An appraisal of the application of the overset method to the CFD simulation of fluid machines

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Scuola di Ingegneria Industriale e dell’Informazione

Corso di Laurea Magistrale in Ingegneria Meccanica

An Appraisal of the Application of the Overset Method

to the CFD Simulation of Fluid Machines

Relatore: Augusto DELLA TORRE

Tesi di Laurea Magistrale di: Graziano SPIRITO Matr. 899419

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Ringraziamenti

Il primo enorme ringraziamento è rivolto al mio tutor, Augusto Della Torre, che, negli ultimi mesi, ha costantemente trovato il tempo e la pazienza di trasmettermi le Sue preziose conoscenze, fondamentali per la finalizzazione di questo lavoro. Un sincero grazie a tutta la mia famiglia, in particolar modo a mia sorella, mia mamma e mio papà, da sempre le mie certezze di vita. La loro inossidabile pre-senza mi ha regalato rari momenti di gioia e mi ha permesso di superare le difficoltà. Grazie alla mia ragazza, che, per quasi due anni, ha avuto la capacità di sop-portare uno studente di ingegneria meccanica e il dono di rendere più leggere le situazioni più gravose.

Grazie, infine, a tutte le altre persone che mi hanno sempre sostenuto e accompa-gnato in questo percorso di crescita. Innanzitutto i miei amici, i miei compagni di avventura e di barca, ma anche coloro che, per un motivo o per l’altro, sento meno frequentemente, non per questo meno importanti. Senza dimenticare i miei compagni di studi, che nel corso degli anni trascorsi al Politecnico sono diventati sempre più numerosi, e con i quali ho condiviso un’ampia gamma di emozioni, dalle risate all’ansia per gli esami, dalle partite a briscola a chiamata agli sforzi per apprendere i concetti spiegati a lezione.

Inoltre, un doveroso pensiero è dedicato a tutti i medici, infermieri e volontari che stanno combattendo contro l’emergenza sanitaria che sta affliggendo il nostro Paese e il resto del mondo, con la speranza che tutto si risolva nel minor tempo possibile.

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List of Figures

1.1 Notation used for writing conservation laws referred to a generic

control volume . . . 6

1.2 Turbulent velocity fluctuations and eddies length scales . . . 11

1.3 The figure shows the inverse proportionality between spectral energy E(k) and wavenumber k and the level of approximation of modeling methods . . . 13

1.4 Generic CV with cell centroid denoted with P, sharing an internal face with a neighboring cell whose centroid is labeled with N. . . . 16

1.5 OpenFOAM structure . . . 22

1.6 OpenFOAM case directory structure . . . 23

2.1 FSI example of a flow interacting with a skyscraper . . . 26

2.2 An immersed boundary method example of a body moving in a stationary non-deforming Cartesian grid . . . 27

2.3 Typical grids adopted for a proper solution of the boundary layer . 27 2.4 Example of merge procedure of two component meshes . . . 28

2.5 Intermediate steps of the overset approach concerning the assignment of the overset patch and zoneID . . . . 29

2.6 cellTypes visualization of the final mesh and its component meshes, each of them identified by a specific zoneID . . . . 31

2.7 Detailed view of an overlapping zone for the overset technique, as reported by Hadzic [11] . . . 32

2.8 Concavity issues in a surface normal vector test . . . 34

2.9 Intersection vector or ray-casting test . . . 35

2.10 Hole-map test . . . 36

2.11 X-ray method . . . 37

2.12 Example of hole fringe point and outer boundary point in the overset technique. . . 37

2.13 Iteration scheme to solve Equation 2.6 for the computational space of a potential donor element evaluated in point P . . . 39

2.14 Computational space of a candidate donor cell for P . . . 39

2.15 Gradient search for the element that bounds P, starting from cell "a" and terminated in "c". . . 41

2.16 Partition boundaries, or "buckets", associated to a spatial partitioning of a curvilinear grid component. . . 42

2.17 Example of overlap matrix considering four blocks. The overlap of different blocks is identified by x symbol. . . . 44

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2.18 ADT process applied on a 2D case: the domain is bisected along x and y directions in alternating order and the new intervals are stored

in the tree data structured. . . 45

2.19 Iterative procedure representing the core of IHC . . . 47

2.20 Cylinder: dynamicMeshDict for static overset cases . . . . 49

3.1 Cylinder: vortex shedding . . . 53

3.2 Cylinder: vortex shedding of a smooth circular cylinder at different Re numbers . . . 55

3.3 Cylinder: Strouhal number for a smooth circular cylinder . . . 56

3.4 Cylinder: effect of refineMesh command . . . . 58

3.5 Cylinder: coarse and fine cylinder meshes . . . 59

3.6 Cylinder: background mesh . . . 61

3.7 Cylinder: zoom on the "global" overset mesh . . . 62

3.8 Cylinder: zoom on the cylinder with cellTypes . . . . 67

3.9 Cylinder: velocity field comparison between overset and snappy approaches . . . 69

3.10 Cylinder: temporal history of drag and lift coefficients . . . 70

3.11 Cylinder: Single-Sided Spectrum of lift coefficient . . . 71

3.12 Cylinder: Single-Sided Spectrum of lift coefficient . . . 72

4.1 MRL Tidal Turbine: real machine and 2D scheme representation . . 76

4.2 MRL Tidal Turbine: AMI and blades STL files . . . 77

4.3 MRL Tidal Turbine: cellZone visualisation for AMI approach . . . 78

4.4 MRL Tidal Turbine: blade layers for the AMI approach . . . 79

4.5 MRL Tidal Turbine: master and slave patches for AMI approach . . 80

4.6 MRL Tidal Turbine: baffles generation scheme for AMI approach . 80 4.7 MRL Tidal Turbine: AMI approach final mesh . . . 81

4.8 MRL Tidal Turbine: detail of one blade mesh for the overset approach 82 4.9 MRL Tidal Turbine: overset approach final mesh . . . 83

4.10 MRL Tidal Turbine: zoneID visualisation . . . . 85

4.11 MRL Tidal Turbine: cellTypes visualisation . . . . 87

4.12 MRL Tidal Turbine: detail of cell size at the overset patch . . . 88

4.13 MRL Tidal Turbine: velocity field . . . 89

4.14 MRL Tidal Turbine: turbulent viscosity field . . . 89

4.15 MRL Tidal Turbine: time evolution of forces . . . 90

4.16 MRL Tidal Turbine: Cp vs BSR variation . . . . 92

5.1 ICE cylinder: cylinder and valve FreeCAD sketches . . . 96

5.2 ICE cylinder: cylinder and valve STL files . . . 97

5.3 ICE cylinder: zoom on the valve mesh . . . 98

5.4 ICE cylinder: meshes for overset and snappy approaches . . . 98

5.5 ICE cylinder: cylinder-valve zoneIDs . . . . 101

5.6 ICE cylinder: cylinder-valve cellTypes visualisation . . . . 103

5.7 ICE cylinder: cylinder-valve overset patch detail with cellTypes visualisation . . . 104

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5.9 ICE cylinder: diffusivity effect on the valve mesh . . . 105 5.10 ICE cylinder: steady-state mass flow rate evolution until β = 1.2 . . 106 5.11 ICE cylinder: average steady-state mass flow rate trend versus β . . 107 5.12 ICE cylinder: limits of a diffusion-based mesh deformation . . . 108 5.13 ICE cylinder: piston mesh and piston crevice detail . . . 109 5.14 ICE cylinder: Final assembly of cylinder, valve and piston meshes . 110 5.15 ICE cylinder: zoneID assignment with the introduction of the piston111 5.16 ICE cylinder: cellTypes assignment with the introduction of the piston112 5.17 ICE cylinder: evolution of the velocity field during piston motion

and valve opening . . . 113 5.18 ICE cylinder: detail of the initial valve opening . . . 114 5.19 ICE cylinder: trend of main flow quantities with respect to the same

