three different mass transfer model were calibrated using the objective function f1.
In order to find the couples of the empirical coefficients which minimized the objective function f1,and thus reduced the numerical errors, two different optimization algorithms were run in sequence.
4.2. Description of the calibration strategy 35
used, starting from the three best solutions obtained using MOGA-II. In this case no significant improvements were observed.
Generate initial set of designs by DOE
ANSYS - CFX Solve Cavitating Flows
Compute objective, function f(X1,X2)
Stop Condition Set current design,
couple (X1, X2) Post processing TRUE FALSE modeFRONTIER Optimizer CFD SOLVER
Figure 4.1: Logic of the calibration strategy
For the sake of clarity Fig. 4.1 shows the logic of the calibration strategy, where the optimizer block stays for both the MOGA-II and the Simplex algorithm, and where (X1, X2) stays for the couple of empirical coefficients of a given mass transfer model. In this study, the calibration strategy was easily implemented in the modeFRONTIER 4.2 optimization system. The details of the reference test case and of the numerical setup adopted to perform the simulations with CFX are given in the following.
4.2.2
Reference test case
The NACA66(MOD) hydrofoil used to calibrate the different mass transfer models was designed with a camber ratio of f /c=0.020, a NACA meanline of a=0.8 and a thickness ratio of t/c=0.9, where f was the maximum thickness, t the maximum camber and c = 0.150 m the chord length of the hydrofoil section.
The experimental measurements were carried out in the High-Speed Water Tunnel of the California Institute of Technology [83]. In order to minimize the scale effects, distributed roughness was applied on the hydrofoil model from the leading edge to 1.5 percent of the chord on both upper and lower surfaces.
36 4. Calibration of masstransfer models
for the sake of simplicity, we preferred to use a completely smooth hydrofoil, assuming a fully turbulent boundary layer from the leading edge of the hydrofoil.
4.2.3
Solution strategy
In order to simulate the flow around the NACA66(MOD) hydrofoil a rectangular domain having the shape depicted in Fig. 4.2 was used.
−3.5c 0 5.5c −2.5c 0 2.5c Outlet Top Bottom NACA66(MOD) Inlet
Figure 4.2: Shape of the computational domain for NACA66(MOD)
The simulations were carried out in 2D and in steady-state conditions. The following boundary conditions were applied: on solid surfaces (Top, Bottom, Hydrofoil) the no-slip condition was set. On the Outlet boundary a fixed static pressure, PREF = 202,650 P a was imposed and on the side faces the symmetry condition was enforced. On the Inlet boundary the values of the free-stream velocity components, and turbulence quantities were fixed. Water and vapour volume fractions were set equal to 1 and 0, respectively. In order to match the experimental setup, during the numerical simulations the same Reynolds numbers were used. Since the water kinematic viscosity was assumed νL = 8.92 × 10−7 m2/s the free stream velocity was set to U
∞=12.2 (m/s). Assuming a turbulence intensity of 1%, the turbulent kinetic energy k and the turbulent dissipation rate ε were set equal to k = 0.0223 m2/s2, ε = 0.1837 m2/s3 on the Inlet boundary. The water and vapour density were formally set constant and equal to ρL=997 kg/m3, ρV=0.02308 kg/m3, but it is important to clarify that the maximum water-vapour density ratio was limited to ρL/ρV =1000 in order to ensure solver stability. Different cavitating flow regimes related to the cavitation (or Thoma) number, σ, were defined varying the value of the saturation pressure PV. This because PREF, U∞, ρLwere kept constant and σ was defined as:
σ = PREF − PV (1/2)ρLU∞2
4.3. Preliminary study (fully wetted flow) 37
All the simulations were performed on a hexahedral grid with 58700 nodes (see Fig. 4.3) which proved to give mesh independent results, in the preliminary studies carried out considering the fully wetted flow conditions.
4.3
Preliminary study (fully wetted flow)
Before applying the optimization strategy, the preliminary mesh independence study was carried out considering the fully wetted flow conditions.
The study was performed by monitoring the influence of the three progressively finer meshes on the values of the lift CLand drag CD coefficients and on the minimum value of the pressure coefficient CPmin according to :
CL= FL (1/2)ρU2 ∞S CD= FD (1/2)ρU2 ∞S − CP = PREF − P (1/2)ρU2 ∞ (4.3.1)
where P is the local absolute pressure, S = c · d planar surface with d equal to the span. During the mesh refinement the distances of the first nodes from the solid surfaces was kept unchanged. This permitted to maintain the same numerical model, in particular the wall model, and thus to perform a consistent mesh independence study.
All the meshes were hexa-structured and they were generated using the ANSYS-ICEM CFD meshing tool. The average value of the y+ on the hydrofoil surface was equal to 23. y+ was defined as y+ = µ
τy/ν where µτ = (τw/ρL)1/2 was the friction velocity, y the normal distance from the wall, ν the kinematic viscosity, τw the wall shear stress.
Table 4.1: Influence of the mesh refinement. (Fully wetted flow)
Mesh Nodes −Cpmin CL CD
HS1 30504 1.92 0.650 0.015
HS2 58734 1.75 0.649 0.017
HS3 122380 1.73 0.648 0.017
Exp. 1.76 0.629 0.018
From the results listed in the Table 4.1, it is possible to note that the differences were minimal between the HS2 and HS3 meshes. For this reason the different mass transfer models were calibrated using mesh HS2, depicted in Fig. 4.3.
38 4. Calibration of masstransfer models
4.4. Application of the calibration strategy 39 0.00 0.20 0.40 0.60 0.80 1.00 −0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 X/C −Cp Exp. HS1 HS2 HS3
Figure 4.4: NACA66(MOD), AoA=4°, Re=2×106. Suction side pressure distribution.
