FACULTY OF AEROSPACE ENGINEERING Master of Science degree in Aerospace Engineering
MASTER THESIS In Collaboration With:
von Karman Institute for Fluid Dynamics
Enviromental and Applied Fluid Dynamics Department
Numerical Simulation and
Development of a Flow Control
Method for Oscillating
Impinging Gas Jets
Candidate: Lorenzo Paoli
Supervisors:
Prof. Jean Marie Buchlin Prof. Maria Vittoria Salvetti
Ir. Miguel Alfonso Mendez Prof. Simone Camarri
One of the most beautiful chapters of my life has now concluded. First, I wish to thank my supervisors at VKI, Jean Marie Buchlin and Miguel Alfonso Mendez, who gave me the chance to work in a dynamic reality such as the von Karman Institute. In particular, I want to thank Miguel, who took care of me and supported me with enthusiasm and friendship. I want to say thanks to my Professors Maria Vittoria Salvetti and Simone Camarri for the supervision of my thesis and to Professor Guido Buresti for the recommendation letter.
Special thanks are due to my family, especially to my parents who supported me all these years. It is not possible to describe with a few words what they have done for me. Thanks for all mamma Vittoria, pap´a Andrea and two my sisters Asia and Serena. Now, it is the time to say thanks to my old and new friends. Nicola ”Alexander”, Francesco (KOS), Andrea ”Laramas”, Daniele and Serena, we go back a very long time and managed to go through many adversities, but here we are, thanks for your constant support in every single aspect of my life. Alessandro ”il Bacci”, I met you in high school. I understood the importance of our friendship because you supported me in first line, whether we were near or far from each other. Special thanks are also due to my university colleagues, especially to Luca, ”lo Spezzino” and to his Rational Mechanics professor; to Federico FADEC (nicknamed ”Senza Ossa”), to Daniel ”MaMMeglio”, to Fabio ”The Big” for supporting me all these years, for our friendship full of laughter and thousands of funny stories and thanks to Pisa where I met a lot of people. Last but not least, thanks to my special friend Luca ”Necking”, I will never forget the crazy things we did together with Vallecchia and Co., we faced many challenges studying at night and we spent a lot of time together. I have to say thanks to a lot of people, but this thesis is not long enough. I cannot fully describe the experience at the VKI, but I will never forget the stager room people: Carmelitano, Javi, Lore ”Buo”, Costanza, Andrea ”Picchia Fuori”, Michele ”il Boss”, Riccardo Mario ”Bi¸cakci”, Onur ”g¨un I¸sigim”, Utkan ”my Bro”, David ”VillaBlanco”, Sandra, Piero, Jiyeon, Domenico, Domenico ”Napuleta”, Maria ”Pizza Girl” and the others, also my ”colocataires”, ”lo Zio” Max, Sthephan and Daniel, I love you all. I cannot forget the soundtrack of my Belgian adventure, ”l’amour toujours” and the song of 2014, the best year of my life, ”Lady (Hear Me Tonight)”. I do not forget to say thanks to the people who never believed in me, since now they have given me the motivation to prove them that they were wrong.
Lorenzo Paoli
Disce pati si vincere voles.
Impinging gas jets in the low Mach number regime assume an important role in both the academic and the industrial environment. This thesis is conducted within the von Kar-man Institute (VKI) research framework on the flow control for gas jets impinging on the curved walls, a configuration which is capable of promoting self-sustained oscillation of the flow. The aim was to drive the pre-design stage of a new control method via Compu-tational Fluid Dynamics (CFD), investigating several designs and operating conditions. The simulations were carried out in OpenFoam 2.3.1 with a k−ω SST URANS approach. Starting from an uncontrolled configuration, the influence of the velocity inlet, the shape of impinging wall and the height of the curved wall has been studied. Once identified the flow mechanism responsible for the jet oscillation, two control strategies were de-fined: the first aiming at hindering its occurrence; the second aiming at compensating its effects. The first approach was implemented by introducing a buffer plate in strategic locations of the flow configuration; the second was implemented via a thrust vectoring technique using a secondary injection and a Coanda surface. The numerical investiga-tion of these configurainvestiga-tions was conducted in parallel with high speed flow visualizainvestiga-tion from other members of the VKI team and has finally lead to the mechanical design and construction of a new nozzle, currently under investigation in the VKI laboratories.
List of Figures ix
List of Tables xiii
List of Symbols xv
1 Introduction 1
1.1 Project BackGround . . . 2
1.1.1 M.A.Mendez: Behaviour of jet impinging . . . 2
1.1.2 M.T.Scelzo: Active Flow Control . . . 4
1.1.3 A.Enache: Passive Flow Control . . . 4
1.1.4 M.Di Nardo and D.Dumoulin: Numerical Validation . . . 5
1.2 Flow Control Literature . . . 6
1.3 Thesis Aims and Outline . . . 8
2 Numerical Methods 9 2.1 Part I: Turbulence Modeling . . . 9
2.1.1 What is the Turbulence? . . . 9
2.1.2 Turbulence, Vortex and Cascade of energy . . . 10
2.2 Turbulence Models . . . 12
2.2.1 URANS . . . 12
2.2.2 Large Eddy simulation and others approaches . . . 18
3 The Finite Volume Discretization in CFD 21 3.1 The Discretisation of Domain . . . 21
3.1.1 Accuracy of Discretization . . . 22
3.2 Discretization Methods and OPENFOAM . . . 23
3.2.1 Spatial Discretization . . . 24
3.2.2 Convective term . . . 31
3.2.3 OpenFOAM and Convection Term . . . 36
3.2.4 Diffusion Term . . . 38
3.2.5 Minimum Correction Approach . . . 39
3.2.6 Orthogonal Correction Approach . . . 39
3.2.7 Over-Relaxed Approach . . . 40
3.2.8 OpenFoam and Diffusive term . . . 41
3.2.9 Source Terms . . . 42
3.3 Time Discretization . . . 42
3.3.1 Time Discretization in OpenFOAM . . . 44
3.3.2 Stability of Numerical Methods . . . 44
3.4 OpenFOAM Solver: Pressure-Velocity coupoling in Incompressible flows (PISO) . . . 45
4 Modelling thrust vector 51 4.1 Simulation Analysis . . . 51
4.2 Modelling thrust vector . . . 55
4.2.1 Flow deflection of primary Jet . . . 56
4.2.2 Secondary jet . . . 56
4.2.3 Model Formulation . . . 58
4.3 Results . . . 59
5 Investigation of the No Control Flow Simulations 65 5.1 Boundary Condition, Geometry and General Settings . . . 66
5.1.1 No Control Case Geometry . . . 66
5.1.2 No Control Case Mesh . . . 69
5.1.3 No Control Case Boundary Condition . . . 71
5.1.4 FFT, Time step and Criteria . . . 73
5.2 OpenFoam: simulations analysis . . . 74
5.2.1 k − ω SST VS DES . . . 74
5.2.2 Flow Field Description . . . 75
5.3 Oscillation Description . . . 85
6 Flow Control Method 89 6.1 Active control: M.Scelzo Design . . . 89
6.2 Active Flow: Ours Nozzle . . . 91
6.2.1 Development of the solution . . . 91
6.2.2 Nozzle Realisation and Experiment . . . 102
6.3 Passive control: A.Enache Design . . . 103
6.3.1 Geometry . . . 103
6.4 Solution Comparison . . . 109
7 Conclusions and Future Work 117 7.1 Conclusions . . . 117
7.2 Future Work . . . 118
A Annex A 119 A.1 OpenFOAM Files . . . 119
A.1.1 No Control Configuration . . . 119
A.1.3 Passive Flat Plate Configuration . . . 132
A.1.4 No Control Configuration: 0 Folder . . . 132
A.1.5 No Control Configuration: Constant Folder . . . 132
A.1.6 No Control Configuration: system Folder . . . 135
A.1.7 Active Control Configuration . . . 139
A.1.8 Active Control Configuration: 0 Folder . . . 139
A.1.9 Active Control Configuration: Constant Folder . . . 150
A.1.10 Active Control Configuration: system Folder . . . 152
A.1.11 MATLAB FILE . . . 156
1.1 Examples of flow confinements producing oscillating planar impinging jet: a) basic (stable) configuration, b) confined jet impingement, c) jet imping-ing on V-shape interface, d) jet discharge into rectangular cavity, e) jet impinging on liquid pool (dimple configuration), f) flow configuration of
this work with varyed height. Figure taken from [13]. . . 1
1.2 Different configuration experiment: four heights from the impinging point ˆ H, three perpendicular distances from the curved wall ˆZ and two config-urations for curved wall. Figure taken from [13]. . . 3