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List of Tables

3.1 Cylinder:cylinderMesh extrusion parameters . . . . 57

3.2 Cylinder: cylinderMesh patches . . . . 59

3.3 Cylinder: Total number of cells for the two cylinder meshes . . . 59

3.4 Cylinder: boundary file of the global mesh . . . . 63

3.5 Cylinder:snappyHexMeshDict layer addition settings . . . . 63

3.6 Cylinder: total number of cells comparison . . . 64

3.7 Cylinder: list of boundary conditions . . . 65

3.8 Cylinder: clock time comparison between snappy and overset ap-proaches . . . 68

3.9 Cylinder: main parameters of interest . . . 72

3.10 Cylinder: average volumetric flow rate with the overset approach . . 73

4.1 MRL Tidal Turbine: boundary patches . . . 78

4.2 MRL Tidal Turbine: layer addition settings for AMI approach . . . 79

4.3 MRL Tidal Turbine: layer addition settings for blades meshes with the overset approach. . . 82

4.4 MRL Tidal Turbine boundary conditions . . . 84

4.5 MRL Tidal Turbine: mean forces for U=1 m/s . . . 91

4.6 MRL Tidal Turbine: mass conservation results for U=1 m/s . . . . 91

4.7 MRL Tidal Turbine: flow velocity and power coefficient with respect to BSR variation . . . . 93

5.1 ICE cylinder: boundary patches of the final cylinder-valve overset mesh . . . 99

5.2 ICE cylinder: main parameters for cylinder and valve meshes . . . . 99

5.3 ICE cylinder: cylinder-valve boundary conditions . . . 100

5.4 ICE cylinder: thermoPhysicalProperties settings . . . . 102

5.5 ICE cylinder: turbulenceProperties settings . . . . 102

5.6 ICE cylinder: list of solvers . . . 103

5.7 ICE cylinder: flow tests mass flow rates results . . . 105

5.8 ICE cylinder: main blockMeshDict parameters for the piston mesh . 109 5.9 ICE cylinder: main numerical results of the intake flow for different piston velocities . . . 114

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Sommario

Il presente elaborato analizza le prestazioni dell’approccio overset, implementato nel codice open-source OpenFOAM, sviluppato per condurre simulazioni di flu-idodinamica computazionale. Lo scopo della tesi è di valutare l’applicabilità del metodo overset per lo studio delle macchine a fluido, sia nel contesto di ricerca che in quello industriale. Tre diversi casi studio, caratterizzati da un crescente livello di complessità, sono stati analizzati e, ad eccezione dell’ultimo, la validazione del metodo overset è stata condotta tramite il confronto con dati sperimentali, oltre che con i risultati ottenuti da altre strategie di simulazione. La natura non conservativa dell’ approccio overset è stata responsabile di una perdita di massa attorno all’ 1% nel caso di un problema comprimibile senza movimento della mesh, mentre quella misurata per i casi incomprimibili, con e senza movimento della mesh, è risultata, rispettivamente, uno e due ordini di grandezza inferiori. In termini di risultati numerici, non ci sono stati vantaggi significativi nella creazione di una mesh apposita per uno specifico componente statico, anche considerando l’incremento del tempo computazionale e i problemi di conservazione della massa derivanti dall’approccio overset. D’altro canto, sempre ragionando su casi statici, la strategia overset potrebbe ancora rivelarsi una soluzione adatta per studi di ottimizzazione del design, grazie alla sua estrema flessibilità nella creazione di mesh specifiche per i componenti, che possono essere sovrapposte in maniera arbitraria. Tuttavia, le vere potenzialità dell’ approccio overset sono sfruttate in caso di ges-tione dinamica della mesh, poichè il movimento di un corpo può essere definito in maniera semplice, senza le complesse strategie di movimentazione della mesh, che coinvolgono deformazione di griglia e cambiamenti topologici, necessari in caso di singola mesh. L’applicazione dell’approccio overset è particolarmente promettente, perchè è l’unico metodo che permette di simulare condizioni di dominio disconnesso, come nel caso di una valvola che si chiude completamente, con distanza nulla dalla sua sede.

Parole chiave: approccio overset, OpenFOAM, conservazione della massa,

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Abstract

The present work analyses the performances of the overset approach, implemented in the open-source code OpenFOAM, developed for conducting computational fluid dynamics simulations. The scope of the thesis is to assess the applicability of the overset method for the study of fluid-machines, in both research and industrial context. Three different case studies, characterized by an increasing level of complexity, were analysed and, except for the last one, the validation of the overset method was conducted by means of the comparison with experimental data, as well as results coming from other simulation strategies. The non-conservative nature of the overset approach was responsible for a deficit of mass around 1% in case of a compressible problem without mesh motion, while the one measured for incompressible cases, with and without mesh motion, was, respectively, one and two orders of magnitude smaller. In terms of numerical results, there were not significant advantages in creating a dedicated mesh for a specific steady component, also considering the increase of computational time and the mass conservation issues that derived from the overset approach. On the other hand, still focusing on static cases, the overset strategy could be a suitable solution for design optimization studies, thanks to its extreme flexibility in creating specific mesh components that can be overlapped in an arbitrary manner. However, the full potentialities of the overset approach are exploited in case of dynamic mesh handling, since body motion can be described in a straightforward way, without the complex mesh motion strategies, involving grid deformation and topological changes, required in case of a single mesh. The application of the overset approach is particularly promising, since it is the only method that allows to simulate conditions of domain disconnection, as in the case of a valve which completely closes, with zero gap from its seat.

Keywords: overset approach, OpenFOAM, mass conservation, dynamic mesh

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Extended abstract

Introduction

Grid generation is a critical and time consum-ing procedure characterizconsum-ing the field of CFD and can become a really challenging task in case of complex geometries and body motion. In most practical applications, more structured meshes are combined together in an unstruc-tured manner, through a multi-block method, to better reconstruct complex regions, while re-taining a relatively high computational efficiency. Nonetheless, the time spent for the mesh gener-ation step could be very expensive and there is no guarantee about the final mesh quality.