4.4
Application of the calibration strategy
After the preliminary investigations carried out considering the fully wetted flow condi-tions the calibration strategy was applied to the three different mass transfer models in order to obtain accurate results also for the cavitating flow regimes.
The empirical coefficients of the three different mass transfer models were tuned within the ranges listed in Table 4.2. Other physical parameters present in the models were kept fixed and equal to the values reviewed in chapter 2. Finally it is important to clarify that for the Kunz model the chord of the hydrofoil was chosen as a length scale, L, according to [45], in order to set the mean flow time scale t∞=L/U∞.
Table 4.2: Ranges set for the calibration of the mass transfer models Model Evaporation coefficient Condensation coefficient
Zwart 30 ≤ Fe≤ 500 5.0 × 10−4≤ Fc≤ 8.0 × 10−2
FCM 0.01 ≤ Ce≤ 1 1.0 × 10−6≤ Cc≤ 1.0 × 10−2
Kunz 100 ≤ Cdest≤ 5000 10 ≤ Cprod≤ 1000
40 4. Calibration of masstransfer models
mass transfer model of CFX, the suggested empirical values have been already found to work well for a wide range of applications.
Table 4.3: Default and calibrated coefficients for the different mass transfer models
Zwart FCM Kunz
Fe Fc Ce Cc Cdest Cprod
Default 50 0.01 0.02 0.01 100 100
Calibrated 300 0.03 0.40 2.3E-04 4100 455
4.4. Application of the calibration strategy 41 0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20 X/C −Cp σ=1.00 Exp. σ=0.91 Exp σ=0.84 Exp. Zwart−tuned Zwart−default
Figure 4.6: NACA66(MOD) at AoA=4°, Re=2×106. Suction side pressure distributions computed using non-calibrated and calibrated Zwart model.
0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20 X/C −Cp σ=1.00 Exp. σ=0.91 Exp σ=0.84 Exp. FCM−tuned FCM−default
42 4. Calibration of masstransfer models 0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20 X/C −Cp σ=1.00 Exp. σ=0.91 Exp σ=0.84 Exp. Kunz−tuned Kunz−default
Figure 4.8: NACA66(MOD) at AoA=4°, Re=2×106. Suction side pressure distributions computed using non-calibrated and calibrated Kunz model.
0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20 X/C −Cp σ=1.00 Exp. σ=0.91 Exp σ=0.84 Exp. Zwart FCM Kunz
4.5. Closure 43
Table 4.4: CL and CDfor NACA66(MOD) at AoA=4°and Re=2×106. Numerical values computed using calibrated mass transfer models
σ CL CD
Exp. Zwart FCM Kunz Exp. Zwart FCM Kunz
1.00 0.645 0.650 0.654 0.654 0.020 0.018 0.017 0.018
0.91 0.670 0.656 0.664 0.666 0.021 0.018 0.018 0.018
0.84 0.699 0.668 0.674 0.673 0.024 0.017 0.021 0.018
Table 4.5: Cavity lengths, Lc/c, for NACA66(MOD) at AoA=4°and Re=2×106. Nu-merical values with calibrated mass transfer models.
σ Exp. Zwart FCM Kunz
1.00 0.20 0.21 0.22 0.21
0.91 0.36 0.33 0.34 0.35
0.84 0.60 0.48 0.47 0.47
From Table 4.4 and Fig.4.9 it is possible to state that the solutions provided by the different calibrated mass transfer models were very close to each other and compared well with the experimental data.
Table 4.5 shows how the numerical cavities were always under-predicted compared to the experimental ones, except for σ=1.00. This probably justify the over-prediction of the numerical lift coefficient for σ=1.00 and its under-prediction for the other two operational conditions.
4.5
Closure
The cavitating flow can be modelled using several approaches. One possibility is that of using the homogeneous (transport equation based) model where the mass transfer rate due to cavitation in modelled by the mass transfer model. In literature there are sev-eral mass transfer models which share the common feature of employing the empirical coefficients to tune the evaporation and condensation processes due to cavitation, which can, affect both the stability and the accuracy of the numerical predictions. Thus, a calibration strategy driven by the modeFRONTIER 4.2 optimization system was devel-oped, for the fair and congruent calibration of different mass transfer models involving empirical coefficients. The strategy was applied to ANSYS-CFX and three widespread mass transfer models were properly calibrated. As a matter of fact the results obtained with the calibrated mass transfer models were very similar to each other and compared well with the experimental data.
5
Assessment of the calibrated
mass transfer models
The calibrated mass transfer models are assessed for the prediction of the stable sheet cavity flow around a hydrofoil. The influence of the turbulence model on the accuracy of the numerical predictions is evaluated. The results obtained with the different mass transfer model are similar to each other but some discrepancies are observed with the available experimental data. Slight differences are observed between the simulations performed using different turbulence models.
5.1
Evaluation of the calibrated mass transfer models
The calibrated mass transfer models were evaluated investigating the stable sheet cavity flow on the NACA66(MOD) hydrofoil at an incidence of 6◦, and on the NACA0009 hydrofoil at an incidence of 2.5◦.
5.1.1
NACA66(MOD) at AoA=6
°
The numerical simulations for the NACA66(MOD) at AoA=6° were carried with the same numerical setup adopted in the calibration process, except for the mesh which was adapted to the higher angle of attack.
The numerical results obtained with the three different calibrated mass transfer models were very close to each other. From Fig. 5.2 it is possible to note that, for σ = 1.48 the differences among the cavities as well as of the flow field details, predicted using the three different mass transfer models were minimal. The same level of congruency was observed also for the other cavitating flow regimes.