1.3 Numerical simulation of the nozzle (a),(b), Nozzle creation (c). Taken from [14]. . . 4
1.4 Different set-up(a)(b)(c) and analysis of the centerline of the jet (d)(e)(f). Taken from [1]. . . 5
1.5 Coanda effect on a curved wall. Figure taken from (https://media. lanecc.edu/users/driscolln/RT112/Air_Flow_Fluidics/Air_Flow_Fluidics8. html). . . 6
1.6 Thrust Vectoring (a), Maison experiment (b). Taken from [15]. . . 7
1.7 Uncontrolled synthetic flux (a), Schematic design of control (b), Synthetic flux controlled (c). Taken from [6]. . . 8
2.1 Difference between three Reynolds numbers (left) Re=9000, (middle) Re=3152, (right) Re=900. Taken from [27]. . . 9
2.2 Boundary layer and different regimes. Taken from http://www.nuclear-power. net/nuclear-engineering/fluid-dynamics/reynolds-number/. . . 10
2.3 Eddy stretching in a 2-D strain field,Tennekes and Lumley(1981): (a)Before stretching (b)After Stretching. Taken from [24] . . . 11
2.4 Vortex Tilt and Streatch . . . 11
2.5 Cascade of energy: Spectrum of energy . . . 12
3.1 Control Volume Discretition. Taken from [11] . . . 22
3.2 Control Volume. Taken from [8] . . . 25
3.3 Calculated Gradient for uniform one dimension mesh. Taken from [8]. . . 25
3.4 CF cross the centroid f of the face Sf. Taken from [8] . . . 26
3.5 First approach: f0 is the point of the intersect between the segment [CF]
and the face Sf. Taken from [8]. . . 27
3.6 Second approach: The position of f0 is in the middle of CF . Taken from [8] 28 3.7 Third approach: The point f0 is on the segment CF and the length of the segment f f0 is minimized. Taken from [8] . . . 29
3.8 Analytical solution of the steady one dimension Convection-Diffusion prob-lem imposed between the nodal point of the uniform grid. Taken from [8]. . . 32
3.9 Analytical solution of the steady one dimension Convection-Diffusion prob-lem varying the Peclet number. Taken from [8]. . . 33
3.10 Upwind Scheme. Taken from [8]. . . 36
3.11 Numerical and Analytical solutions (a), Numerical and Analytical solu-tions of the Convection-Diffusion term variyng the P eclet number in 1D (b). Taken from [8]. . . 37
3.12 Orthogonal and Non-Orthogonal Diffusion Term. Taken from [19]. . . 39
3.13 Decomposing Sf via the minimum correction approach. Taken from [19]. . 40
3.14 Decomposing Sf via the orthogonal correction approach. Taken from [19]. 40 3.15 Decomposing Sf via the over-relaxed approach. Taken from [19]. . . 41
3.16 Uniform grid. Taken from [19]. . . 46
4.1 Main control variable parameters. Taken from [17] . . . 51
4.2 The three region of pressure profile. . . 52
4.3 R = 2mm hs= 0.5mm hp = 1.5mm: (a)(b) β = 0.4, up = 30m/s; (c)(d) β = 0.4, up = 38m/s. . . 53
4.4 R = 2mm hs= 0.5mm hp = 1.5mm: (a)(b) β = 0.5, up = 30m/s; (c)(d) β = 0.5, up = 38m/s. . . 54
4.5 R = 5mm hs= 0.5mm hp = 1.5mm: (a)(b) β = 0.4, up = 30m/s. . . 55
4.6 Control Volume Primary Jet Model. Taken from [17]. . . 56
4.7 Control Volume Secondary Jet Model. Taken from [17] . . . 57
4.8 Function of the angular coordinate θ for the all the test cases : The radial pressure gradient P (θ) along the Coanda surface (a) and the averaged velocity G(θ) (b). Taken from [17]. . . 60
4.9 The four configuration tested with the extracted velocity profile along the Coanda surface. Taken from [17]. . . 61
4.10 Thrust vectoring coefficient CZ as a function of the momentum ratio βM for the three geometrical configurations The primary jet opening is fixed at hp = 1.5mm. Taken from [17]. . . 62
5.1 Bump(a), Half-Bump (b), Smoothed step(c). . . 66
5.2 No Control Domain, Circle A: Bump, Square B: Half Bump. . . 67
5.3 Patches name No Control Domain. . . 68
5.4 Patches name No Control Domain. . . 69
5.5 Initial Course Mesh∼7000 cells (a), Box Refinement on Mesh (b), Detail of Inlet (c), Final Integer Mesh∼170.000 cells (d). . . 70
5.6 FFT Di Nardo Configuration 18ms (a), Results of Di Nardo ARO Project
(b). . . 71
5.7 No Control Low Velocity Bump Pressure Field: Detached Large Eddy Simulation (a)(b)(c)(d), k − ω SST Simulation (e)(f)(g)(h). . . 75
5.8 No Control Low Velocity Bump Velocity Field: Detached Large Eddy Simulation (a)(b)(c)(d), k − ω SST Simulation (e)(f)(g)(h), the FFT of the cases (i)(g). . . 76
5.9 No Control Bump Configurations Pressure Field: Low Velocity Case (a)(b)(c)(d), Inter-medium Velocity Case (e)(f)(g)(h), High Velocity Case (i)(j)(k)(l). . . 78
5.10 No Control Bump Configurations Velocity Field: Low Velocity Case (a)(b)(c)(d), Inter-medium Velocity Case (e)(f)(g)(h), High Velocity Case (i)(j)(k)(l). . 79
5.11 FFT No Control Bump Configurations: 30ms (a), 90ms (b), 150ms (c). FFT Di Nardo Smoothed step Configuratio: 18ms(d). . . 80
5.12 No Control Half Bump Configurations Pressure Field: Low Velocity Case (a)(b)(c)(d), Inter-medium Velocity Case (e)(f)(g)(h), High Velocity Case (i)(j)(k)(l). . . 81
5.13 No Control Half Bump Configurations Velocity Field: Low Velocity Case (a)(b)(c)(d), Inter-medium Velocity Case (e)(f)(g)(h), High Velocity Case (i)(j)(k)(l). . . 82
5.14 Di Nardo Smoothed Step Configurations Pressure field: 18ms (a)(b)(c)(d), Low Velocity (e)(f)(g)(h), High Velocity (i)(j)(k)(l). . . 83
5.15 Di Nardo Smoothed Step Configurations Velocity field: 18ms (a)(b)(c)(d), Low Velocity (e)(f)(g)(h), High Velocity (i)(j)(k)(l). . . 84
5.16 Correlation between Vortex Pressure, Velocity Magnitude and impinge-ment point jet. . . 85
5.17 Correlation between Velocity Magnetude and the Impingement Point. . . 85
5.18 Correlation between the Velocity Magnetude of the all cases with dimen-sionless time. . . 86
5.19 Correlation between Vortex Pressure, Velocity Magnetude and impinge-ment point jet at 30ms. . . 87
5.20 Correlation between Vortex Pressure, Velocity Magnetude and impinge-ment point jet 90ms. . . 87
5.21 Correlation between Vortex Pressure, Velocity Magnetude and impinge-ment point jet 150ms. . . 88
6.1 Free-jet domain dimension (a), Geometry of M.Scelzo design (b). . . 89
6.2 M.Scelzo simulations with differentβ ratios. β = 0.1 (a), β = 0.3 (b), β = 0.5 (c) . . . 90
6.3 Old Nozzle Design. . . 91
6.4 Dimension of new nozzle domain . . . 91
6.5 Cylinder narrows the Inlet Active(a), Cylinder is leaned on(b) . . . 92
6.7 Cylinder narrows the Inlet Active β = 0.3 (a), Velocity Scale(b), . . . 93
6.8 Cylinder narrows the Inlet Active(a), Cylinder is leant on(b), Cylinder is tangent to Inlet Active (c) . . . 94
6.9 Dimension of new nozzle Detail D. . . 95
6.10 Detail D simulations with different beta: β = 0.3(a), β = 0.5(b), β = 1(c). 95 6.11 Dimension of new nozzle Detail E the Spoon. . . 96
6.12 Detail E simulations with different beta ratios: β = 0.3(a), β = 0.4(b), β = 0.55(c). . . 96
6.13 Dimension of new nozzle Detail E the Spoon. . . 97
6.14 Dimension of new nozzle Detail E the Spoon inclination. . . 98
6.15 Detail E inclined geometry simulation β = 0.5: Velocity field, maxi-mum(a) and the minimum(b) oscillation. Pressure field (d)(e) . . . 98
6.16 Old Nozzle Design. . . 99
6.17 Nozzle Plate Cross Dimension. . . 99
6.18 Detail F Nozzle Geometry (a), the different departure angles (b)(c)(d). . . 100
6.19 Free Stream Simulation of Detail F: up = 30ms, β = 0.5 . . . 100
6.20 Detail F Simulation Inclined Geometry up = 30ms, β = 0.5: Velocity Field (a)(b), Pressure Field (c)(d). . . 101
6.21 Realized Nozzle mounted . . . 102
6.22 Passive Control Flat Plate Domain . . . 104
6.23 Passive Control Flat Plate Patches . . . 105
6.24 Passive Control Simulation Low Velocity uinlet = 30ms: Velocity Field (a)(b)(c)(d), Pressure Field (e)(f)(g)(h), Dimensionless FFT (i) . . . 107
6.25 Passive Control Simulation High Velocity uinlet = 150ms: Velocity Field (a)(b)(c)(d), Pressure Field (e)(f)(g)(h), Dimensionless FFT (i) . . . 108
6.26 Bump No Control Pressure Gradient Low Velocity uinlet = 30ms . . . 110
6.27 Passive Control Pressure Gradient High Velocity uinlet= 150ms . . . 111
6.28 Passive Control Pressure Gradient Low Velocity uinlet = 30ms . . . 112
6.29 Passive Control Pressure Gradient High Velocity uinlet= 150ms . . . 113