In the context of mesh generation strategies, the overset mesh or Chimera grid [1] greatly simplifies the mesh generation process, being a particular type of block structured matrix based on the superposition of two or more independent grids that communicate by means of interpola-tion. This simple idea has immediate benefits in terms of high quality meshes generation and, especially, their eventual dynamic handling in presence of body motion. However, the need of establishing a robust and computationally efficient domain connectivity between the over-lapping meshes is still a challenging aspect under investigation. Another drawback to be consid-ered is the violation of the mass conservation of finite-volume methods, which cannot be guar-anteed by an interpolation-based grid coupling and must be limited by dedicated corrections.

The aim of this thesis was to validate the overset approach when applied to the simulation of fluid-machines, in the OpenFOAM environ-ment, reasoning in terms of its possible future implementation in the industrial context, as well as in the research field. Three different case studies, characterized by an increasing level of complexity, were analysed and results obtained by the overset approach were compared with the ones coming from other simulation strate-gies and also exploiting the experimental liter-ature, whenever possible. Particular attention was given to the evaluation of the mass

conser-vation deficit, accuracy of the results and also to the computational time, since they are all key factors for the assessment of the applicability of the overset method to the common industrial field.

State of the art

The actual methodology, followed by the ma-jority of existing industrial applications, for the simulation of fluid-machines relies on very ef-ficient single mesh strategies, typically based on a multi-block grid to exploit, at the same time, the efficiency of structured mesh solvers and the flexibility in shape reconstruction of unstructured grids. However, approaches based on a single mesh are strongly limited in case of body motion and complex mesh motion strate-gies are introduced: mesh deformation based on diffusion of internal points, addition/removal of layers of cells based on layering techniques, mesh sliding interface for stator-rotor interac-tion, GGI/AMI strategy based on interpolation techniques, automatic mesh generation or other topological changes. Sometimes, these dynamic mesh handling strategies require some simplifi-cations, either geometrical and conceptual, to perform a specific simulation, but it is easy to understand that results are not representing the real conditions of the problem. Moreover, the manual effort to obtain a sufficiently high mesh quality could be significant.

The main issues concerning the dynamic mesh handling are almost eliminated by the ex-ploitation of the overset approach, which, how-ever, is still considered an open topic of research. The main reasons are the continuous search for more robust and efficient domain connectivity algorithms, able to assemble the information coming from disconnected mesh regions, and the development of correction schemes for the minimization of the deficit of mass due to the overset interpolation. Moreover, despite the idea of overset mesh was conceived around the 80’s, its support in OpenFOAM is very recent [2].

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Overset mesh

In the overset grid approach, the computational domain is made of at least two different grids or sub-domains, which are meshed independently, using any meshing utility, and then overlapped with each other in an arbitrary manner. The steady mesh component is called background mesh and is the only component grid which is not characterized by the presence of the

over-set patch, the user-defined interpolation fringe,

used to provide, via interpolation, a boundary condition for the considered component mesh. Two major steps are used to establish mesh to mesh communication:

• Chimera Hole-Cutting: it involves the identification of so-called hole cells, those cells which do not belong to the computa-tional domain and so must not be taken into account in the solution of discretized governing equations.

• Donor search procedure: it’s involved in the interpolation process that allows the establishment of domain connectiv-ity among the disconnected mesh compo-nents. Based on the concept of acceptor and donor cells, solution is interpolated at the overset patch and at the interpo-lation fringe around hole cells, which is automatically created by the overset algo-rithm. After all donors have been deter-mined, the general interpolation formula, linking acceptor cells with the respective donors, has the following form:

φpi=

ND

X

k=1

αwkφDk (1)

expressing the interpolated function value at acceptor node Pi, called φpi, in terms

of the function values at donor points Dk,

named φDk, with αwk the interpolation

weights. In order to increase the efficiency of the interpolation process, it is impor-tant to have a similar cell size, at the overset patch, between the overlapping component meshes, since the accuracy of the transfer of flow variable information is maximum when the geometric proper-ties of donor and receptor elements are similar.

Three different types of cells are so characteriz-ing the overset method: hole/passive cells, in-terpolated cells and calculated/active cells, the only ones for which solution is computed.

A detailed view of the overlapping zone for the overset mesh technique is proposed inFigure I.

Present contribution

The present work contributed to the continuous search for alternative dynamic mesh handling strategies in the field of CFD simulations for fluid-machines, by testing the more recent devel-opments of the overset approach implemented in the free-source OpenFOAM software. Three different case studies, characterized by an in-creasing level of complexity, were built up from scratch and analysed in detail, in order to have a full comprehension of nowadays overset method capabilities, from the mesh generation to the setup and final run of the simulation.

First case study

This preliminary case study analysed the well-known phenomenon of vortex shedding behind a static cylinder, under the hypothesis of incom-pressible and two-dimensional flow. It is a typi-cal example of flow interaction with a bluff body, which was deeply studied during the years to increase knowledge of vortex-induced vibrations structural problems, with many experimental results available.

A laminar flow condition was considered to anal-yse the Von Karman vortex street, considering a Reynolds number equal to 100, calculated with respect to a unitary cylinder diameter; no tur-bulence models were used.

Simulations were performed following two differ-ent strategies, namely the overset and snappy approaches, and a preliminary mesh sensitivity analysis was made, by considering two different levels of refinement at the near-wall region. The parameters of interest are drag and lift force coefficients, vortex shedding frequency and the associated Strouhal number.

A similar near-wall mesh resolution was pre-served among the two different strategies to make a consistent comparison of numerical re-sults. Particular attention was given to the radial increase of cell size of the cylinder compo-nent mesh, which was limited by the exploitation of refineMesh utility, such that it was possible to respect the condition of similar cell size at the overset patch, as well as to keep a sufficient distance from the wall to avoid discontinuity of flow quantities from mesh to mesh. The fine cylinder mesh is shown inFigure II.

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Figure I

Figure II

The mesh quality resulting from the merge of cylinder and background component meshes was clearly higher than the one obtained by the snappy method, which is affected by mesh deformation.

Dealing with a laminar incompressible prob-lem, the setup of the case was straightforward: a unitary flow velocity was imposed at the inlet patch, pressure was fixed to ambient value at the outlet patch and the kinematic viscosity of the fluid was set according to the Re.

Second case study

The second case study dealt with a more com-plex problem,the simulation of the dynamics

of a Momentum Reversal Lift (MRL) tidal tur-bine, under the hypothesis of incompressible two-dimensional flow. The machine is made of three blades with symmetrical profile and exploits drag and lift forces to extract fluid power from blades rotating motion, developing about both their own axis and the machine rotor. When the turbine blade is in the upper part there is only drag contribution, while in the other angu-lar positions there is a combination of lift and drag forces. A k-ωSST turbulence model was selected, to properly solve the blades bound-ary layer (condition of y+< 1), without losing

accuracy in the wake free-stream region, The setup and simulation of this case study was already performed, using an Arbitrary Mesh Interface (AMI) approach, for the final project of the course “Modeling Techniques for Fluid Machines”. The same task was then repeated for the scope of this thesis, hence exploiting the overset method.

The fact that AMI patches cannot overlap led to a necessary decrease of the real blades chord and it was deemed as one of the main causes of some discrepancies between numerical and experi-mental results. Although the overset method does not have problems in terms of overlapping patches, the reduced blades chord was kept, in order to make a consistent comparison between results coming from the two approaches. The rotational speed of the shaft was 100 rpm, twice that of the counter-rotating turbine blades around their own axis.