6.30 Active Control Pressure Gradient High Velocity up= 30ms, β = 0.5. . . . 114
3.1 OpenFOAM discretization mode . . . 23
4.1 Differt configurations . . . 63
5.1 k − ω SST No Control Cases . . . 66
5.2 DES k − ω SST Cases No Control Case . . . 67
5.3 DES Boundary condition . . . 72
5.4 Boundary condition No-control and Di Nardo Cases Formula . . . 72
5.5 Boundary condition No-control case . . . 73
6.1 Different configuration for M.Scelzo Nozzle . . . 90
6.2 k − ω SST Passive Control Cases . . . 103
6.3 Boundary Condition Low Velocty Flat Plate Simulation. . . 103
6.4 Boundary Condition High Velocty Flat Plate Simulation . . . 106
Acronyms
CFL Courand Number
STP Short Training Programs RM Research Master
FV Flow Visualization
PIV Particle Image Velocimetry
POD Proper Orthogonal Decomposition IPT Image Process Thresholds
mPOD multi-scale Proper Orthogonal Decomposition
URANS Unsteady Reynolds Average Navier-Stokes Equations LES Large Eddy Simulation
DES Detached Large Eddy Simulation SST Shear Stress Transport
CV Control Volume FVM Finite Volume Method
PISO Pressure Implicit with Splitting of Operators VKI von Karman Institute
Abbreviation
Cz Thrust Vectoring Coefficient Re Reynolds number
St Strouhal number Pe Peclet Number Ti Turbulence Intensity
Roman symbols
f Frequency 1s k Turbulent Kinetic Energy 1s
p Pressure P a u Velocity ms h Inlet Opening m R Coanda Radius m Z Impinging Distance m Y Impinging Point m Greek symbols
β Thrust Vectoring Ratio -ω Turbulence Specific Dissipation -ν Eddy Viscosity
-λ wavelength µm
ρ density kg/m3
φ Generic Scalar Variable various
Sub- and Superscripts p Primary
s Secondary v vortex ref Reference
Introduction
The impingement of jet flow on a flat surface is a common configuration in fluid dy-namics, with a several industrial applications. The flow impact is used to the high convective fluxes and pressure gradients it produces on the impinged surface. The main fields are cooling, heating and drying [28],[3],[26]. In figure (1.1) several configurations of impinging jets are shown. The basic configuration Fig.(1.1a) is characterized by a symmetric bifurcation of the flow on the impingement wall at the stagnation point O and a quasi-parallel entrainment flow of surrounding fluid.
Figure 1.1: Examples of flow confinements producing oscillating planar impinging jet: a) basic (stable) configuration, b) confined jet impingement, c) jet impinging on V-shape interface, d) jet discharge into rectangular cavity, e) jet impinging on liquid pool (dimple configuration), f) flow configuration of this work with varyed height. Figure taken from [13].
The dynamics of the impinging jets change according to the confinement or the shape of the surface. When these two charateristics are altered, the impinging jet un-dergoes self-sustained oscillation as well known in several configuration types recalled in Fig.(1.1(a-f)). This includes entreatment confinement (b), jet impingement on V shape(c), jet flow in a cavity, jet impingement on liquid and the configuration on curved wall(f). Contrary to the convective instabilities of a free jet, confined jets manifest well organized oscillation patterns [7]. In the literature, it is largely accept that the appear-ance of this globally organized phenomenon is due to a hydrodynamic feedback [10]. The feedback mechanism is not clearly understood but, due to the strong interest for impinging phenomena, their fluid dynamics properties are well known. In particular, the characteristic dimensionless parameter of the oscillation is the Strouhal number:
St = f Z
U (1.1)
where f is the oscillation frequency, U the center-line jet velocity and Z the stand-off distance between the nozzle and the surface. The existence of a constant Strouhal number regardless of the Reynolds number is retrieved in these configurations. In the next section 1.1 are listed the works previously done at VKI. The section 1.2 is dedicated to the Flow Control Literature and the last one 1.3 is for the thesis aims and outline.
1.1
Project BackGround
This thesis project is in the framework of a Von Karman Institute (VKI) research project, which involves a PhD thesis [21] and several short training programs (STP) and research master (RM) projects. A summary of these works here proposed.
1.1.1 M.A.Mendez: Behaviour of jet impinging
Mendez [18] studied experimentally the jet impinging responses according to several ge-ometrical configuration, namely the dimensionless impinging distance ( ˆZ = Z/d) (aper-ture inlet), the adimensional height ( ˆH = H/d) of the stagnation point and two interface shapes (M). The investigation was carried out by means of high speed Flow Visualiza-tion (FV) and the time-resolved Particle Image Velocimetry (PIV) acquisiVisualiza-tion with two image processing algorithms and Proper Orthogonal Decomposition (POD) analysis [13].
For the different interfaces several conclusions were drawn: • Interface
– The size of interface (M) has a strong impact in the amplitude of the oscilla-tion but not on the dimensionless frequency. In all the cases, the jet response starts with a deflection towards the bump side, followed by a almost sym-metrical oscillation with the amplitude growing as the interface deformation evolves. Instead, the position of the jet flow origin with respect to the inter-face (parameters ˆZ and ˆH) determines whether the self-sustained mechanism is triggered or not.
Figure 1.2: Different configuration experiment: four heights from the impinging point ˆH, three perpendicular distances from the curved wall ˆZ and two configurations for curved wall. Figure taken from [13].
• Frequency
– When the self-sustained mechanism is triggered, the frequency of the oscil-lations remains unchanged as the interface moves within the range of the critical position (the highest membrane). The frequency scales so that a con-stant Strohual number StZ = f ZUj = 0.055 is found which is independent of
any operating conditions. Moreover, neither amplitude nor the dimensionless frequency depend on the Reynolds number. The frequency has been mea-sured by Image Process Thresholds (IPT). The grey-scale images acquired from the PIV, are filtered with a particular algorithm, which is capable to find the centerline of the jet. The IPT has been developed from Mendez [13].
• Oscillation
– The oscillation amplitude is a function of the bump height. In all the cases, the trigger of the oscillation is a downward deflection of the jet, which slowly evolves into an oscillation with a monotone increase of amplitude.
• Behaviour of the Jet
– Through the POD and mPOD (multi-scale Proper Orthogonal Decomposi-tion), two physical mechanisms responsible for the hydrodynamic feedback have been captured at largely different scales. These large and fine scale mechanisms have been separated via wavelet filtering on the temporal corre-lation matrix, allowing for distinguishing the main frequencies associated to each modal structure.
1.1.2 M.T.Scelzo: Active Flow Control
Starting from Mendez’s work, which investigated on the role of vortex and the confine-ment after a literature review, Scelzo combined co-flow secondary injection and a Coanda surface to develop a flow control method based on jet flow vectoring [14]. This config-uration has been studied both in free conditions, that is with the jet released far from walls, and in the impinging conditions. For the latter the jet appears to be stabilized by the effect of the low pressure zone formed above the jet, due to the attachment of the secondary injection to the Coanda surface Fig.(1.3a). The concept has been confirmed by means of additional simulations with the nozzle inclined Fig.(1.3b). These numerial tests have shown that depending on the mass flow ratio between the two injection β, the jet can be forced to a stable position. These simulations have guided the new noz-zle pre-design by identifying the operating conditions leading to suitable flow control. Finally, a first prototype of the nozzle has been manufactured at the VKI Fig.(1.3c)
(a) (b) (c)
Figure 1.3: Numerical simulation of the nozzle (a),(b), Nozzle creation (c). Taken from [14].
1.1.3 A.Enache: Passive Flow Control
The passive flow control method was applied in order to limit the entrainment flow that comes from below the jet. A 20 mm plate was positioned at different distances D as can be seen in the figure (1.4). This plate limits the space below the jet so the recirculation structure does not have space to form, and it does not destabilize the jet anymore. For very small distances D, the Coanda effect can be seen and the jet flow attaches to the plate [1]. From these results, it can be seen that the passive control can be used to stabilize the jet flow, and that the amplitude of the oscillations decrease when the plate is positioned closer to the jet flow [1]. A.Enache studied the effect of the vertical distance on the behaviour of the jet, for the smallest distance D, the jet flow is
completely stabilized.
Figure 1.4: Different set-up(a)(b)(c) and analysis of the centerline of the jet (d)(e)(f). Taken from [1].
1.1.4 M.Di Nardo and D.Dumoulin: Numerical Validation
M. Di Nardo [16] proved by simulating the impinging jet that the Strouhal number is constant regardless of the impinging distance (Z) between the jet opening and the walls. Concerning the vertical distance (Y), the sensitivity of the jet oscillation to the impinging point on the curved wall has been verified: the jet does not oscillate when the impinging point is beyond two times the opening (H) from the curved edge of the wall, in agreement with the experimental results in [14]. However, Di Nardo reports a significant discrepancy between the simulations and the experiments results in terms of oscillation frequency. To validate the numerical results, a comparison between Numeca and OpenFoam has been done. The first used a compressible and the second, an incompressible solver (PISOfoam, pressure based algorithm). Many simulations have been performed with OpenFoam varying principally 3 parameters: the impinging distance (L), the impinging point locations (Y ) and the inlet velocity (u0). The constancy of the Strouhal number in
the different configurations simulated suggests that: in the range of instability (i.e. L = 7H-16H for Y = 0), the flapping frequency is directly proportional to the inlet velocity and inversely proportional to the impinging length (L). The oscillation decreases as the misalignment Y is increased.
1.2
Flow Control Literature
Flow control strategies can be passive, where no external energy is required, or active, where additional energy is spent. Passive techniques modify the geometrical confinement of the flow so as to change the turbulent structure (on thus the pressure field) responsible for the jet oscillation [12]. In her work, Scelzo [14], has analysed the manipulation of the pressure field, using the Coanda effect as a powerful method proposed in literature to deviate the jet centreline [2]. For incompressible inviscid two dimensional flows, the Coanda effect (the tendency of the flow to follow the nearest surface) is due to the balance between the centrifugal forces and the radial pressure. As the flow leaves the slot Fig.(1.5), it remains attached to the curved surface until the surface pressure equals the ambient pressure and the flow detaches [2]. In an inviscid fluid, the wall pressure remains below the ambient pressure. In real viscous flows, however, the flow entrainments results in the thickening of the jet and a reduction of the mean velocity. As mean velocity decreases, the surface pressure along the wall increases and eventually equals the ambient pressure. When this occurs, the flow separates from the curved surface. Therefore, inviscid flows may attach themselves according to the balance of centrifugal forces, but viscous effects are the cause for jet separation from the curved wall.
Figure 1.5: Coanda effect on a curved wall. Figure taken from (https://media.lanecc. edu/users/driscolln/RT112/Air_Flow_Fluidics/Air_Flow_Fluidics8.html).
Active control schemes can be pre-determined or interactive: in an interactive method the actuator (controller) is continuously adjusted based on some feedback information, provided by a sensor element, contrary to the pre-determined techniques in which the control is activated independently from the flow response. There are several strategy to actively control the flux. For example Piezoelectric actuator that are used locally for flow-induced cavity oscillation [5] to introduce mechanically the perturbation in the flow. Another strategy is Fluidic actuators, in particular synthetic jets [6], which do not require external flow source and, in principle, are capable of producing complex
waveforms using a variety of transduction schemes [4] or another technique like thrust vectoring in aircraft application [15]. Mason performs an experimental and numerical campaign on the use of a Coanda surface and a secondary coflow injection for jet thrust vectoring for aircraft application [15], as depicted in figure (1.6).
(a) (b)
Figure 1.6: Thrust Vectoring (a), Maison experiment (b). Taken from [15].
In [25], Smith and Glezer studied the possibility of jet vectoring using synthetic jets. Methods of jet vectoring can be divided into two distinct groups, namely, approaches that rely primarily on extended surfaces, and approaches that are based on fluidic actuation. In the former case, the flow direction of a planar jet can be substantially altered either by exploiting the adherence of the jet to a curved surface that is a smooth extension of its nozzle, or by the reattachment of a separate jet to an adjacent solid surface [25]. The latter case is an attractive possibility since it offers the combination of suction and blowing from the same orifice without the need of an extra flow, but it requires the development of an actuator capable of delivering the oscillatory flow at the desired frequency and amplitude [14]. Another method has been proposed by Newman [23]. He noted that a second mechanism through which the flow direction of a plane jet can be altered is the attachment of a separate jet to an adjacent solid surface that extends to the edge of the nozzle. This attachment is induced by the formation of a low-pressure region between the jet and the surface blowing to entrainment and is known as the Coanda effect.
(a) (b) (c)
Figure 1.7: Uncontrolled synthetic flux (a), Schematic design of control (b), Synthetic flux controlled (c). Taken from [6].
1.3
Thesis Aims and Outline
This thesis continues the previous work at the VKI and aims at extending the numerical dataset available, as well as supporting the final design of a control scheme by meams of the Computational Fluid Dynamics (CFD). At the scope, several flow configurations have been studied using the URANS (Unsteady Reynolds Average Navier-Stokes equa-tions) finite volume solver Pisofoam from OpenFoam. The URANS strategy and more generally the challenges involved in the numerical simulation of turbulent flows are re-called in Chapter 2. This includes turbulence modelling. The Chapter 3 contains a review of Finite Volume discretization implemented in OpenFoam. Chapter 4 analyses the jet flow vectoring performances as a function of several operating parameters, intro-duces the Image Processing Threshold (IPT) technique used to measure the inclination of the jet and a simple analytical model to understand the jet vectoring mechanism. Chapter 5 reports on the analysis of the oscillation mechanism, investigating the influ-ence of the jet flow velocity, interface shape, interface confinement with incompressible solver, comparing Detached Large Eddy Simulation (DES) and URANS. The results and the conclusions shown from these two investigations finally lead to the design of a new active flow control method, which is tested in several configurations in Chapter 6, and compared to other approaches considered in the framework of VKI research on the sub-ject. The proposed method is capable of achieving better stabilization than the previous nozzle design, opening the possibility to control the inclination of the jet vectoring. The development of this active control strategy has been compared to the previous nozzle and to a passive technique designed and tested by Mendez and Enache [1]. Chapter 7 presents the conclusions. The last chapter summarized the considerations on the influ-ence of the jet flow velocity, interface shape, interface confinement on the simulations tested for the no control configurations. Finally, comparing the weak and strong point of the control method strategies, it has been answered which solution is better to stabilize the jet from an economical and application point of view.
Numerical Methods
This Chapter gives an overview of the theory behind turbulence and the challenges involved in its description. Section 2.1 describes the theory of turbulence. Section 2.2 presents the main models, which have been developed in many years of research and can be classified three categories: URANS, LES and DES (hybrid model RANS/LES).
2.1
Part I: Turbulence Modeling
2.1.1 What is the Turbulence?
Firstly, it is important to take a brief look at the main characteristics of turbulent flows. The Reynolds number of a flow gives a measure of the relative importance of inertia forces (associated with convective effects) and viscous forces. In fluid systems experiments, it is observed that at values below the so called critical Reynolds number Recrit the flow
is smooth and made of adjacent layers (Fig.2.1 on the right). This type of flow is known as Laminar flow.
Figure 2.1: Difference between three Reynolds numbers (left) Re=9000, (middle) Re=3152, (right) Re=900. Taken from [27].
At values of the Reynolds number above Recrit a complicated series of events takes
place and eventually leads to a radical change of the flow characteristics. If Re increases, the inertia of the fluid becomes greater and the instabilities that occur cannot be damped by viscous forces. In the final state the flow behaviour is random and chaotic, it is called Turbulent flow.