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Figure III

Figure IV

The advantages of the overset method were immediately evident from the mesh generation process, where it was easy to realise proper meshes for each blade, that were subsequently merged with the steady background mesh, and to assign the motion laws through the

dynam-icMeshDict. On the contrary, the mesh

gen-eration for the AMI approach was much more expensive, due to the contemporary meshing of all blades and the four additional AMI patches (three for the blades and one for the turbine shaft) used to decouple the rotating mesh re-gions from the stationary domain. Moreover, the resulting AMI mesh was of inferior quality

compared to the overset one, as can be seen by

Figure III, representing the whole meshes, and

Figure IV, showing a detail of the mesh of a single blade.

The work was carried out in two steps: first, an operating condition characterized by a flow speed of 1 m/s was considered and simulation results were compared; in the second part, in-stead, it was analysed the effect of the variation of the blade to speed ratio BSR on the power coefficient Cp, trying to reproduce the trend of

the machine experimental curve proposed by M. Berry and G.R. Tabor scientific paper [3]. Obviously, due to the reduced blades chord, an

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underestimation of experimental values was ex-pected.

Third case study

The last case study consisted in a preliminary application of the overset approach to the impor-tant field of internal combustion engines (ICEs) simulations, where the correct handling of multi-body motion is a fundamental aspect for which specific CFD packages were developed during the years. In this perspective, as long as mass conservation issues are minimum, the overset approach should represent a very interesting so-lution to easily handle mesh motion, although it was rarely applied to ICEs simulations.

The overset method was applied on a basic ICE cylinder with flat head and a single valve and the work was divided into two different sub-cases: preliminary steady-state flow tests for different pressure expansion ratios and intake flow due to piston motion during valve opening. Again, a two-dimensional grid was chosen, to lighten the computational burden, but the com-pressible nature of the working fluid (a pure mixture of air) was considered.

A k-ωSST turbulence model was selected. The simplified cylinder and valve geometries were drawn in the FreeCAD environment and exported as STL files to be subsequently meshed via snappyHexMesh utility.

The same component meshes were used for the two sub-cases, such that the last one was simply provided with an additional component mesh for the piston, created by means of blockMesh and

topoSet utilities. The final assembly of cylinder,

valve and piston component meshes is shown in

Figure V.

The first half of the work focused on steady-state flow tests, at the condition of maximum valve lift, for different pressure expansion ratios. The overset approach was compared to a single mesh strategy based on snappyHexMesh utility and particular attention was given to the evalu-ation of the mass deficit, in case of the overset method, to make a preliminary estimation of its possible applicability to the simulation of ICEs.

The second part of the work was, instead, dedicated to the analysis of the intake flow, gen-erated by a moving piston, starting from the critical condition of valve closed.

Unlike previous case studies, the complexity of the dynamic mesh handling characterizing this problem prevented the possibility of developing an alternative strategy to the overset approach. In fact, large body motion was involved and

basic mesh deformation based on diffusion of internal points would easily fail due to mesh deterioration. On the other hand, the extreme flexibility of both mesh generation and multi-ple body motion, proper of the overset method, allowed to easily handle such a problem, pre-serving a sufficiently high mesh quality.

Last but not least, it is important to under-line that the valve was set up to start from a condition of zero gap from the valve seat, to un-derstand if the overset approach could manage a condition of domain disconnection, that cannot be handled by any other meshing strategy.

Figure V

Results and discussion

The first case study, about vortex shedding be-hind a circular cylinder, showed that the per-centage deficit of mass was limited in the order of 10−2and that a cylinder mesh refinement had an overall beneficial effect on results, due to the reduced weight of both mesh and interpolation effects when adopting a fine mesh. The com-parison with the grid based on snappyHexMesh utility showed that a vortex shedding frequency of 0.175 Hz was uniquely captured by both meth-ods and was in good agreement with the typical experimental value of about 0.17 Hz [4]. Compa-rable numerical results were also obtained from the analysis of force coefficients, but the overset approach paid a higher number of employed cells and a more expensive computational time.

The simulation of a MRL Tidal Turbine offered the opportunity to analyse the real ad-vantages of the overset approach: the extreme easiness and flexibility in the creation of optimal component meshes and their successive dynamic

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handling.

Positive outcomes were obtained from the com-parison of the mean calculated forces with re-spect to an inlet flow velocity of 1 m/s: 0.7% of error for the horizontal force component, 8% for the vertical one and just 0.48% for the torque, directly related to the power, the useful effect of the machine. Moreover, the overset deficit of mass for such dynamic incompressible problem was equal to 0.13%, still quite limited for a free stream problem, but it increased of about one order of magnitude from the previous case study because of dynamic mesh motion.

Regarding the sensitivity analysis on non-dimensional parameters, BSR vs Cp, both

nu-merical methods were able to reproduce the trend of the machine experimental curve, as shown inFigure VI. According to the reduced blades chord, simulations underestimated the real power coefficient, but were characterized by similar results for intermediate values of BSR, the region of maximum power extraction, while higher discrepancy was observed for extreme

BSR values. 0 0.2 0.4 0.6 0.8 1 BSR 0 0.1 0.2 0.3 0.4 0.5 Cp Cp vs BSR overset AMI exp Figure VI

The exploitation of the overset approach had also beneficial effects from the computational time point of view, since, in average, it was about two times faster than the AMI method, limited in the time step by the interpolation at the cellZones interfaces.

The preliminary application of the overset method to ICEs immediately evidenced the in-creasing mass conservation issues when a com-pressible problem is considered. In fact, the deficit of mass that was observed in simple steady-state flow states, for increasing expansion ratios, exceeded, in average, the usual maximum value of acceptability for the simulation of an ICE, set to 1%.

On the other hand, very promising results were

observed for the dynamic case, where the full potentialities of the overset approach, combined with a slight modification of the CVW overset interpolation algorithm, allowed to rather easily simulate a complex problem characterized by large body motion and domain disconnection, due to the initial valve closed.

An intermediate phase of the intake flow process is shown inFigure VII, where flow recirculation is visible.

Figure VII

Reasonable results were obtained in terms of both in-cylinder pressure and inlet mass flow rate evolution, for increasing piston velocities. Simulation time was strongly influenced by the efficiency of the interpolation method adopted to transfer information between background and foreground meshes: for example, the adoption of efficient AABBTree method allowed to reduce the computational time by a factor of 10 with respect to the standard method.

Conclusions

This work highlighted the significant potential-ities of the overset approach, exhibiting excep-tional advantages for both mesh generation and complex multi-body motion, advantages that may find a place in a real industrial context, especially if the overset approach proves to be a suitable alternative solution with respect to standard methods based on multiple meshes and mesh-to-mesh interpolation.

The principal drawback of the method relies in the mass conservation, critical for the simulation of closed domain problems, like ICEs, but not so relevant for incompressible cases. Moreover, the overset method could still represent a suitable solution for external aerodynamics applications and for preliminary design optimization studies. For counteracting the strong increase of com-putational time associated to compressible dy-namic cases, it should be always recommended

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a parallelization of simulations and the adoption of more efficient interpolation algorithms. Future works should focus on the test of other

overset interpolation algorithms, or on the mod-ification of the already existing ones, with the aim to minimize mass conservation issues.