2.1.2 Turbulence, Vortex and Cascade of energy
Turbulence can be considered to be composed by eddies of different sizes. The largest eddies are characterized by the length-scale l0, velocity u0 = u(l0) and time-scale τ0
(Kolmogorov Theory), that respectively can be compared with the flow scale Lf, velocity
Uf and time-scale τf. From this comparison, the Reynolds number of eddies can be
derived as the Reynolds number of flow.
Re0=
u0l0
ν = UfLf
ν = Re
If each Reynolds number is similar, this means that the largest eddies are governed by inertial forces.
Figure 2.2: Boundary layer and different regimes. Taken from http://www. nuclear-power.net/nuclear-engineering/fluid-dynamics/reynolds-number/.
The Fig.2.2 shows that in the turbulent boundary layer the large eddies are aligned with the flow. They interact and extract energy from the mean flow and transfer it to the smaller and smaller eddies. It is possible to explain this mechanism in way:
• Tennekes and Lumley (1981): explained this mechanism of energy transfer with Fig.(2.3). If two eddies normal to each other are subjected to a two-dimensional strain field s11 = s22 = s, the eddies aligned with positive strain (ω11) will be
stretched and reduced in diameter D since in an incompressible flow the vortex filament must conserve its own volume as the second Helmotz theorem reads. Un-like the first Helmotz theorem reads ”The strength of a vortex filament is constant along its length”.
Strength = Z Z S ~ ω dS (2.1) dS = π(D 2) 2 (2.2)
This means that if diameter D of the eddy is reduced and its strength must be constant, the vorticity ω1 must increase. This is the mechanism of amplification
of vortex without any source. The vortexes are Stretched and Tilted, if they are sufficiently stretched, they present a mechanism of instability, then they are broken and become more smaller and smaller. The small eddies are also able to interact with the strain field of the larger eddies causing energy to be transferred from the largest to the smallest eddies. This phenomenon is known as Cascade of Energy.
Figure 2.3: Eddy stretching in a 2-D strain field,Tennekes and Lumley(1981): (a)Before stretching (b)After Stretching. Taken from [24]
Cascade of energy
The turbulence energy transfer can be represented in the energy spectrum, where the turbulence energy per unit wave number is plotted against the wave number. The wave number of turbulence fluctuation is inversely proportional to the eddy size. Fig.(2.5) shows the energy of the turbulence fluctuation as a function of wave number. Energy is mainly concentrated in the large eddies (low wave number), which are the ones that generate turbulence energy, by removing it from the mean flow. The energy is then transferred across eddies of intermediate sizes until it reaches eddies of a size small enough for the dissipation process to take over.
Figure 2.5: Cascade of energy: Spectrum of energy
2.2
Turbulence Models
The following descriptions are brief summaries of [8], in which some sizes of turbulence are defined:
• The turbulent kinetic energy k = √2
u2+ v2+ w2= 3 2(TiV∞)2 • Dissipation of turbulence ε = C( 3 4) µ k 3 2 l • Dissipation rate ω = kε
with l the Prandtl mixing-lenght, depending on the geometry, Titurbulence intensity
and Cµ= 0.09.
2.2.1 URANS
Substituting in Navier-Stokes equation, the velocity with Reynolds decomposition writ-ten as: u = ¯U + u’ (where ¯U is the time-averaged quantity and u’ is the temporal fluctuation), it obtains the Reynolds Averaged Navier-Stokes equations:
Mass Flow:
∂ ¯Ui
∂xi
Momentum Equation: ∂ ¯Ui ∂t + ¯Uj ∂ ¯Ui ∂xj = −1 ρ ∂ ¯p ∂xi + 1 ρ ∂ ∂xj (τij + τijt |{z} Reynolds−Stress ) (2.4)
To close the equations the Reynolds-Stress τijt has to be modeled. Boussinesq pro-posed in 1877 that Reynolds stresses are proportional to the mean rates of deformation [8], similarly to the shear stress produced by molecular diffusion in Newton’s law.
τij = µ ∂Ui ∂xj +∂Uj ∂xi = 2µSij (2.5) τijt = µt ∂ ¯Ui ∂xj +∂ ¯Uj ∂xi −2 3ρkδij | {z } Deviatoric P art = 2µtS¯ij (2.6)
Where k is turbulent kinetic energy. To model the eddy viscosity µt, different
ap-proach of growing complexity have been proposed in the literature and briefly reviewed in this section.
Model 0 Equations
These models express the eddy-viscosity with an algebraic model as a function of the shear rate in the averaged flow. The first model was proposed by Prandtl in 1925:
• Mixing Length Model
The turbulent kinematic energy is contained in the largest eddies, those which have length scale lm (m) (dimension of eddies), velocity scale ϑ[m/s] and interact with
the mean flow. For these reasons, the Reynolds-Stress is described with properties of the mean flow. Through the dimensional point of view the eddy-viscosity can be written like µt = ρvtlt, where vt and lt have respectively a velocity and a
length dimension. In fact, eddies are linked to main flow : ϑ = lm
∂ ¯u ∂y . Prandtl considered a simple bi-dimensional flow like Shear flow, starting from the three main properties:
– ¯U is the mean velocity. – ¯U ¯V
– ∂ ¯∂yU is the only not negligible gradient component
– τijt = −ρu0v0 is the only not negligible component of Reynolds-Stress
Prandtl proposed this dimension analysis:
Turbulent viscosity
Cinematic turbulent viscosity
µt= ρϑlm [Kg/m s]
But since ϑ is known, the result is:
τijt = −ρ ¯u0v0 = µ t ∂ ¯U ∂y = ρlm 2∂ ¯U ∂y ∂ ¯U ∂y (2.7)
The Reynolds-Stress is proportional to shear of the mean flow ∂ ¯∂yU and the length lm, which is a function of the flow topology and must be empirically determined for
each flow configuration. For example in the incompressible boundary layer there is a different eddy-viscosity for each layer:
1. In the logarithmic region, lm = κy where κ is the universal constant of Von
Karman (κ = 0.41).
2. In the viscous sub-layer, the logarithmic law and the mixing length model does not work. Van Driest proposed in 1956 a modified Prandtl model called Van Driest correction, that is validated also in the viscous sub-layer.
lm= κy h 1 − e− y+ A0+ i
These models give optimum results for flow characterized by thin shear layers such as jet, mixing layers and boundary layer. The advantages are low computational cost and the fact that they have largely been studied and optimized. Instead they are not applied for separate flow which involves recirculation.
Model 1 Equations
From the Reynolds transport equation, it is possible to obtain the kinetic turbulent transport equation. Resolving it, it is possible calculate k and then the eddy-viscosity with Prandtl-Kolmogorov model:
k = 1
2uiuj νt= lt √
k
• Spalart-Allmaras Model
More recently in 1992, Spalart-Allmaras proposedd a model which included 8 ar-bitrary coefficients and 3 empirical dumping functions. This model involves one transport equation for kinematic eddy viscosity parameter ˜ν and a length scale by means of an algebraic formula. The (dynamic) eddy viscosity is related to ˜ν by :
The equation contains the wall-damping function fν1 = fν1 ˜ ν ν = ( 1 Re → ∞
0 near the wall , but this quantity is calculated by a transport equation:
∂ρ˜ν ∂t + ∇ · (ρ˜νU) = 1 σν ∇ · h(µ + ρ˜ν)∇˜ν + Cb2ρ ∂ ˜ν ∂xk ∂ ˜ν ∂xk i + Cb1ρ˜ν ˜Ω −Cw1ρ ˜ν κy fw (2.9)
where κ is Von Karman’s constant, ˜Ω = Ω+(κy)˜ν 2fν2, Ω =p2ΩijΩij = mean
vortic-ity Ωij = 12 ∂ ¯Ui ∂xj+ ∂ ¯Uj ∂xi
= mean vorticity Tensor and fν2 = fν2
˜ ν ν fw = fw ˜ ν ˜ Ωκy2 wall-damping function. The model has been modified by the wall-damping function to be adapted to hybrid simulations like RAS/LES for separate fluxes. Compared with zero model equation, it has another partial differential equation to solve plus empirical damped function, so the computational cost is higher.
Models 2 Equations
These models are based on the eddy-viscosity hypothesis and they consist in two partial differential equations, written for two characteristic magnitudes of the tur-bulence; the first one usually for the kinetic turbulent energy per mass unit (k) and the second one is different depending on the model. The eddy-viscosity is obtained with dimensional considerations.
• URANS : k − ε Standard (Jones & Launder, 1972)
This turbulence model is the most commonly used in CFD to account for history effects like convection and diffusion of turbulent energy. The first transported equation is for turbulent kinetic energy, k per mass unit. The second transported equation in this case is for the turbulent dissipation, ε. The terms of this equation are very difficult to model because they are hard to interpret from a physical point of view and to obtain experimentally. For these reasons, the ε equation is highly empirical. ε determines the scale of the turbulence, whereas k, determines the energy in the turbulence.
Starting from a dimensional consideration, the eddy viscosity is written in eq.(2.10):
νt= Cµ k2 ε (2.10) ∂ρk ∂t + ∇ · (ρkU)| {z } Convective−term = 2µtS¯ij · ¯Sij | {z } P roduction−term Pk − ρε |{z} Dissipation−term +∇ ·hµ + µt σk ∇ki | {z } Dif f usive−term (2.11)
∂ρε ∂t + ∇ · (ρεU) = C1ε ε k2µtS¯ij· ¯Sij− C2ερ ε2 k + ∇ · h µ + µt σε ∇εi (2.12) where ¯Sij = 12 ∂ ¯Ui ∂xj + ∂ ¯Uj ∂xi
is the strain rate, µt = 0.09ρk
2
ε is the turbulent
vis-cosity, C1ε = 1.44, C2ε = 1.92, σk = 1.00, σε = 1.30 are adjustable constants.
These closure terms are determined from the calibration with known properties of canonical flux. The ε-equation is not well posed near the wall, where ε and k tend to zero (in particular the destruction term (Eq.2.12) εk2 could diverge. Where the boundary layer with high adverse gradient of pressure is present the model does not work and in general it has to be set for each new application.
• URANS : k − ω (Kolmogorov, 1942, Wilcox, 1988)
In the second equation model k − ω is exactly the same, the first equation is to k and the second transport equation is for the turbulence specific dissipation, ω[s−1]. This type of model arise from the k−ε substituting ε with ω, the advantage is which the model has a good behaviour near the wall and it does not require wall function for High Reynolds applications. The eddy-viscosity is written in the eq.(2.13):
νt= k ω (2.13) ∂ρk ∂t + ∇ · (ρkU)| {z } Convective−term = 2µtS¯ij· ¯Sij | {z } P roduction−term Pk − ρβ∗kω | {z } Dissipation−term +∇ · h µ + µt σk ∇ki | {z } Dif f usive−term (2.14) ∂ρω ∂t + ∇ · (ρωU) = α ω k2µtS¯ij · ¯Sij− βρω 2+ ∇ · hµ + σµ t ∇ωi (2.15)
Where the coefficients values are α = 59, β = 403 , β∗= 1009 , σk= 2 and σ = 12.
This model is sensible to the free-stream flux, where ω tends to zero.
• URANS : k − ω in k − ε
The basis of this technique is the transformation of the k − ε model to a kω formulation. This is an exact conversion, except for small contributions from the diffusion term due to the difference in the diffusion coefficients of the k and ε equations [19]. ∂ρk ∂t + ∇ · ρkU = ∇ · h µ + µt σk ∇ki+ Pk− β∗ρkω (2.16)
∂ρω ∂t + ∇ · (ρωU) = ∇ · h µ + σµt ∇ωi+ Cα2 ω kPk− Cβ2ρω 2+ 2σ ω2 ρ ω∇K · ∇ω (2.17) The differences between this formulation and the original k − ω are in the last term of the new formulation in the eq.(2.17), which is an additional cross-diffusion and the modeling constants that are given by Cα2 = 0.4404, Cβ2 = 0.0828, σk2 = 1.0,
σω2= 0.856.
• URANS: k − ω SST (Menter 1993 )
The term that appears in the previous model, is taken to create this new model. The SST k −ω turbulence model is a two-equation eddy-viscosity model. The shear stress transport (SST) formulation combines the best of two turbulence models. The use of a k − ω formulation in the inner parts of the boundary layer makes the model directly usable all the way down to the wall through the viscous sub-layer, hence the SST k − ω model is used as a Low-Re turbulence model without any extra damping functions. The SST formulation also switches to a k − ε behaviour in the free-stream and thereby avoids the common k − ω problem that the model is too sensitive to the inlet free-stream turbulence properties. The SST k − ω model does produce a bit too large turbulence levels in regions with large normal strain, like stagnation regions and regions with strong acceleration. This tendency is much less pronounced than with a normal k − ε model though.
∂ρk ∂t + ∇ · (ρkU) = ∇ · h µ + µt σk ∇ki+ Pk− β∗ρkω (2.18) where Pk = 2µSij · Sij − 23ρk∂U∂xjiδij
is the production rate of turbulent kinetic energy (δij is the Kronecker delta), µt= ρωk and β∗ = 0.09 an adjustable constant.
∂ρω ∂t + ∇ · (ρωU) = ∇ · h µ + µt σω,1 ∇ωi+ γ2 2ρSij · Sij− 2 3ρω ∂Ui ∂xj δij −β2ρω2+ ρ σω,2 ω ∂k ∂xk ∂ω ∂xk (2.19)
with the costants: σk = 1.0, σω,1 = 2.0, σω,2 = 2.0, γ2 = 0.44, β2 = 0.083. The
seventh term contains a blending function which allows for switching between the k − ε and the k − ω model:
Where FC = FC
lt
y, Rey
is a function of the ratio between lt = √
k
ω and the
distance y from the wall, together with the turbulence Reynolds number Rey = y
2ω
ν .
Fc=
(
Fc→ 0 At the wall , k − ω
Fc→ 1 F ar away f rom the wall, k − ε
The blending function ensures a smooth transition between the two models.
Eddy Viscosity Limitation The eddy viscosity is limited to improve the perfor-mance in flow with adverse pressure gradients and wake region
Pk= min 10β∗ρkω, 2µtSij· Sij − 2 3ρk ∂Ui ∂xj δij µt= a1ρk max(a1ω, SF2)
with S =p2SijSij, a1 = constant and F2is a blending f unction
2.2.2 Large Eddy simulation and others approaches
Navier-Stokes equations are spatially filtered ( ¯u0) in a region with the same
dimen-sion of the discretization numerical size and they resolve directly only the big scale of turbulence (bigger than the discretization used).
LES
The aim of LES is to study the behaviour of the largest eddies, which are considered more anisotropic and not well modelled contrary to the small eddies, which have a universal behaviour, are isotropic and easier to model. This is the essence of the Large Eddy Simulation (LES) approach to the numerical treatment of turbulence. Instead of time-averaging, LES uses a spatial filtering operation to separate the larger and smaller eddies. The method starts with the selection of a filtering function and a certain cut-off width with the aim of resolving in an unsteady flow computation of all those eddies with a length scale greater than the cut-off width.
∂ ¯ui ∂xi = 0 (2.21) ∂ ¯ui ∂t + ∂( ¯uiu¯j) ∂xj = −1 ρ ∂ ¯p ∂xi + ν ∂ 2u¯ i ∂xjxj − ∂τij ∂xj |{z}
sub−grid scale stress
(2.22)
Smagorisky Model τij − δij 3 τkk= −2νsgsSij = −νsgs ∂ ¯ui ∂xj +∂ ¯uj ∂xi (2.23)
The length scale of the subgrid motions is made proportional to the filter width, lsgs = Cs∆ where Cs is the Smagorisky constant and needs to be specified. The
subgrid viscosity 2.24 is obtained by assuming that the subgrid scales are in equi-librium (production and dissipation subgrid energy are in balance). This leads to this expression, νsgs = (Cs∆)2 S , S = (2 ¯SijS¯ij) 1/2 (2.24)
Integrating the eq.2.23 with eq. 2.24, it is possible to derive:
τij− δij 3 τkk= (Cs∆) 2 S , S = (2 ¯Sij ¯ Sij)1/2 (2.25)
The constant C in eq.2.24 has to be assigned a priori, it changes with the flux type. Near the wall, the constant is not well posed, to solve this problem other models have been invented.
DES
DES models attempt to treat near-wall regions in a RANS-like manner, and to treat the far-field regions in an LES-like manner. DES was designed to keep a lower computational cost than LES approach, by avoiding simulation the need of a fine grid next to the wall (where velocity gradients and turbulence production dominates). Several turbulence models can be used in the DES formulation. The first approach was the Spalart-Allmaras (S-A) one equation model. More recently DES couples with Menter k − ω SST , two equation model. More details can be found in the following equations.
1. Spalart-Allmaras Model (S-A)
The one-equation Spalart-Allmaras model eq.(2.9) (for νt) is used for the
RANS part. In this model the length scale is modified to control the distri-bution of νt, in this mode:
δ = min(d, CDES∆) (2.26)
where δ is the distance from the wall, CDES is arbitrary parameter, ∆ is
the maximum local dimension (in three directions) of the mesh. δ is the discriminant to decide which models to use.