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Introduction

Fluid mechanics is the branch of science concerning the study of fluids and their motion. At the beginning, this field of study was limited to aeronautics and astronautics, however, it has rapidly become more and more widespread among other important fields like automotive and infrastructure, where the study and prediction of fluids behaviour play a key role in the design optimization. This effect was mainly driven by the growing importance of CFD simulations in industrial applications. In fact, favored by the continuous increase of computational resources and the development of more efficient mathematical models and simulation strategies, CFD has experienced a strong evolution in the last forty years and now represents a powerful tool for the simulation of different problems (fluid dynamics, combustion, electromagnetics, solid mechanics, etc...) that, if addressed with an experimental approach, would typically result in a much higher expense, considering both time and money invested. The most relevant improvements in the history of CFD mainly focused on the meshing strategy, since it is well known that results are extremely dependent on the mesh quality, not only in terms of number of cells used to replace the continuity of the real space, but also their shape, which has issues related to the numerics. For these reasons, grid generation is a critical and time consuming step in the CFD analysis, which can become a really challenging task for complex geometries, especially in case of body motion.

There are different meshing strategies available, each type of mesh having some advantages and drawbacks with respect to features like structure of the solution matrix, flexibility in shape reconstruction, total number of cells needed and their acceptable level of deformation. As written by Hirsch [5], the simplest kind of mesh is the structured grid, which also allows to obtain a regular structure of the solution matrix; on the other hand, an unstructured grid strategy is characterized by a higher flexibility in shape reconstruction, but the solution matrix is no longer regular. Hence, structured mesh solvers are inherently faster than unstructured ones but, often, a single structured mesh cannot adequately define the geometry of interest. That is why, usually, more structured meshes are combined together in an unstructured manner, following a multi-block method, to better reconstruct complex regions, while maintaining a relatively high computational efficiency. Nonetheless, the time spent for the mesh generation could be huge if the geometry becomes more complex or mesh motion is involved; moreover, there is no assurance that the final mesh would be of sufficient high quality with respect to the analysed problem.

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Scope of the thesis

The present work deals with an alternative approach, called overset mesh technique, that could result in an interesting, practical solution towards the main issues concerning the mesh generation step. This method, in fact, relies on a meshing strategy which greatly simplifies the mesh generation process, by overlapping different grids in an arbitrary manner. This simple idea has immediate benefits in terms of high quality meshes generation and, especially, their eventual dynamic handling in presence of body motion, as will be better explained inChapter 2. Despite these advantages, the need of establishing a robust and computationally efficient domain connectivity between meshes is still a challenging aspect under investigation, which can be threatened by complex multi-body motion.

The aim of this thesis was to validate the overset approach when performing fluid dynamics simulations, reasoning in terms of its possible future implementation in industrial applications. The validation of the overset method was performed by comparing the results of simulations obtained, respectively, with classical and overset methods and by exploiting the experimental literature, whenever possible. In this context, classical or standard methods refer to all the other meshing strategies working on a single mesh domain, which can be either structured, unstructured or created in a multi-block way. In particular, the validation process was conceived to be carried out step by step, starting from simple cases and, then, moving towards more complex case studies, of a more practical interest. Each case study was started from scratch, to have a full comprehension of both mesh generation and setup of the overset method. Since the different component meshes interact with each other through interpolation, mass is not conserved, so the deficit of mass was always checked and deemed to be acceptable or not.

The task was carried out using OpenFOAM (Open Field Operation and Ma-nipulation), which is a free open source CFD software, based on FVM and C++ libraries, that is able to simulate anything from complex fluid flows involving chemi-cal reactions, turbulence and heat transfer, to solid dynamics and electromagnetics. Despite the idea of overset mesh was conceived around the 80’s, its support in OpenFOAM is very recent1_{. In order to keep up with the most important} improve-ments about this approach, an updated version of OpenFOAM was installed and used for running simulations.

Thesis structure

The work is structured into five chapters:

• Chapter 1 introduces the foundations of CFD: Navier-Stokes equations, tur-bulence models, FVM discretization. The main structure of OpenFOAM is also presented.

• Chapter 2opens with an introduction of the overset approach and its novelties with respect to other existing methods; then, it is followed by an in-depth

pre-1_{Boger et al.(2010) [}₂_{] first developed the FoamedOver library to implement the overset}

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sentation of the fundamental algorithms and the main rules and peculiarities in the setup required by this method.

• Chapter 3 deals with the first case study: the simulation of vortex shedding behind a circular cylinder in regime of Von Karman street. It was chosen as first case study because it is a static case, with incompressible, two-dimensional flow assumptions. The overset method was compared with a single mesh strategy based on snappyHexMesh utility and with experimental results found in the literature.

• Chapter 4 introduces the second case study: the simulation of a MRL Tidal Turbine. The assumptions of incompressible, two-dimensional flow are still valid but the problem is now dynamic, due to the rotating turbine blades. The overset mesh was compared with the AMI strategy and with the experimental results of Berry and Tabor’s report [3].

• Chapter 5 deals with the final case study, a preliminary application of the overset method to ICEs. In this case, the compressible nature of the working fluid (a pure mixture of air) was considered. The work was subdivided into two steps: first, it was performed an introductory steady-state analysis about flow tests on a basic ICE cylinder-valve configuration, for different pressure expansion ratios, in the condition of maximum valve opening; after that, an unsteady problem was investigated, which can be described as the intake flow due to the piston motion during valve opening. The last sub-case represents the most relevant application of the overset approach which was tested in this work, since large, multiple body motion is involved; moreover, a very critical condition, that would make all other meshing strategies fail, was rather easily simulated by the overset method: the valve opening starting from the condition of zero gap with the valve seat.

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Chapter 1 Basics of CFD

Computational Fluid Dynamics (CFD) is the numerically-based branch of fluid mechanics which analyses and solves problems involving fluid flows. It is an al-ternative, usually cheaper, solution to the experimental branch which introduces mathematical models and numerical methods to simulate fluid dynamic phenomena, struggling with their complex non-linear properties (turbulence, flow unsteadiness, shock waves and compressibility effects).

There is always a constant communication between experimental and numerical approach, in order to, for example, tune coefficients or install instruments for performing better measurements; as consequence, CFD models are always char-acterized by a certain level of empiricism, coming from real world observations. Both mathematical models and numerical schemes play a key role for the correct description of the flow evolution and all its physical properties, such as velocity, pressure, temperature, density and viscosity.

In this chapter, a brief recall of fluid mechanics governing equations, turbu-lence modeling and finite volume method is proposed. The details about these fundamental topics are voluntarily neglected and can be easily found in any book of computational fluid dynamics. The main reference book is represented by the Versteeg-Malalasekera [6].

A section is also dedicated to a general description of the OpenFOAM structure, according to the OpenFOAM User Guide [7].

1.1 Fluid dynamics governing equations

CFD is based on the formulation of conservation laws. Conservation equations are written according to a Eulerian approach, which, unlike the Lagrangian point of view, does not track the motion of the single fluid particle, but develops balances for a stationary fluid element. Hence, properties are not expressed per unit of mass (Lagrangian approach), but per unit of volume; this is particularly convenient when

dealing with CFD codes, which are based on the finite volume method.

Conservation equations are built according to the fundamental concept of flux: "The variation of the total amount of a quantity inside a given domain is equal to the balance between the amount of that quantity

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Figure 1.1: Notation used for writing conservation laws referred to a generic control

volume

entering and leaving the considered domain, plus the contribution from eventual sources generating that quantity".

With reference to the notation used in Figure 1.1, the integral formulation of the general conservation equation for a generic scalar per unit volume φ is:

∂ ∂t Z ΩφdΩ = − I S ~ F · d ~S+ Z ΩQvdΩ + I S ~ Qs· d~S (1.1)

The previous general conservation equation can be read in this way: the unsteady variation of a given quantity φ is equal to the variation of flux terms plus the contribution of volume and surface source terms.

Surface integrals of Equation 1.1 can be transformed into volume integrals by applying the Gauss theorem. After that, since the balance is valid for an arbitrary volume Ω, it must be valid in any point of the flow. It is thus possible to write the so-called convection-diffusion differential form of conservation equation:

∂φ

∂t + ~∇ · ~F = Qv + ~∇ · ~Qs (1.2) The convection-diffusion equation takes the name from the two contributions to the flux, which can be either convective or diffusive.

• The convective flux represents the amount of φ that is carried away or transported by the fluid flow: ~FC = φ~U. It appears as a first order partial

derivative term and represents a non linear term, as the velocity field depends on transported variables.

• The diffusive flux represents the amount of φ that is carried away or trans-ported by the presence of its gradient: ~FD = −kρ~∇φ. Diffusion appears as a

second order partial derivative term, called Laplace operator.

The form of previous equations does not change when the conserved property is a vector ~φ = (φx, φy, φz); in this case the flux becomes a tensor, as well as the surface

source term (stress tensor), whereas the volume source term is a vector.

The conservation equations that completely determine the behaviour of the fluid system, without the need of any additional dynamic law, are presented hereafter:

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• Continuity equation

Considering an infinitesimal control volume, in absence of nuclear reactions, the variation of mass is equal to the variation of entering and leaving convective flux through the element surfaces. In integral form:

∂ ∂t Z V ρdΩ + Z S ρ ~U · d ~S = 0 (1.3) In differential form: ∂ρ ∂t + ~∇ · (ρ~U) = 0 (1.4) • Momentum equation

It derives from Newton’s law, as the balance between the variation of momen-tum of a fluid element and the forces exerted on it. Forces can be divided into surface and volume forces:

– Assuming a Newtonian fluid, surface forces are described by the stress tensor σ, which expresses the deformability of the fluid. It is characterized by an isotropic pressure component pI and by the viscous shear stress tensor τ, representing the internal friction force of fluid layers against each other.

σ = −pI + τ (1.5)

– Volume forces like gravity, Coriolis acceleration and other applied forces are represented by a specific source term.

The integral momentum conservation equation, written in vectorial form, is: ∂ ∂t Z V ρ ~U dΩ + I S(ρ~U ~U) · d~S = Z V ρ ~fedΩ + I S σ · d ~S (1.6) And in differential form:

∂(ρ~U)

∂t + ~∇ · (ρ~U ~U + pI − τ)dΩ = ρ ~fe (1.7) • Energy equation

The energy equation derives from the first principle of thermodynamics, stating that the total variation of the energy of a fluid particle is equal to the thermal power introduced plus the mechanical power acting on the surface of the particle.

The conserved quantity is the total energy, defined as the sum of fluid internal energy plus its kinetic energy per unit mass:

E = e + 1 2U~2

According to this definition, the gravitational potential energy contribution is regarded as a body force, which does work on the fluid element as it moves

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through the gravity field.

For energy, there are both types of fluxes: convective ~

FC = ρE ~U

and diffusive (k being the thermal conductivity of the fluid) ~

FD = −k~∇T

The volume source terms are the work of the volume forces fe and the heat

sources (radiation, reactions or electrical resistance): Qv = ρ~fe· ~U + qH

The surface sources ~QS are the result of the work done on the fluid by the

internal shear stress acting on the surface of the control volume: ~

QS = σ · ~U = −p~U + τ · ~U

Grouping all the contributions, the energy equation in integral form becomes: ∂ ∂t Z V ρEdΩ + I S ρE ~U · d ~S = I S k ~∇T d~S+ Z V(ρ ~fe · ~U+ qH)dΩ + I S(σ · ~U) · d~S (1.8) Which becomes in differential form:

∂ρE

∂t + ~∇ · (ρE ~U) = ~∇ · (k~∇T ) + ~∇ · (σ · ~U) + ρ ~fe· ~U + qH (1.9) By expressing the stress tensor in its isotropic and viscous shear stress component and by introducing the specific enthalpy h = e +p

ρ, it is possible to rewrite the energy balance in terms of total enthalpy, which is particularly suitable when dealing with combustion processes:

∂ρH

∂t + ~∇ · (ρH ~U − k~∇T − τ · ~U) = ∂p

∂t + ρ ~fe· ~U + qH (1.10) To provide a closure to the system of equations, a set of constitutive relations for compressible Newtonian fluids are necessary:

• Equation of state:

Except for flows with shock-waves, the fluid can be considered in local equi-librium, so its state can be described by means of two equations of state, in the form:

p= p(ρ, T ) and e = e(ρ, T ) (1.11) They enable to link the energy equation to mass and momentum conservation for compressible flows; in fact, the distribution of mass is affected by density variation caused by local changes of pressure and temperature in the flow field. In case of perfect gas: p = ρRT and e = cvT.

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• Newton’s law of viscosity:

The nine components of the viscous stress tensor τ can be expressed as functions of the strain rate (or local deformation rate), given by the sum of two contributes:

the linear deformation rate ij = 1 2 ∂ui ∂xj +∂uj ∂xi ! (1.12) and the volumetric deformation rate

∂u ∂x + ∂v ∂y + ∂w ∂z = ∇ · ~U (1.13)

With the addition of two constants of proportionality, the first (dynamic) viscosity µ and the second viscosity λ, the viscous stress components are expressed as:

τij = 2µij + λ(∇ · ~U)δij (1.14)

where δij being the Kronecker delta. For gases it is a good approximation to

consider λ = −2

3µ(Schlichting, 1979).

The dynamic viscosity µ can be calculated according to Sutherland’s formula: µ= 1.45T

3 2

T + 11010

−₆ _(1.15)

Specific heat capacity is usually expressed as a fifth order or higher polynomial. • Fourier’s law

The heat flux is related to the gradient of the temperature field by Fourier’s law:

q= −k∇T (1.16)

The fluid mechanics governing equations for an unsteady, compressible, three-dimensional flow, are called Navier-Stokes equations and can be summarized here-after, in differential form:

                             Continuity ∂ρ ∂t + ~∇ · (ρ~U) = 0 Momentum ∂(ρ~U) ∂t + ~∇ · (ρ~U ~U + pI − τ)dΩ = ρ ~fe Energy ∂ρE

∂t + ~∇ · (ρE ~U) = ~∇ · (k~∇T ) + ~∇ · (σ · ~U) + ρ ~fe· ~U + qH Eq. of state p = p(ρ, T ) and e = e(ρ, T )

(1.17) Unfortunately, there’s still no general analytical solution in closed form for such equations, except for very simple cases of limited practical interest, where non-linear terms can be neglected under the main hypothesis of steady, laminar flow and simple geometry. Due to mathematical limitations and the huge costs related

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to the experimental branch, great focus was given to the numerical approach, in order to properly simulate fluids behaviour, based on the solution of discretized Navier-Stokes algebraic equations. The discretization of such equations, and so their implementation in CFD softwares, will be described in Section 1.3.

The general set of Navier-Stokes equations can, sometimes, be simplified by, mainly: reducing the dimensions of the problem, assuming incompressible or steady flow, neglecting viscosity (Euler equations).

Under the hypothesis of incompressible flow, the density is constant and, hence not linked to the pressure by the equation of state. In this case, a pressure-velocity coupling issue arises, since pressure must adapt to the evolving velocity field. Mass conservation becomes a constraint for the velocity field and, if combined with the momentum, can be used to derive an equation for the pressure, called Poisson equation:

1

ρ∆p = −~∇ · (~U · ~∇)~U + ~∇ · ~fe (1.18)

1.2 Turbulence modeling

Turbulence is a highly complex phenomenon whose physical origin lies in the non-linearity of the flow governing equations. Main non-non-linearity is provided by the convective term of momentum conservation and by compressibility effects (e.g. shock waves).

One of the most effective definitions of turbulence states that:

"Turbulence is a three-dimensional, unsteady, rotational fluid motion with broad-banded fluctuations of flow quantities (velocity, pressure, temperature, etc...) occurring in both time and space."

Experience has shown that turbulence is the final state of a process composed by a succession of non-linear instabilities, called transition, occurring above a certain critical Reynolds number, defined as:

Recrit =

U L

ν (1.19)

where U is the flow velocity, L a suitable length scale and ν is the kinematic viscosity of the fluid.

Turbulent flows show an opposite behaviour to the laminar regime, since they are characterized by: chaotic, stochastic property changes, high momentum convection and rapid variation of pressure and velocity in space and time, even in presence of time-invariant boundary conditions.

The random nature of turbulent flows makes them almost impossible to be predicted in detail. On the other hand, nearly all flows of practical engineering applications are turbulent (e.g. turbomachinery and ICEs), because they greatly enhance the rates of momentum, heat and mass transfer. This is the reason why it is of fundamental importance to be able to mimic and predict, as accurately as possible, turbulent effects, by means of modeling techniques.

From the modeling perspective, a statistical approach resulted to be most effective and an exhaustive description can be found in the Kundu [8]. Without

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entering into details, a dimensional analysis highlighted that turbulent flows are constituted by highly unstable vortical structures of different scales, called eddies. The length scale of eddies is connected to the fluctuating component of a given quantity, obtained by the subtraction of its steady mean value. An example of the different eddies length scales is shownFigure 1.2.

Figure 1.2: Turbulent velocity fluctuations and eddies length scales

In the first half of the XX century, an interesting thory was developed by Richardson and Kolmogorov, named as Energy Cascade, which was experimentally validated some years later. This theory states that momentum is transferred from larger to smaller eddies in inviscid way, until viscous dissipation occurs at the level of the smallest scale. This fact is of paramount importance because, in order to numerically solve turbulent flows, it is necessary to have a grid resolution which is, at least, equal to the smallest eddies length scale, otherwise viscous dissipation is neglected.

Eddies are characterized by a size l, a velocity u(l) and a time scale τ(l) = l u(l). Largest eddies have a so-called integral length scale lI, which is comparable to

the flow length scale L. The same is valid for the corresponding Reynolds number, ReI =

uIlI

µ , which is large enough that viscosity is negligible with respect to inertia. Large eddies extract energy from the mean flow by means of shear stresses caused by velocity gradients. This process is named vortex stretching and is responsible of the energy transfer from larger to smaller eddies. Since angular momentum is conserved, smaller eddies have a higher rotation rate, increasing the velocity gradients and the viscous shear stresses; this enhance a break-up process of the eddies into smaller ones, which are stretched in turn. Finally, at the smallest scales, called Kolmogorov microscale, the Reynolds number is sufficiently small that the kinetic energy of turbulence is converted into thermal energy by viscous dissipation.

The energy content of different eddies is function of their fluctuation frequency f, which is typically expressed in terms of wavenumber, k:

k = 2πf

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Smaller eddies spin faster (higher wavenumber) but have lower spectral energy E(k)1 than larger ones. In fact, the latter directly interact with the mean flow, while the former have a higher dissipation rate, because inertial and viscous effects have similar importance. This concept is illustrated in Figure 1.3.

The highest wavenumbers belong to the eddies in the Kolmogorov scale. These vortexes have characteristic lengths, times and velocities (respectively lk ,τk and uk)

whose magnitude can be estimated by means of dimensional analysis, comparing them with the integral scales (lI, τI and uI):

                   Length-scale ratio lk lI ≈ Re−_I3/4 Time-scale ratio τk τI ≈ Re−_I1/2 Velocity-scale ratio uk uI ≈ Re−_I1/4 (1.21)

The above relations show that the higher the Reynolds number, the higher the difference between macro and micro scales. This is a very bad result from a computational point of view, because, to properly solve turbulence and capture flow variations, extremely small time and space intervals are necessary, even for simple cases of engineering interest.

There are three macro approaches to deal with turbulence modeling, presented hereafter in ascending order of level of approximation:

• Direct Numerical Simulation (DNS)

No turbulence model is used. Navier-Stokes equations are entirely computed from the largest to the smallest scales of turbulence. It requires prohibitive computational power and time, due to the very small time step and cell size of the mesh. It is employed in a limited number of research studies with limited Reynolds number.

• Large Eddy Simulation (LES)

It simulates directly the largest scales of turbulence and models the small scale vortexes. In this way, it is possible to accurately predict the evolution of large scale eddies, characterized by a higher energy content, while decreasing the computational demand required by a DNS method.

• Reynolds Averaged Navier-Stokes equations (RANS)

It is the most popular strategy to deal with turbulence, but it also introduces the highest level of approximation. It models all scales of turbulence, by solving time averaged equations obtained using statistic operators (Reynolds Averaged Navier-Stokes equations). Thanks to its reduced computational time, it is suitable for industrial applications.

The level of approximation of the different turbulence modeling methods, with reference to the energy cascade, is shown in Figure 1.3.

1_{The spectral energy E(k) is the kinetic energy per unit mass and per unit wavenumber of}

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Figure 1.3: The figure shows the inverse proportionality between spectral energy E(k)

and wavenumber k and the level of approximation of modeling methods

In the present work, RANS simulation types are treated, adopting a k-ωSST turbulence model. Hence, a brief description of the general idea behind the RANS approach is given, together with the development of the k-ωSST turbulence model. A detailed mathematical description of the RANS method and of all other turbulence models can be found in reference [8].

1.2.1 RANS turbulence models

RANS equations are obtained by so-called Reynolds averaging procedure. It is a linear operator which is applied on every governing equation, in which flow quanti-ties are written as the sum of their mean and fluctuating components (φ = φ + φ0_). According to the Reynolds averaging properties, the equation’s linear terms are replaced by the corresponding mean terms, while the non-linear advective term is given by a mean component and a non-linear combination of fluctuating components, that do not vanish. These fluctuating non-linear components, appearing in diver-gence form in the momentum equation, act like additional stresses to the molecular one and are known as Reynolds stresses. They physically enhance diffusivity and mixing rate in turbulent flows and belong to the symmetric Reynolds stress tensor:

r= −ρ~u0_u~0 (1.22)

In a similar way to the Reynolds stresses, the non-linear convective term in the energy equation causes the presence of a combined fluctuating term, called turbulent heat flux:

− ρcT0_u_~0 (1.23)

Since both Reynolds stresses and turbulent heat flux are additional unknowns, introduced by Reynolds decomposition, a closure problem arises.

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A remarkable step of the modeling effort was represented by Boussinesq’s hypothesis, that expressed the Reynolds stress tensor as sum of an isotropic and a deviatoric anisotropic component:

r= −ρ2

3k −2µTD (1.24)

where D = sym(∇~U) is the mean strain rate tensor, meaning that the deviatoric component was modeled with the Newton’s stress-strain rate law.

In this way the original six unknows introduced by the Reynolds stress tensor are reduced to just one scalar coefficient, called turbulent viscosity µT.

A similar strategy was adopted also for the turbulent heat flux, introducing the turbulent diffusivity kT:

− ρcT0_u_~0 = k

T∇T (1.25)

Boussinesq’s hypothesis drastically reduced the closure problem to the evaluation of the turbulent viscosity only2_{, which, by dimensional analysis, was found to be} proportional to the product between a turbulent length scale lT and a turbulent

velocity uT:

µT ∝ lT · uT (1.26)

The problem was so translated into an appropriate modeling of the turbulent length scale lT, through the so-called mixing length concept, providing very good results

for the boundary layer problem (shear flows). Some empirical formulations of the turbulent length scale were proposed (e.g. Van Driest model) to provide a closure to RANS equations.

The mixing length concept was then replaced by equation models, introducing transport equations for modeling quantities which are related to the turbulent viscosity by links of proportionality. The most popular equation models, also selected for the present work, are the ones based on two balance equations: the first one is dedicated to the turbulent kinetic energy k, for which Prandtl, in 1945, derived an equation in analytical closed form; the second transport equation, instead, refers to either the turbulent dissipation rate ε or frequency ω and, in both cases, it is entirely modeled, characterized by the presence of typical terms appearing in balance equations (unsteady, production, dissipation and transport terms). The names given to the couple of basic two equation models is rapidly derived: k-ε and k-ω.

The main properties of these two models are summarized hereafter: • k-ε

The model solves a modeled equation for the dissipation rate ε, defined as ε = 2µ(D : D). This turbulence model is problematic at the wall due to singularity issues, necessarily requiring the use of so-called wall functions, but it provides good performance in the free-stream region.

Even though wall functions should represent a universal trend of velocity

2_{The turbulent diffusivity can be determined by a turbulent Prandtl number: P r}

T = c

µT

kT

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profile, they are not universal at all, since they do not consider pressure gradients due to curved geometry or compressibility effects like shock waves. They are applied when the resolution of the boundary layer is not needed, reducing calculation burden, but they require a proper cell clustering near the wall region: the mesh should be coarse enough to have y+ _>_{30 (logarithmic} region of the boundary layer).

The y+ _{is the dimensionless wall distance, defined as:} y+= yUτ

ν (1.27)

Where y is the distance to the nearest wall, ν is the local kinematic viscosity of the fluid and Uτ is the friction velocity, which is proportional to the shear

stress at the wall and defined as: Uτ =

s

τw

ρ (1.28)

When running simulations, it is important to keep in mind that it is not possible to evaluate the y+ _{value a priori, because it depends on the shear} stress at the wall τw, which is part of the solution itself.

• k-ω

This model solves a modeled equation for the turbulent frequency ω, expressed as ω = k

ε. Contrary to k-ε, k-ω model is able to solve properly the boundary layer, given a sufficient mesh refinement, without the need of wall functions, that can be used to save computational time. It also behaves better in presence of pressure gradients and prediction of separation, but it is less accurate in the far-wall region.

In order to overcome the principal drawbacks of the previous models, the k-ωSST (Shear Stress Transport) turbulence model was introduced: it is a hybrid model that

combines positive aspects of k-ε and k-ω models, through a blending function that varies according to the y+_{. To correctly describe the fluid-dynamics in the near-wall} region it is required a y+_<_{1, which means a strong mesh refinement at the wall.} It is probably the most used turbulence model for industrial turbomachinery flow calculation, so it was selected also for this work.

1.3 The Finite Volume Method

The finite volume method (FVM) is the discretization strategy adopted by CFD codes, included OpenFOAM, to translate mathematical models and Navier-Stokes governing equations into numbers that can be managed by computers.

The first step consists in the discretization of the domain of interest into sub-domains, called control volumes (CVs), by means of a computational grid or mesh. Then, conservation equations in integral form are applied to all CVs and global conservation of the method is guaranteed. In particular, variable values are calculated at the CVs centroids and, by means of interpolation, they are referred to

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Figure 1.4: Generic CV with cell centroid denoted with P, sharing an internal face with

a neighboring cell whose centroid is labeled with N.

the surface variable of each CVs nodal point. An example of cell element is shown inFigure 1.4.

Consider a scalar φ and its generic unsteady transport equation written in differential form:

∂(ρφ)

∂t + ~∇ · (ρφ~U) = ~∇ · (Γ~∇φ) + Qφ (1.29) where the above terms are, respectively: unsteady term, transport convective flux, transport diffusive flux and source term.

By integrating over the volume V of a generic CV, in a finite time step ∆T and applying Gauss theorem on the transport terms, the equation becomes:

Z V Z t+∆T t ∂(ρφ) ∂t dt dV + Z t+∆T t Z S(ρφ~U) · ~ndS dt = =Z t+∆T t Z S(Γ~∇φ) · ~ndS dt + Z t+∆T t Z V QφdV dt (1.30) The aim of the discretization process is to obtain a system of algebraic equations in the form:

aPφnP +

X

N

aNφnN = Qu (1.31)

In the left-hand side there are the unknown variables φ, at the new time step n, both referred to the current cell center P and to neighboring cell centroids N. The coefficients relative to the value of quantity φ is denoted with a, while the right-hand side term represents the source term contribution of the previous time step.

All the discretization schemes must satisfy some fundamental properties in order to be physically realistic. The main ones are:

• Conservativeness: a scheme must be consistent. It means that the flux of a general quantity φ exiting from the surface of a cell CV must have the same value of the flux entering the adjacent cell.