(a) Near the wall, where δ is equal to d, there is the classic Spalart-Allmaras Model.
(b) Far from the wall, the LES Model is used, whe δ is equal to CDES∆.
2. Shear Stress Turbulent Model (SST)
∂ρk ∂t + ∇ · ρk − → U = ∇ · µ + µt σk ∇k] + Pk− β∗ρkω (2.27) Pk= 2µSij · Sij− 23ρk∂U∂xi jδij ∂ρω ∂t + ∇ · ρω − → U = ∇ · µ + µt σω,1 ∇ω] + Pω− βρω2+ ρ σω,2 ω ∂k ∂xk ∂ω ∂xk (2.28) Pω = γ2 2µSij · Sij − 23ρω∂U∂xjiδij with : σk = 1, σω,1 = 2, σω,2 = 2, γ2 = 0.44, β = 0.083
Following the same aim of the previous equation for S-A model, it is possible to define the SST-RANS model with the length scale expressed in terms of k and ω:
l(k−ω)=
k11
β∗ω
And with the same focus of S-A model, the DES length scale will be expressed as: ˜ l = min(lk−ω, CDES∆) D˜ν Dt = c| {z }b1S ˜˜ν P roduction T erm − cω1 hν˜ ˜ d i2 | {z } Distruction term + 1 σν h ∇(ν + ˜ν)∇˜ν + cb2|∇˜ν|2 i cb1S ˜˜ν = cω1 hν˜ ˜ d i2 and consequently ν ∝ ˜˜ S ˜d2 ˜
d = min(d, CDES∆), ∆ = max(∆x, ∆y, ∆z)
(
d ∆ S − A Rans M odel
The Finite Volume Discretization
in CFD
This Chapter addresses the problem of discretization. The purpose of any discretization practice is to transform one or more partial differential equations into a system of alge-braic equations. The solution of this system produces a set of values which correspond to the solution of the original equations at some pre-determined locations in space and time[20][9]. The discretisation process can be divided into two steps: the discretisation of the solution domain (Mesh) and the discretization of the equations, shown respec-tively in the Sections 3.1 and ??. In the Section 3.2, the discretized equations are shown from an analytical point of view and with their implementation in openFOAM In the last Section 3.4, the Pressure Implicit with Splitting of Operators (PISO) solver is explained.
3.1
The Discretisation of Domain
The discretization of the solution domain produces a numerical description of the compu-tational domain. The space is divided into a finite number of discrete region, called cells. Then the conservation equations are solved on these discrete non-overlapping cells that completely fill the computational domain to yield a grid or mesh system. These meshes are classified according to several characteristics: structure, orthogonality, blocks, cell shape, variable arrangement, etc. A distinction for time and space discretisation must be done. In fact, for time discretisation, it is sufficient to set the size of the time-step used during the simulation.
The space discretisation for the Finite Volume Method (FVM) is obtained with a subdivision of the domain into control volumes (CV) Fig.(3.1).
Z
V p
(~x − ~xp)dV = 0 (3.1)
The control volume is bounded by flat faces in common with the other neighbouring CV. The cell faces in the mesh can be divided in two groups
Figure 3.1: Control Volume Discretition. Taken from [11]
• Internal: the faces between the CV
• Boundary: the faces which coincide with the boundary of domain
For each face area, too, a vector called S is defined, which is normal to the face and has the magnitude of the area of the abovementioned face.
3.1.1 Accuracy of Discretization
The Navier-Stokes equation is a second order equations, according to the second space derivatives from diffusion term. On the one hand, good accuracy means to discretise it with equal or higher order schemes, on the other hand, in certain parts of the discretiza-tion, it is necessary to relax the accuracy requirement for two principal reasons: mesh irregularity and preservation of the boundedness of solution. The degree accuracy of the discretisation method depends on the assumed variation of the function φ = φ(~x, t) in the space and time at point P. In order to obtain a second order accurate method, this variation must be linear in both space and time, for practical reason a linear first order variation along the x direction has been considered around the point P:
φ(x) = φ(xP) + (x − xP) · ∂φP ∂x (3.2) φ(t + ∆t) = φ(t) + ∆t∂φ ∂t t (3.3)
Take into the consideration the Taylor expansion series in space of a function around the point x along x direction, it is possible to increase the order of the discretization:
φ(x) = φ(xi) + (x − xPi) ∂φ ∂x i+ (x − xPi)2 2! ∂2φ ∂x2 i+ ...(x − xPi) 3 3! ∂3φ ∂x3 i+... + (x − xPi)n n! ∂nφ ∂xn i+ H (3.4) t= (∆x)n n! (3.5)
Depending on the order of the scheme chosen, there is an error of truncation eq.(3.5) proportional to the following term after the order chosen. That error is proportional to the mesh dimension in the case of space discretisation.
3.2
Discretization Methods and OPENFOAM
OpenFOAM is an object-oriented C++ framework that can be used to build a variety of computational solvers for problems in continuum mechanics with a focus on finite volume discretization [19]. OpenFOAM provides a Finite Volume method to solve transport equations for a generic scalar φ of the form :
∂ρφ ∂t | {z } U nsteady−term + ∇ · (ρUφ) | {z } Convective−term − ∇ · (ρΓφ∇φ) | {z } Dif f usive−term = Sφ |{z} Source term (3.6)
is basically written in OpenFOAM as (3.2)
( fvm::ddt(phi) + fvm::div(mDot,phi) - fvm::laplacian(Dphi,phi) == fvm::Sp(p,phi) - fvc::(c) );
Table 3.1: OpenFOAM discretization mode
Objects Type of Data OpenFOAM Class Interpolation Differencing Schemes surfaceInterpolation Explicit discretisation:
Differenzial operator ddt,div,grad fvc:: Implicit discretization:
1. FVC:: The explicit operator fvc, named ’finite volume calculus’, returns an equiv-alent field based on the actual field values. For example the operator f vc :: div(φ) returns an equivalent geometricField (Field + mesh + boundaries + dimension) in which each cell contains the value of the divergence of the variable (φ) [19].
2. FVM:: The fvm implicit operator, instead, defines the implicit finite volume dis-cretization in terms of matrices of coefficients. For example fvm::laplacian(/) re-turns an fvMatrix in which all the coefcients are based on the finite volume dis-cretization of the Laplacian [19].
3.2.1 Spatial Discretization
To understand the discretization first, the steady equation (3.6) has to be integrated in the control volume.
Z VC ∇ · (ρUφ) − Z VC ∇ · (ρΓφ∇φ) = Z VC Sφ (3.7)
The generalised form of Gauss’s theorem will be used to transorm the volume inte-grals of Convective and Diffusive term in surface inteinte-grals, this process involving these identities: Z V ∇ · a dV = I ∂V a dS Z V ∇φ dV = I ∂V φ dS (3.8)
So the eqn. () will become: I ∂Vc ∇ · (ρUφ) · n dS = I ∂Vc ∇ ·ρΓφ∇φ· n dS + Z V c SφdV (3.9) I ∂Vc ∇ · (ρUφ) · n dS = X f aces(V c) Z f (ρUφ) · n dS (3.10) I ∂Vc ∇ ·ρΓφ∇φ· n dS = X f aces(V c) Z f (ρΓφ∇φ) · n dS (3.11) The integral of all the volume surfaces will become, the sum of the surfaces flux trough the faces belong to the CV as shown in Fig.(3.2). When the flux has calculated on the face, it must be known the values on all the points of the face but it can be done cell centered approximation on the face in other words the approximation has done with the center values of the cells that share the face. The same has been done for the volume integral, in fact to discretize it, the mean theorem value occurs:
QC =
Z
VC
QφdV = ¯QφV
Figure 3.2: Control Volume. Taken from [8]
Where QC is the value of Q in the volume of the center.
Gradient Schemes
The gradient discretization is important, in fact it is used to calculated in the momen-tum equation ∇p. For the different types of mesh, the discretization of the gradient is different. Taking a uniform mesh, it is possible discretized the gradient simply with a Central-Difference scheme: dφ dx f= φF − φC xF − xC
Figure 3.3: Calculated Gradient for uniform one dimension mesh. Taken from [8].
Gauss-Green Gradient
This method is the most used for a general typology of mesh. The value of the gradient in the center (C) of the cell is evaluated with the Mean-Theorem value:
∇φC = 1 VC X f aces(C) φfSf (3.13)
Where f refers to the face and Sf is the vector belongs to the face. So the value φf
on the face must be defined.
Figure 3.4: CF cross the centroid f of the face Sf. Taken from [8] .
This approach used by OpenFoam is based on the face with a compact stencil. This method is extendible in two or three dimension, the value of φf is the average of the
value of the cells center which share the face where the value will be calculated as in Fig.(3.3):
φf = λCφC+ (1 − λC)φF (3.14)
Where λC is the geometric weight factor, otherwise the ratio of the distance between
the point F and the centroid of the face f and the distance between the points C and F . λc= ||rF− rf|| ||rF− rC|| = dF f dF C (3.15)
It can happen which the mesh is unstructured, so non-orthogonal. So the segment CF could be cross the face centroid f , therefore a different approach has to be adopted Fig.(3.3). In this case, the value needs a correction, doing an interpolation of the value in the point f with φf0 to obtain φf.
The expression above can be written in the extended form:
φf = λC[φC+ (∇φ)C· (rf − rC)] + (1 − λC)[φF + (∇φ)F · (rf − rF)] (3.17)
The term λC is a function of the position f0 on the segment CF . The point f0 can
be considered in three different position:
a. In the first option, the point f0has been taken as the intersect between the segment CF and the surface Sf. Defining the vector n = ||SSff||, as the unitary vector of
the face Sf and the vector e = ||CF||CF of the segment CF , the position of f
0 can be
found supposing the orthogonality between the vector n and the segment f f0 : (rf − rf0) · n = 0 (3.18)
Figure 3.5: First approach: f0 is the point of the intersect between the segment [CF] and the face Sf. Taken from [8].
To find the point f0 on the vector CF, the vector Cf0 can be expressed in term of the vector e:
Cf0 = (rf0− rC) = ke (3.19)
where k is a scalar quantity. Combined the equations (3.18) and (3.19), it has been obtained:
rf0 = rf
· n
e · ne (3.20)
Then, it can be possible to calculate:
λc= ||rF− rf0|| ||rF− rC|| = dF f0 dF C (3.21)
There is a procedure with it is possible to calculate the value of φf, which is listed
here:
Calculate φf0 using φf0 = λcφC+ (1 − λC)φF
Calculate ∇φC using ∇φC = V1C Pf aces(C)φf0Sf
Now, the correction of the gradient has done using:
φf = φf0 + λC(∇φ)C· (rf − rC) + (1 − λC)(∇φ)F · (rf − rF)
Finally, the value calculated before, it is used again: ∇φC = V1C Pf aces(C)φfSf
b. The position of the point f0 is chosen in the center of the segment CF as shown in the Fig.(3.6). This choice simplified the equations, since the stencil is immediately calculated because the position of the point f is known.
Calculate φf0 using φf0 = φC+φF
2
Calculate ∇φC using ∇φC = V1C Pf aces(C)φf0Sf
Now, the correlation of the gradient has done using:
φf = φf0 + λC(∇φ)C· (rf − rC) + (1 − λC)(∇φ)F · (rf − rF)
Finally, the calculated value is used again ∇φC = V1C
P
f aces(C)φfSf
Figure 3.6: Second approach: The position of f0 is in the middle of CF . Taken from [8] c. The position of the point f0 can be chosen minimizing the perpendicular distance of the segment f f0 from CF Fig.(3.7). So it is possible to write the position of f in the following mode:
rf0 = rC+ q(rC− rF) (3.22)
where 0 < q < 1. The length of vector f f0 is a function of the previous equation. To write the distance:
Figure 3.7: Third approach: The point f0 is on the segment CF and the length of the segment f f0 is minimized. Taken from [8]
d2= (rf − rf0) · (rf − rf0)
= (rf − rC− q(rC− rF)) · (rf − rC− q(rC− rF))
= (rf − rC) · (rf − rC) − 2q(rf − rC) · (rC− rF) + q2(rC− rF) · (rC− rF)
(3.23)
Minimizing the distance d means:
∂(d2)
∂q = 0 ⇒ = −2(rf − rC) · (rC− rF) + 2q(rC− rF) · (rC− rF) = 0 (3.24) So it is obtained the value q that minimized the distance:
q = −rCf· rCF
rCF· rCF (3.25)
Also in this case, it is possible to use a procedure to calculate the value φf:
Calculate rf0 using rf0 = rC − rCf·rCF
rCF·rCF(rC− rF)
Calculate λC using λC = ||rF− rf0||/||rF− rC||
Now, the correction of the gradient is: φf0 = λCφC+ (1 − λC)φF
So, the calculated value will be used again: ∇φC = V1CPf aces(C)φf0Sf
Calculate ∇φf0 using ∇φf0 = λC∇φC+ (1 − λC)∇φF
found the valuo of φf0 upgrade φf = φf0 + ∇φf0 · (rf − rf0) So, finally it can
use again the formula:
∇φC = 1 VC
X
f aces(C)
φfSf
In OpenFOAM, the gradSchemes sub-dictionary contains gradient terms.
• Gauss linear: it means to specify the standard finite volume discretisation of Gaussian integration which requires the interpolation of values from cell centres to face centres. The interpolation scheme is then given by the linear entry, meaning linear interpolation or central differencing.
gradSchemes {
default Gauss linear }
• cellLimited Gauss linear: The discretisation of specific gradient terms is over-ridden to improve boundedness and stability in cases that are involving poorer quality meshes, adding the cellLimited for the grad(U). In fact the cellLimited scheme which limits the gradient such that when cell values are extrapolated from the faces using the calculated gradient, the face values do not fall outside the bounds of values in the surrounding cells.
gradSchemes {
grad(U) celllimited Gauss linear 1; }
where the coefficient (
1 guarantees boundedness 0 applies no limiting Not used frequentely:
• leastSquares: a second-order, least squares distance calculation using all neigh-bour cells.
• Gauss cubic: third-order scheme that appears in the dnsFoam simulation on a regular mesh.
Here, the standard finite volume discretisation of Gauss integration with linear in-terpolation (central difference) has been chosen.
3.2.2 Convective term
The convective term, referring to the general expression (3.8) is:
Z V p ∇ · (ρUφ) dV = X f aces ρUφ fS = X f aces ρU fφfS = X f aces F φf (3.26)
Where F in the eq.(3.26) rapresents the mass flux through the face:
F = (ρU)f· S (3.27)
The flux is calculated from interpolated values of ρ and U. The interpolation mode and the order of accuracy for ρ and U have to be chosen in OpenFOAM. In fact, in addition to div(phi,U) and Gauss linear, which indicates the Gauss theorem method, the interpolation scheme has to be specified. The eq. 3.26 also requires the face value of the variable φ calculated from the values in the cell centres, which is obtained using the convection differencing scheme. A convection term peculiarity is not to produce minor or major values compare with the initial values imposed, so as to preserve the boundedness (criterion for sufficient condition for a convergent iterative method).
Numerical Diffusion
The convective term discretization must be analysed as a result of the problem of numer-ical diffusion. To study that it has been compared the analytnumer-ical solution of the steady one dimension Convection-Diffusion problem with the numerical one:
d(ρu) dx = 0 d(ρuφ) dx − d dx Γφdφ dx = 0
Integrating both the equations:
ρu = cost
ρuφ −Γφdφ dx
= c1
Where c1 is an integration constant, which is function of the boundary condition.
dφ dx = ρuφ Γφ − c1 Γφ
Therefore with a variable manipulation, it is possible to write:
Φ = ρuφ Γφ − c1 Γφ dΦ dx = ρu ΓφΦ
Then the partial derivative equation has this analytical solution: dΦ Φ = ρu Γφdx ⇒ ln(Φ) = ρu Γφx + c3 ⇒ Φ = c2e ρu Γφx
Now, it is possible to express al the equation as a function of previous variable φ substituting Φ with the previous equation:
φ = c2Γ
φeΓφρux+ c 1
ρu
Now, the analytical solution can be imposed between two point as depicted in Fig.(3.8)
Figure 3.8: Analytical solution of the steady one dimension Convection-Diffusion prob-lem imposed between the nodal point of the uniform grid. Taken from [8].
so, the analytical solution becomes:
( φ = φWat x = xW φ = φEat x = xE ⇒ φ − φW φE − φW = e P eLx−xWL − 1 eP eL− 1 (3.28)
where P eLis the local Peclet number, which is the ratio between the convection term
contribution of the variable φ and the diffusion term contribution of the variable φ:
P eL=
ρuL
Γφ L = xE − xW
As shown in the Fig.(3.9), the solution of the variable φ between the nodal point W and E changes: it is a linear profile for P eL= 0 ( the problem is purely Diffusive) to a
smoothed step for P eL= 100.
Soluzione Analitica VS Soluzione Numerica
Firstly, the numerical solution has solved then compared with the analytical one. Start-ing from a steady one dimension Convection-Diffusion problem without any source term, it has been obtained with the Gauss theorem:
