Politecnico di Milano
Scuola di Ingegneria Industriale e dell'Informazione
Corso di Laurea Magistrale in Ingegneria Aeronautica Dipartimento di Scienze e Tecnologie Aerospaziali
A CFD's modeling of a screen considering
as a porous media
Relatore: Prof. Alberto Zasso Correlatore: Ing. Paolo Schito
Tesi di laurea di:
Luca Buscemi Matr. 883321
I dedicate this paper to my missing grandparents, which always have believed in me.
Acknowledgments
I would rst like to thank my thesis supervisor and co-supervisor Prof. Alberto Zasso and Ing. Paolo Schito of the Politecnic of Milan.
The door to Prof. Alberto Zasso and Ing. Paolo Schito o ce was always open whenever I ran into a trouble spot or had a question about my research or writing.
They consistently allowed this paper to be my own work, but steered me in the right the direction whenever he thought I needed it.
I would also like to thanks all my friends, in particular Simone for the support in this years of study.
I would also like to thanks also my uncles and my cousin, in alphabetic order Annalisa, Davide, Marino, Matteo, Patrizia for the constant presence.
Finally, I must express my very profound gratitude to my parents Claudio and Ivana and to my girlfriend Federica for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this thesis. This accomplishment would not have been possible without them. Thank you.
Abstract
The possibility of using a CFD simulation to solve a real case could be very interesting for predicting the behaviour of the wind in proximity of some structure. In particular in case in which it isn't possible to test the real case in a wind tunnel. In this discussion will be treated the modellation of a screen using OpenFoam, since the screen are fundamental in some applications.
An innovative application is to place before of facades of buildings a grid. The grid is positioned in front of facades in order to produce a pressure drop and consequently to reduce the aerodynamic stress on the building. At the same way the aerodynamic forces are partially transferred to the grid's supporter, which means a lower required overextimation of stress and strain on the buildings. In terms of cost the full structure will be certanly cheaper.
The positioning of these types of grid isn't strictly in front of facedes of buildings. There are a lot of elds of application in which could be usefull its positioning in particular in places where the strong wind is a fundamental variable of engineering design. This type of screens are used also in wind tunnel in order to change the ow direction.
The scope of this Thesis is coding a new CFD's solver on OpenFoam which will model all types of screens, starting from an existing one: porousSimpleFoam.
Actually pourosSimpleFoam is used to model porous media, for example the phenomena of water penetration into the ground or to simulate some material with porosity proprietes as gauzes.
It's avaiable a wide range of dierent algebraic models within OpenFoam, which are used to model the added source term of Navier-Stokes equations. In this case Darcy-Forchheimer model is used as starting point.
The current Darcy-Forchheimer model is 3D and evaluates only the stress on the principal axis of inertia.
Obviously when the wind pass through the grid it's necessary to take account of the components of velocity out of principal axis of inertia, for this reason the Darcy-Forchheimer matrices (F and D) won't be diagonal.
The values of matrices terms are obtained in two ways : Openfoam simulations of ow through screens ( sempli ed or modelled on CAD) or from experimentally
The advatage of modelling a ow through a grid with a solver ,that just incor-porate the porosity eects of the grid, is principally the lower computational cost.
In fact real case the screens could be of the size of buildings, which means that the requested cells to generate a mesh is very huge. If it's not necessary to model a grid, the amount of cells used will be very smaller with respect to a case in which it's necessary a modellation of the grid.
In the end the solver will be validate with a comparison of the obtained results, both experimental and from CFD simulations. And nally the model is tested on a real case, a building, in order to compare the two results : with the physical grid and the new solver.
Sommario
La possibilità di utilizzare una simulazione CFD per risolvere un caso reale potrebbe essere interessante al ne di predire il comportamento del vento attorno a strutture. In particolare nei casi in cui non è possibile testare il caso reale in galleria del vento.
In questa trattazione verrà spiegato come poter modelare una griglia utilizzando OpenFoam, poichè le griglie hanno un importante ruolo in molte applicazioni.
Una delle applicazioni piu innovative è posizionare una griglia davanti alle facciate di edi ci.
La griglia è posizionata davanti alle facciate di edi ci al ne di produrre una caduta di pressione e di conseguenza ridurre gli sforzi sull'edi cio. Allo stesso modo le forze aerodinamiche sono trasferite parzialmente ai supporti della griglia, il che comporta una minor sovrastima di sforzi e deformazioni sull'edi cio.
In termini di costi l'intera strutture avrà un costo minore.
Il posizionamento di questi tipi di griglie non è strettamente davanti a facciate di edi ci. Ci sono parecchi campi di applicazione in cui potrebbe essere utile il loro posizionamento, in particolare dove il vento forte è una variabile fondamentale della progettazione. Questo tipo di griglia viene anche utilizzato in galleria del vento per cambiare la direzione del usso.
Lo scopo di questa Tesi è programmare un nuovo solutore CFD in OpenFoam, partendo da un solutore già esistente : porousSimpleFoam.
Attualmente porousSimpleFoam è utilizzato per modellare mezzi porosi, come per esempio il fenomeno di penetrazione dell'acqua all'interno del del terreno oppure per simulare alcuni materiali con proprietà porose come garze.
E' disponibile un vasta gamma di modelli algebrici all'interno di OpenFoam, i quali sono utilizzati per modellare il termine sorgente aggiuntivo delle equazioni di Navier-Stokes. In questo caso il modello algebrico usato come punto di partenza è Darcy - Forchheimer.
Il modello di Darcy - Forchheimer attuale è 3D e valuta solamente gli sforzi sugli assi di inerzia principali.
Ovviamente quando il vento attraversa la griglia sarà necessario tener conto delle componenti di velocità al degli essi di inerzia principali , per questa ragione le matrici
I valori dei termini delle matrici sono ottenuti in due modi : simulazioni Open-Foam di ussi attraverso griglie (sempli cate o modellate su CAD) o dalle forze misurate sperimentalmente in galleria del vento.
Il vantaggio di modellare un usso attraverso una griglia con un solutore, che ha già incoporato al suo interno gli eetti di porosità della grglia, è principalmente il minor costo computazionale.
Infatti in casi reali le griglie potrebbero avere le stesse dimensioni dell'edi cio, il che comporta un'enorme quantità di celle necessaria a generare la mesh. Se non fosse necessario modellare la griglia, la quantità di celle utilizzate sarebe sicuramente minore rispetto al caso in cui dovrebbe modellare anche la griglia.
In ne il solutore verrà validato tramite una comparazione dei risultati ottenuti, sia sperimentalmente che tramite simulazioni CFD.
Per concludere il modello sarà testato su un caso reale, un edi cio, al ne di confrontare i due risultati : con la griglia sicamente presente e con il nuovo solutore.
Contents
1 Introduction 1
1.1 The physic of a ow through a screen . . . 1
1.1.1 The shape of the real screen . . . 3
1.2 The actual model on OpenFoam : Darcy-Forchheimer . . . 4
1.2.1 The OpenFoam's structure of the model . . . 6
1.3 The Forchheimer coe cient . . . 9
1.4 The experiment setup . . . 12
1.5 Building's phenomena . . . 14
2 The two-dimensional case 19 2.1 The corrections on the OpenFoam's model . . . 19
2.2 The OpenFoam's setup for the 2D case . . . 26
2.2.1 Results of the 2D simulations . . . 29
2.3 The calculation of the terms of inertial tensor Fij for the 2D case . . 32
2.3.1 Matlab implementation of LSM on the overdetermineted sys-tem of equations . . . 34
2.4 Results . . . 35
3 The three-dimensional case 37 3.1 The 3D model of a screen . . . 37
3.1.1 Setup of SnappyHexMesh : mesh generation around the model 41 3.2 Validation of the screen . . . 43
3.3 Calculation of entering values of 3D inertial tensor Fij . . . 47
3.3.1 Matlab implementation: solution of the 3D system of equations 53 3.4 Results . . . 57
4 A real application case 61 4.1 Setup and B.C used to reproduced the wind tunnel test . . . 61
4.2 Results and postProcessing . . . 63
Bibliography 74
A First appendix 75
A.1 The relation between the head loss coe cient Kij and the inertial one
Fij . . . 75
List of Figures
1.1 Section of a screen: generic wind direction and forces . . . 1
1.2 Convention in a generic case . . . 2
1.3 real screen: expanded metal sheet . . . 4
1.4 porous test case diagram . . . 10
1.5 Section view of the wind tunnel setup . . . 12
1.6 forces calculated from wind tunnel experiments: section 600 × 600mm2 13 1.7 A simpli ed wake over a building . . . 15
1.8 Re-circulation zone on the rear of the building :entertainment . . . . 15
1.9 History of pressure measured by taps on the facade . . . 16
1.10 ENI building . . . 17
2.1 scheme of the B.C used for the 2D setup . . . 27
2.2 2D case setup: the shape of the test case . . . 28
2.3 mesh discretization of the test case . . . 28
2.4 pressure and velocity elds at dierent α, 2D cases negative α . . . . 29
2.5 pressure and velocity elds at dierent α, 2D cases positive α . . . . 30
2.6 streamlines in two dierent cases . . . 31
2.7 Results of the 2D cases . . . 36
3.1 . . . 38
3.2 pro le of the CAD screen . . . 39
3.3 mirroring of the initial sketch . . . 40
3.4 lateral view . . . 40
3.5 front view . . . 41
3.6 Screen obtained after snappy and castellated . . . 43
3.7 Results from the validation . . . 45
3.8 Zoom of the central part of the cut screen 600 × 600mm2 . . . . 46
3.9 bounding box : setup of the 3D case . . . 48
3.10 3D run cases: xy plane, positive angle . . . 49
3.11 3D run cases: xy plane, negative angle . . . 50
3.13 3D run cases: xz plane, negative angle . . . 52
3.14 Results of the forces calculated from ow directions on xy plane . . . 57
3.15 Results of the forces calculated from ow directions on xz plane . . . 58
3.16 polar diagram . . . 59
4.1 Setup used to reproduce the wind test experiment . . . 62
4.2 Setup of the porosity walls . . . 62
4.3 Comparison of pressure elds . . . 64
4.4 Comparison of pressure elds . . . 65
4.5 Width of the wake in the xy plane, 0, 2m from the ground . . . 68
4.6 Wall shear stress magnitude in z direction on the front facade . . . . 69
List of Tables
1.1 values of ∆p and u2 selected to test the solver porousSimpleFoam . . 10
4.1 Forces distributions on the building : not porosity case . . . 66 4.2 Forces distributions on the building : porosity case . . . 66 4.3 Velocity measurement using the probes function in postProcess: not
porosity . . . 67 4.4 Velocity measurement using the probes function in postProcess: not
List of Algorithms
1.1 abstract of DarcyForchheimerTemplate.C . . . 7
1.2 abstract of DarcyForchheimer.C . . . 8
2.1 Generic case that use the Darcy-Forchheimer modeling . . . 20
2.2 abstract of modi cation in porousSimpleFoam . . . 21
2.3 modi cation on DF3D.H le . . . 23
2.4 modi cation of DF3D.C . . . 24
2.5 The role of adjustNegativeResistence . . . 25
2.6 les le of Porosity3D . . . 25
2.7 options le of porousSimpleFoam . . . 26
2.8 Matlab function used to implement the LSM in 2D case . . . 34
3.1 optimized snappyHexMeshDict: snappy and castellated controls. . . 42
3.2 core of the Matlab algorithm . . . 56
Chapter 1
Introduction
1.1 The physic of a ow through a screen
For sake of simplicity it's possible to visualize a screen just looking at its section. As illustrated in the Figure 1.1 the section of a screen is an array of at plates positioned ones above the others.
In the showed case the screen is oriented around 45 degrees and passing through it the wind is de ected.
Obviously the forces generated on the screen are the same of which the ow is subjected. For this reason, as it is simply to image, the ow change direction and the new one will be strongly dependent on the boundary conditions of the ow. A detailed approach, both analytical and experimental, could be found in the following papers [3], [11], [4].
In order to be more general, in the Figure 1.1 the boundary conditions aren't speci ed, but in the following chapters it will be treated very carefully. For example, if the ow was a channel ow, as into wind tunnel, the wind direction will tend to be parallel to the walls after the interaction with the screen.
The ow is incompressible in all the applications that will be considered, this approximation is due to very low speed of the wind. In particular the analyzed ow will be an atmospheric one and therefore the maximum speed will not exceed the 100 − 150kmh , that corresponds to a Mach number around 0, 1.
The forces that act on the ow are responsible of a change of momentum balance and since the ow could be treated as an incompressible ow, the pressure before and after the screen will be dierent.
The pressure after the screen will be larger than the upstream one if the screen is oriented as in the Figure 1.1 and the wind direction generates a relative negative angle of attack (where the convention for the sign of the angle of attack is positive if clockwise and negative if counterclockwise).
When the ow passes through the screen, as said before, a force is produced which intensity and direction is strictly dependent on the upstream ow proprieties, in particular the module and the angle of the velocity at the inlet. Obviously an other fundamental propriety is the B.C (boundary conditions), though not intrinsic in the ow, but in an idealize ow where the boundaries are considered to be far away from the screen surely its aren't so important.
As is possible to see in the Figure 1.1 the force produced on the screen, that is the same to which the ow is subjected, is decomposed in two components Fx and
Fy. These two components are always chosen in the direction of x and y axes, so
its don't change direction with the orientation of the inlet velocity. The use of this convention will be more compatible and intuitive when the forces will calculate with OpenFoam. An other important speci cation concerned with the resultant force that is referred to all the screen, therefore isn't the sum of any single airfoil of the screen. For sake of simplicity in the follow gure 1.2 all the conventions are clari ed with a very generic case.
1.1. The physic of a ow through a screen Observing gure 1.2 it is noted that direction of velocity are positive if it is considered the previously introduced convention, and in a generic case the three angle α, β, γ are dierent one from the other.
The angle α is the input of the case and in the reality is the wind direction, which could change since is a random variable.
The angle β is strictly dependent on the screen orientation, in fact in gure 1.2 where, as it is possible to see from the section, the principal direction of the plates is 45 degrees also the angle β will have a value around 45 degrees.
The last angle γ is strictly dependent on the B.C therefore its could be walls or simulate an atmospheric condition.
In a ideal case the ow will be de ected from the screen and should be maintain the direction since the aren't forces that act after the ow pass through the screen and for this reason the momentum has to be the same. Conversely if the are walls the ow will align with its.
The application of a screen in order to generate a pressure drop is very innovative. There aren't example of the same use for the same scope, for this reason the principal issue was to choose how to model screen and which B.C are the best to simulate a real case in OpenFoam.
The application of screens, or perhaps better, perforate plates is recently em-ployed in study of renewable energy from sea waves, as treated by the article [8].
Other applications that concerned with porous media are used to model gauzes, but there aren't applications in OpenFoam that are similar to the idea of model a screen as a porous media.
1.1.1 The shape of the real screen
In order to clarify what is the purpose of this Thesis, it is surely better to show a real screen and in this way understanding how its could work.
As is possible to see from gure 1.3, the screen is a very thin metal sheet with a particular shape. It is made up starting from a metal sheet cut horizontally and then it is stretched until it takes on a form similar to that it is shown in the following gures. Obviously one could get a principal orientation dierent from 45 degrees as in this case.
(a) front view
(b) lateral view
Figure 1.3: real screen: expanded metal sheet
If it is observed carefully the lateral view it isn't so di cult to note that the section of the screen is very similar to the simple sketch of gures 1.1 and 1.2. In fact a screen could be simpli ed as an array of airfoils or better thin plates.
In the following chapters this approximation will be used to understand the be-havior of the screen in a 2D case, but a more detail explanation will be done to the reader.
1.2 The actual model on OpenFoam : Darcy-Forchheimer
The scope of modeling a screen introducing a sort of porosity in the OpenFoam solver has the great advantage: the computational cost.
Therefore the only reason for made up a model of a screen is to save a lot of time and consequently to reduce the economic impact.
In all the discussion what is intrinsic is that the model will be a good approxi-mation of the screen, but it will not be perfectly the same of a physic screen.
The meaning of this state isn't that the model will be a rough approximation of the reality, but conversely it means that the goal of this application is to estimate the behavior of the screen nding the best compromise between the the reality and the saved computational cost.
The actual OpenFoam solver works adding a sink term in the momentum balance of the Navier-Stokes [6], this term has the role of accounting for the presence of a
1.2. The actual model on OpenFoam : Darcy-Forchheimer porous media. In the cited article is proposed an other parameter that modi ed the temporal derivative, but in the following discussion will be considered of unit value and cause of it will be omitted.
To better understanding how is implemented is possible to observed the equation of momentum balance: ∂ ∂t(ρui) + uj ∂ ∂xj (ρui) = ∂p ∂xj + µ∂τij ∂xj + Si (1.1)
the 1.1 is the usual equation of conservation of momentum balance, but the only dierence is the presence of the sink term Si that takes on the role of modeling the
porosity media.
Once it is understood how to include the eect of the screen, viewed as a porous media, the attention is shifted on how to model the sink term.
Fortunately into OpenFoam there are several analytic method which is possible to use to characterize the term Si.
The models that are more used, as is denoted by [6], are the Darcy-Forchheimer and the power law equations.
In order to better understanding the intrinsic physic of the model it is chosen the Darcy-Forchheimer modulation, since it is composed of two parts a viscous loss term and an inertial loss term as shown in the following equation:
Si = −(µDij+
1
2ρ|u|Fij)ui (1.2)
where the two tensors Dij and Fij represent the viscous and the inertial parts,
its dimensions are respectively [ 1
m2]and [m1]. There are two important observations
about 1.2: the rst is that the tensors are implemented as diagonal matrix which means the two index i, j are equal; the second is that the pressure drop generated by the sink term is linearly proportional to the velocity for the viscous term and squared proportional for the inertial term.
The rst observation points out that the model concerns about the only principal inertia axes and consequently there aren't interaction between x, y, z axes. Then, for example, the porosity in the direction x hasn't any eect on the porosity on y, z direction. The three momentum equations are completely decoupled.
In some cases, when the porous media is homogeneous, the terms D and F becomes scalars.
As anticipated, an other way to model the sink term Si is the power law:
Si= −ρC0|u|(C1−1)/2 (1.3)
where C0, C1 are empirical coe cients de ned by the user. The fact that the
understanding the intrinsic physic of the equation. In fact using the power law model is very di cult understand which parts is related to viscous eects and which to inertial ones. In particular, for the scope of this application the viscous part will be neglected because a screen isn't a properly porosity media and it doesn't produce relevant viscous stress, conversely for example to the ground when is irrigated by the rain water.
1.2.1 The OpenFoam's structure of the model
The solver that is used to solve a case with a modeled porosity is porousSimple-Foam that has the same structure of the standard simpleporousSimple-Foam. Remember that is a steady-state solver for turbulent ow of incompressible uids with implicit or explicit porosity treatment [6].
The implicit porosity solver is suppose to be more robust and is needed if the resistances are large, or strongly anisotropic.
The directory where the solver is located is the same folder of simpleFoam, more precisely : $FOAM_SOLVERS/incompressible/simpleFoam/poroussimpleFoam.
In this folder is possible to nd the Make folder and the following les: • UEqn.H
• pEqn.H
• porousSimpleFoam.C • createFields.H
Into the header le UEqn.H and pEqn.H are constructed the momentum equations for p and U.
The great dierence between the simpleFoam and the porousSimpleFoam solver is the presence,at the latest row of the le UEqn.H, of the member function ad-dResistence that has the role to add the source term Si in the momentum equation.
Remark that all the used member functions, as addResistence, are de ned in the porosityModel folder which is located in the src folder, in particular following the niteVolume/cfdtools/general/porosityModel... directory.
The core of the solver is the choice of the model, in the current case Darcy-Forchheimer. In order to make more understandable the discussion in the the fol-lowing chapters, it is analyzed in detail the structure of the model.
1.2. The actual model on OpenFoam : Darcy-Forchheimer Algorithm 1.1 abstract of DarcyForchheimerTemplate.C
In the Darcy-Forchheimer folder there are three les: • DarcyForchheimer.C
• DarcyForchheimer.H
• DarcyForchheimerTemplate.C
The DarcyForchheimer.H is simply the header le, where all the variables are de-clared.
The DarcyForchheimerTemplate.C contains the core structure of the model, in fact in the Algorithm 1.1 it is recognizable the implementation of the equation 1.2.
The rst part is the declarative one, in the second part the code identi es the cells where the porosity was de ned and in these cells it is applied the correction to the source term of the momentum balance.
It's also important to observe that the term U is decomposed in Udiagand Usource,
which contain the momentum equations. The reason of the decomposition is only of reducing the computational cost, in fact the Usourceterm isn't used in the pEqn.H le
cause of Usource is referred to the extra diagonal terms of the momentum equation.
The pressure has only components on principal axes of inertia.
The DarcyForchheimer.C is the executive le where takes place the calculation of forces, the reading of D and F from the porosityProprieties le, into constant folder, as it could be see from algorithm 1.2.
Algorithm 1.2 abstract of DarcyForchheimer.C
The tensors D, F are lled on the principal diagonal of the matrix, that means the three equations are decoupled.
Finally to sum up, the user could use this solver simply selecting the type of mod-ulation (Darcy-Forchheimer, power law...) and entering the coe cients into porosi-tyProprieties le. After the de nition of the porosity zone and the usual blockMesh the case is ready to be run.
At least is important to observe that the input coe cients which the software is waiting for has to be a vector of only three components, the model is set to ll only the diagonal of the tensors D, F .
At now the last question of the reader could be how to determine the value to enter into the tensors D, F , but in this discussion the modulation of the viscous term isn't discussed since for a screen surely can be neglected. Remain unresolved how to ll the inertia tensor F .
1.3. The Forchheimer coe cient
1.3 The Forchheimer coe cient
There are a lot of empirical modulation of the Fij coe cients as reported in the
articles [2], [12] as for example the Erdun one.
But rst of all it is useful a dimensional analysis of the added term Si. As de ned
in the equation 1.2 the sink term Si has the same dimensions of the ∂x∂pi[P am], that it
isn't a great discovery since is a term belonging to the momentum equation.
At now to better understand the way of seeing the equation 1.2 could be useful a comparison with the usual formula of pressure drop [13], seeing that as a function of dynamic pressure and a head loss coe cient:
∆p = qK (1.4)
where the dynamic pressure is de ned as q = 1
2ρu2, and and K is adimensional
head loss coe cient which value depends on the geometry, Reynolds where the ow is passing through.
If the viscous term of the 1.2 is negleted, the equation becomes very similar to the equation 1.4, that means the K coe cient is related to the F coe cient of the Forchheimer law.
The relation isn't just in terms of similarity of the two equations, but if analyzed in detail after some very simple algebraic passages is possible to nd:
K = 2∆xF (1.5)
where ∆x is the length crossed by the ow in the porous media, in A are explained all the algebraic passages.
The relation is linear and the link between K and F is yielded by 2∆x since the equation 1.2 de nes a gradient of pressure, while the 1.4 is referred to a pressure drop.
Therefore if the gradient is approximated using nite elements, ∆p
∆x, the presence
of ∆x in the relation is explained.
This brief introduction has just the role of interpreting the physical meaning of the Forchheimer coe cient, but it doesn't suggest any on what are the possible value to enter in F or K, at now the only way is empirical.
Fortunately the OpenFoam documentation gives an explanation about the Darcy-Forchheimer formulation, which is brief reported in 1D case.
The pressure is suppose to be proportional to the square of the velocity, remember that in the discussion the viscous term is always neglected.
so if it is substituted in the 1.2, the inertial term F assume the following value:
F = 2a
ρ∆x (1.7)
where a is the linear regression of the ∆p and u2.
Therefore with a set of data of ∆p and u2 the value of F could be determined.
In the gure 1.4 is shown the pressure drop, in a very simple test case. The entering values used to calculate the Forchheimer term F are reported in the table 1.1.
Using these values and a ρ = 1, 225kg
m3 and a ∆x = 0, 5m, the value of F obtained
is 2, 211 m.
The results con rm that the value of ∆p is very similar to the one is expected from calculation, in fact the pressure drop is around 58, 8P a when the real one is 60P a. Remind that the pressure that calculate OpenFoam has to be multiplied by the density.
In order to obtain these results it is important to remark that the values entered in the porosityProprieties le are (2, 21 0 0), where the meaning of the values 0 in y and z axes is that there isn't a porosity in these directions.
∆p u2
2,5 15
5 30
7,5 45
10 60
Table 1.1: values of ∆p and u2 selected to test the solver porousSimpleFoam
1.3. The Forchheimer coe cient In the previously treatment the fundamental hypothesis is that F is a scalar, or better a vector of scalars, since the case is monodimensional.
In practice case would be di cult to nd the value of pressure in all the principal directions, if the case is multidimensional.
For this reason it is chosen an approach through the forces, which surely are measurable in a simple way as for example positioning some balances on the screen. An other reason that explains this choice could be nd in the Openfoam imple-mentation of the calculation of forces. In fact in 2D the forces case are calculated as reported in the following equation:
Fi = V ρ|ui| " fxx fxy fyx fyy # ui (1.8) where Fi = " Fx Fy #
is the vector of forces, V is the volume of the porous media and the inertial matrix is written in explicit way as Fij =
"
fxx fxy
fyx fyy
# .
For all the treatment the principal axes are always oriented as the global reference system, they don't change their orientation with the velocity direction at the inlet.
At the end is possible to shown the direct dependency on the direction of inlet velocity: " Fx Fy # = V ρ|u|2 " fxx fxy fyx fyy # " u x |u| uy |u| # (1.9) where " u x |u| uy |u| # = " cosα sinα #
and α is the angle between the inlet velocity direction and the x axis, as reported in gure 1.2.
Then the equation 1.9 becomes: " Fx Fy # = V ρ|u|2 " fxx fxy fyx fyy # " cosα sinα # (1.10) At least it is important points out that the actual Openfoam model have only the possibility to enter three values in the porosityProprieties le, which are the elements on the principal diagonal in a 3D case. So the equation 1.10 can't be implemented at now. Or better it is possible to implemented only the case in which the extra diagonal terms are null.
1.4 The experiment setup
The experiment will use for the validation of the solver, for these reason all the procedure is explained in details.
First of all the dimensions of the wind tunnel where the screen is tested are 2000 × 1500mm2.
The screen is positioned with an angle of 90 degrees with respect to the walls of the wind tunnel and it has the same dimensions of its section. So all the ow pass through the screen since the section of the wind tunnel is entirely taken by the screen.
To better understand the positioning of the screen see gure 1.5:
Figure 1.5: Section view of the wind tunnel setup
As an careful observer could note, the gure 1.5 is exactly the same of gure 1.1 but the screen is rotated 90 degrees around the x axes.
In order to measure the forces and the moments on the screen, from the full section was cut a central part of dimension 600 × 600mm2. This central part of the
screen remain attached to the full screen, but between four balances that measure the forces on the cut screen.
The reason of this setup is measuring the forces only in the central part of the screen, but with the presence and interaction of the full screen. In this way the forces measured could be represent an ideal case where the boundary eects are neglected. In a real building the eects on the inner part of the facade are sure major that one on the border. Then it is important that the validation of the model will be done considering only the central part of the screen.
1.4. The experiment setup Finally the results from wind tunnel experiment are reported in the following gures:
(a) Force in x direction :Fx
(b) Force in y direction :Fy
(c) Force in z direction :Fz
As it is expected the forces Fxand Fzbecome larger in module with the increasing
of ow speed (U). Since their role is respectively of drag and lift measure on the screen and if it isn't cosinder particular condition the forces increase with the ow speed.
At least seeing gure 1.6b the Fy isn't constant, but its module doesn't change
a lot. For this reason the screen could be consider symmetric and it is possible to neglect the small force's uctuations with the ow speed.
1.5 Building's phenomena
In this last section will be brie y discuss about the some aerodynamic behavior of a building.
First of all a building is a blu body, then the characterization of the wake could be very similar to any type of blu body as for example a sphere.
The wake dimensions change a lot with the Re number and as a consequence the blockage produced by the wake decreases with the increasing of the Re number.
The de nition of Reynolds number is Re = ρU L
ν where L is the characteristic
length of the body.
In the following treatment, as just said in the previous section, the ow is consider incompressible, that means that the Re number isn't aect by the density ρ of the air. This is a good approximation since the Mach number is surely less of 0, 3.
In fact in the following test the ow speed is set at 10m
s, that it is an high speed
in fact in a real case in which the atmospheric conditions are standard the ow speed is absolutely less than 10m
s.
In very tall building or in atmospheric conditions in which a drastic scenario is taking place the ow speed could increase a lot, but in each of these cases the approximation of incompressible ow is very good.
As previously introduced, the wake of a blu body ,in this case a building, in u-ences a lot the forces that are generated on the body itself.
The analysis of the wake, also only qualitative analysis, is a way to better under-stand if the forces on the body are greater or less in two compared cases. In fact in the chapter 4, where a real application of a screen around a building is simulated, it will be shown a comparison between the wakes and the dierences of forces on the building are qualitative visible also in the wake dimensions.
In the paper [7] is modeled the building wake and are studied the eect in a not isolated building case.
In the following gure 1.7 is reported a very simpli ed example of wake behind a building, further is shown where the ow speed has a positive or negative sign.
1.5. Building's phenomena
Figure 1.7: A simpli ed wake over a building
As is possible to see in the gure 1.7, also if it is a very simpli ed case, the behavior of the wake over a building is better understandable and the dimension of the wake that is reduced in case of less forces on the building is shown in the upper part of the gure.
In the lower part of gure is shown the wake above and on the rear of the building, which is the re-circulation or cavity zone.
This zone is of particular interest in case in which there is a source of pollutants. In fact as treated by [5], there could be a cases in which the dimension of the re-circulation zone allows to the pollutants of remaining in this zone, obviously as shown in the gure 1.8 it is formed a zone in which the concentration of pollutants is greater than the outside wake zone.
Then, if an application of the screen around the building will atten the rear wake could be a positive behavior in terms of pollutants agents and foul-up.
An other phenomena that is very relevant in term of stress on the facades of a building is the peaks of pressure.
This behavior is carefully explained in the two thesis [1], [10] in which are reported respectively the wind tunnel simulations and the LES simulations.
In wind tunnel experiments the distribution of pressure on the facedes is obtained using pressure tap. Analyzing the temporal history of the measured pressure it was possible to observe some negative peaks.
The positioning of these peaks is strictly near the corner of the building's facade. An example of history measured from the taps is reported in the gure 1.9, where is clari ed what means negative peaks of pressure and above all is possible to noted that the temporal distance of one from an other is very small (in some cases).
Figure 1.9: History of pressure measured by taps on the facade
The chaotic nature is a characteristic of the turbulence, for this reason a LES analysis is proposed by [10]. Through the LES analysis is possible to visualize some coherent structure near the building.
Probably the nature of the negatives peaks is related to some very small structure which life time is also very small. The presence of this small structure is observable near the walls, where the viscosity is relevant and where the characteristic length of the structure is proportional to its life.
This small structure could cause the negative peaks of pressure.
In the following chapter won't be treated this phenomena, but it is a possible future study in order to verify the what could change with the application of a screen.
1.5. Building's phenomena Finally a real case application is reported in the following gure 1.10, the new ENI building.
Chapter 2
The two-dimensional case
After the introduction of the previous chapter, in the following one will be treat the 2D case.
The reader should have a question that concerns about the implementation of the inertial matrix F in OpenFoam. Then in this chapter will be simpli ed the discussion considering the two-dimensional case, which as the role of understanding the behavior of the inertial matrix. The values that will be entered in the inertial matrix will be found through the resultant forces of an OpenFoam's simpli ed case.
2.1 The corrections on the OpenFoam's model
First of all, as said in the previous chapter, the tensor Fij is expected to be a
diagonal matrix.
In order to consider the interaction between the three principal directions, it is important that the extra-diagonal terms of the tensors will be entered.
But the actual model hasn't this possibility, in fact into the porosityProprities le has to be entered as input a vector of scalars, in particular only three scalars. These input are the values on the diagonal of the inertial matrix.
In the algorithm 2.1 it is reported a generic case that use the porousSimpleFoam solver with the Darcy-Forchheimer modeling.
Algorithm 2.1 Generic case that use the Darcy-Forchheimer modeling
The vector d in the algorithm 2.1 has only zeros as input, because the scope of the discussion is to model a screen that doesn't generate relevant viscous components.
The vector f is formed by three scalars. The order of the value depend on the coordinate system that is described in the last part of the algorithm 2.1 , in the reported case is : x, y, z.
The meaning of the input values is that there are two directions which are con-sidered as very resistant, conversely the x direction is the porous one. Then the ow will feel a low resistance in x direction and consequently the its pressure will decrease. In the other two direction the ow will feel something similar to a wall, a blockage that doesn't allow to it to change its direction in the y and z direction.These consideration are valid in the particular case reported previously cause of the value of the input in the y and z direction is too big compared with the x direction.
In this special case it is possible to consider the porous media as monodimen-sional.
In the following rows all the details about the correction and modi cation of the OpenFoam solver are treated.
2.1. The corrections on the OpenFoam's model case, since the 2D case is made up considering the third row and the third column of the tensor Fij formed by all zeros.
First of all it is necessary to copy all the folders and les linked with the Darcy-Forchheimer model in a separately folder, that will become the user folder.
The procedure consists in :
1. to nd the les where the modi cations will be done and renamed its. 2. to understand how to modify the les
3. to change all the link in the caller les. 4. to recompile
For reasons of order and better understand what are the modi cation, the choice is to give a list of the modi cation divided by the folder of belonging.
The folders that needed a modi cation are : • applications (solver)
• functionObjects (calculation of forces) • platforms (library)
• porosityModel (that is renamed Porosity3D)
In the folder applications are included all the solvers and the momentum equation for p and U, in the les Ueqn.H and peqn.H. Obviously it isn't necessary to have all OpenFoam's solvers in the applications folder, but it will be copied and renamed only the necessary ones. In this case porousSimpleFoam.
Starting with the modi cation of the solver porousSimpleFoam. The only modi- cation done is to change the included header les that are modi ed.
In order to have an idea of how to implement this simply modi cation is reported, in the algorithm 2.2 the example of the included les in porousSimpleFoam.
Algorithm 2.2 abstract of modi cation in porousSimpleFoam
As is possible to see in the algorithm 2.2 the only input header le modi ed is IOporosityModelList.H that becomes IOporosityModelList3D.H . The meaning of this correction is to include the new model into the solver. In fact when the executive le is run, it searches in the porosity models the one selected into porosityProprieties le and the model that the solver has to nd is the modi ed one.
The folder functionObjects contains only the forces in this case, since it is the only le that is linked to the model. In the same way as before the included les are modi ed with the renamed ones, but in this case there are same member functions that are used in the forces le. Then all the functions linked to the modi ed model are renamed in the same way in which are renamed in the Porosity3D folder.
Before entering in the core of the discussion, or rather the les contained in the Porosity3D folder, a brief description of platforms is done.
In this folder there are all the user libraries, that means the new solvers and the new modeling build by the user. In order to recompile the only les that are modi ed it is important to write in the les options and les if the library is take from the OpenFoam library or from the user library.
So, for sake of simplicity the folder platforms has only the role to contain the libraries and the new solvers, or better all it is recompile from the user.
Finally the description of the Porosity3D folder. In this folder there are three folders:
• DF3D (that is the renamed Darcy-Forchheimer folder) • porosityModel3D ( that is the renamed porosityModel) • Make
Also in this case the description is in the same order of the list.
DF3D is the core of the model, in fact contain the three fundamental les : • DF3D.H (renamed executive le)
• DF3D.C (renamed header le)
• DF3DTemplates.C (renamed template le)
The header le contain all the de nition, it is the declarative part of the model. In this discussion it is said that the actual model expects it as an input two vectors of three elements Fi and Di. For the scope of the treatment it is necessary
2.1. The corrections on the OpenFoam's model The modi cation is reported in the following algorithm 2.3.
Algorithm 2.3 modi cation on DF3D.H le
As it is possible to note the Fij and Dij are de ned as dimensionedTensor and
consequently as tensor eld. In the original le its are de ned as vectors.
It is understood to the reader that in this le all the referement to the modi ed les is renamed, as shown in the rst row of the algorithm 2.3. In fact porosity-Model3D is the renamed le porosityModel.
Now it is analyzed the executive le DF3D.C. In this le are corrected the tensors Fij and Dij since its were implemented considering only the principal diagonal. But
the interest of the treatment are the terms exstradiagonal. In this way all the terms of the tensors are asked to the user. Then in the porosityProprieties le the input becomes nine and the tensor is ll row by row.
Algorithm 2.4 modi cation of DF3D.C
Into the executive le there is an other correction. In the original le is used a member function, adjustNegativeResistence, that as the reader could suppose has the role of taking the norm of all the negative input and to substitute it with the maximum of all the norms. The meaning of this function is that aren't accepted negative values cause of the modeling has the role of a resistance. But in this treatment it is necessary to consider also the negative input because there could be a direction in which the ow accelerates.
Remark that a screen isn't a porous media as for example the ground, since it will not be only a resistant eect.
The structure of the Fij will be strongly dependent on the shape of the screen, in
particular the principal orientation of the screen will be the fundamental parameter of the characterization of its behavior.
For the reasons explained before the adjustNegativeResistence function is com-mented in the modi ed code.
2.1. The corrections on the OpenFoam's model Algorithm 2.5 The role of adjustNegativeResistence
At least in order of description there is DF3DTemplates.C le. But in this case are modi ed only the including part and the caller functions are renamed.
At the same way all the les in the folder porosityModel3D are modi ed only in the including part and in the caller functions, therefore aren't reported examples.
Finally it is brie y explained the Make folder.
In this folder there are two les options and les. The options contains all the libraries that are used to compile and these could be both de ned from the user and an just existent.
In the le les are written all the les that are linked one with the other and the name chosen for the new library.
An example of these two le is reported in the following algorithms 2.7 and 2.6. Algorithm 2.6 les le of Porosity3D
Algorithm 2.7 options le of porousSimpleFoam
2.2 The OpenFoam's setup for the 2D case
After the modi cation, the new model is ready to be used.
In fact the tensor Fij ,that is the only one used since Dij will be always null,
could be implemented as a 3 × 3 tensor.
The unsolved problem is how to nd the values to enter in the tensor Fij.
A brief introduction to the problem is done in the chapter 1, section 1.3 where the relation between the forces and the inertial tensor is reported.
Because that equation it is very important for the treatment is rewritten in 2D case: " Fx Fy # = V ρ|ui|2 " fxx fxy fyx fyy # " cosα sinα # (2.1) reminding that V is the volume of the porous media, ρ the density of the air and ui the ow speed.
The are two issues to solve, the rst is how entering the values of the forces, or better how to nd these values, and the other issue is that there are only two equations and four unknown in eq. 2.1. In this section will be discussed the procedure adopted to solve the rst issue, in the following one will be explained how to solve the indeterminate system of equations.
First of all it is used a 2D model in order to simplify the geometry of the screen, since it will be used OpenFoam for nding the forces's values.
The implemented case is very similar to the sections proposed in the chapter 1, an array of simple airfoils or better plates one above the other.
The solver used to solve the case is simpleFoam, since the ow speed is low and the phenomena related to compressibility are neglected.
2.2. The OpenFoam's setup for the 2D case κ − ωare very similar then the choice of the model in this case it isn't a an important variable of the case.
Conversely the B.C are a fundamental choice for the setup of the case both in terms of convergence and in terms of obtained results.
For the 2D case the B.C chosen for the top and the bottom are represented in the scheme of gure 2.1.
Figure 2.1: scheme of the B.C used for the 2D setup
The meaning of the cyclic is that the same ow speed and direction outgoing from the top boundary re-enters in the bottom patch. In other word it is how there was a direct link from the top to the bottom patch, as its are neighbor patches.
The idea it is to represent a condition in which the corners of the screen doesn't in uence the result, it is an ideal condition in which the screen is considered in nitely extensive. For these reason it is expected that the center part of the screen will be good approximated and it will feel the presence of the full screen.
The other B.C used for velocity and pressure at the inlet and the outlet are : p: at inlet zero gradient, at the outlet xed value (0P a )
u: at inlet xed value (10m
s), at outlet inletOutlet
The B.C for the back and the front aren't specify since the patches are type empty. The geometry of the case is a simple slice with a very thin thickness, there is only one cell in the direction of the thickness in order to simulate a 2D case.
Indicatively the length of the case is ve times its height. In the gure 2.2 is shown the shape of the case.
Figure 2.2: 2D case setup: the shape of the test case
In order to nd the values of the forces at dierent alpha, the case is run with dierent inlet ow directions. The procedure used to have the desired values of forces in x and y directions is to change the ow direction from +75◦ to −75◦ with a step
of 15◦. Where the convention used to de ne positive and negative angles is the same
reported in chapter 1, gure 1.2.
A nal remark is a consideration about the angle α in the eq. 2.1.
It isn't an unknown cause of is the inlet ow direction, that means the modi ed model is linear and doesn't depend on the the α. Then the values that will compose the inertial tensors will be scalars and won't depend on α. It is a strong hypothesis but it will be supported by the obtained results.
The mesh discretization is shown in the gure 2.3, it is formed by unstructured tetrahedron cells that become smaller as its approach to the screen section.
2.2. The OpenFoam's setup for the 2D case 2.2.1 Results of the 2D simulations
In the following gures will be reported some run case, in particular from +45◦
to −45◦, in order to have a general view of the eects of a screen in a ow. Both the
pressure and velocity elds are shown in the gures 2.4, 2.5.
(a) α = −45◦: pressure eld (b) α = −45◦: velocity eld
(c) α = −30◦: pressure eld (d) α = −30◦: velocity eld
(e) α = −15◦: pressure eld (f) α = −15◦: velocity eld
(a) α = 0◦: pressure eld (b) α = 0◦: velocity eld
(c) α = 15◦: pressure eld (d) α = 15◦: velocity eld
(e) α = 30◦: pressure eld (f) α = −30◦: velocity eld
(g) α = 45◦: pressure eld (h) α = 45◦: velocity eld
2.2. The OpenFoam's setup for the 2D case For the all negative angles the screen is completely in a stall condition, the relative angle between the airfoils and the ow direction is too high in order to guarantee an attached ow. Further, in this condition the blockage of the ow is very high and then the pressure drop is greater than in the positive cases, or better in the cases where the angle of attack is less than 15◦.
Increasing the inlet angle the ow becomes an attached ow until the condition in which it doesn't fell the presence of the screen, since it is aligned with the airfoils. These considerations are clari ed and veri ed with the visualizations of the streamlines in two dierent cases : 30◦ and −15◦ which are reported in gure 2.6
(a) inlet angle : −15◦
(b) inlet angle : 30◦
In fact in the gure 2.6a is shown a re-circulation zone that is a stall condition, it is a condition in which the drag force is the principal force on the plates. Conversely in the gure 2.6b the streamlines follow the shape of the pro le. An other important observation is that the airfoils are very simple, are plates, cause of the presence of a sharp angles the ow tends to separate from the pro le. Then also in a no stall condition the ow generates re-circulation zones.
2.3 The calculation of the terms of inertial tensor F
ijfor
the 2D case
In this section the scope is to calculate the entering values of inertial tensor. In the previews chapter was done an introduction of the problem and the relative issues to solve, the principal one is that the system of equations is indeterminate.
First of all the system of equations is rewritten: " Fx Fy # = V ρ|ui|2 " fxx fxy fyx fyy # " cosα sinα # (2.2) at now the only unknown is the tensor Fij, since the forces are calculated from
the postProcesseing of OpenFoam cases, reminding that the forces are considered always oriented as the reference coordination system.
In order to calculate the values of the inertial terms all the forces are used, where for all the forces means all the values calculated from the OpenFoam cases so from −75◦ to 75◦.
Using this approach the system becomes overdeterminated and a least squared method (LSM) will be used for obtaining the four inertial values that better approx-imate the real forces's curve, the one calculated from OpenFoam's cases.
In the following part of the section the procedure is explained in details and the implemented Matlab code will be reported and described.
The system of equations is linear and for sake of simplicity could be seen in a standard formulation:
Aijxj = bj (2.3)
where Aij is a matrix, xj is the vector of unknown and bj is the vector of forces.
Going into details of the speci c case of this treatment, it will be analyzed term by term.
Taking the eq. 2.2 and performing the calculation, it is possible to observe that the vector bj is formed by the forces, in particular a column vector with the same
number of force in x and y direction. In this case the forces are positioned in the following way :
2.3. The calculation of the terms of inertial tensor Fij for the 2D case bj = Fx−75◦ Fx−60◦ ... Fx75◦ Fy−75◦ Fy−60◦ ... Fy75◦ (2.4)
and the vector of unknown xj is formed by the four unknown:
xj = fxx fxy fyx fyy (2.5)
in this case is a vector of only four terms, since the case is 2D case.
As consequence, the matrix Aij is composed by trigonometric functions and has
four columns and the same number of row of vector b. Performing the equation 2.2 the matrix A becomes :
Aij = V ρ|ui|2 cos(α1) sin(α1) 0 0 cos(α2) sin(α2) 0 0 ... ... ... ... 0 0 cos(α1) sin(α1) 0 0 cos(α2) sin(α2) ... ... ... ... (2.6)
where αi for i = −75◦ : 15◦ : 75◦ are the tested angles. At now the system is
written in a standard linear form.
For solving the system will be used the pseudo-inverse method, in fact the matrix Aij is rectangular, where its the de nition is :
A+ = (ATA)−1AT (2.7)
where A+ is the pseudo-inverse matrix, a generalization of the inverse matrix.
The system is solved in the following way :
2.3.1 Matlab implementation of LSM on the overdetermineted sys-tem of equations
The Matlab implementation is just an application of the theory.
The only value that it isn't described is the volume V , but is simply the volume that is considered to contain the porosity eects. In the treated case is a ideal volume that contains the screen's section. For the previously tested case is V = 7, 34·10−8m3,
it is so small since in 2D case the third direction z is almost 0 (but in OpenFoam it is necessary to assign to it a value).
For sake of simplicity the case was reported considering only three angles, −45◦, 0◦, 45◦.
In the algorithm 2.8 is reported the core of the Matlab implementation, the function that found the values of the terms of the inertial tensor.
Algorithm 2.8 Matlab function used to implement the LSM in 2D case
The Matlab function take as input all the known terms: α, b, V, q. Reminding that q is the dynamic pressure.
First of all the function generate the matrix A, which result is also commented in the algorithm. After the matrix is composed, the system is solved using the method of pseudo-inverse matrix and the four values are found and positioned in the matrix C2Dthat is the inertial tensor Fij.
Obviously in the main Matlab code the values of the OpenFoam's forces and the other data are loaded.
The values calculated for the inertial tensor are :
Fij = 66, 65 −45, 76 0 −59, 0585 53, 62 0 0 0 0 (2.9)
where all the terms referred to the third direction are null, since is a 2D case. The results are plotted and commented in the following section.
2.4. Results
2.4 Results
Finally in this section will be shown the results obtained in 2D case. In the following plots are represented three curves:
• DragLift means the values obtained from the simulations
• PorousSimple means the values obtained from the new model, entering the values reported in the eq. 2.9
• Analytic means the values obtained from the analytical calculation : " Fx Fy # = V ρ|ui|2Fij " cosα sinα #
using the same inertial matrix calculated in the previous section. Observing the gure 2.7 some comments could be done.
First of all it is possible to note that the curves are very similar, then the linear modeling could be considered a good choice.
The dierences in the curves are principally located in the tails, where the angles are very high, therefore these dierences aren't important cause of the fact it is expected a direction of the wind around 0◦. Only in a extreme cases the angle's ow
direction could be greater than 45◦ so the tails of the curves could be neglected.
Considering only the central part of the gures, in all the plots the DragLift curves is very good approximated from the PourosSimple curves even better than the analytic curves.
It is expected that the analytic curve should be almost the same of PorousSimple curve, since the model is constructed on the analytic curve.
The better approximation of the PorousSimple curve is due to the angle α, in fact the model is linear and for the analytic curve the ow angle is xed. Conversely, if it is consider the implemented model, the angle α is the local angle and for this reason change its direction when pass through the porous volume.
Entering in the code and calculating the angles and comparing its with the ana-lytic ones, it is possible to note that cells by cells the angles have a dierent values from the analytic ones. These dierences are accountable of the best approximation of the Porous curves.
Further, in order to be more precisely in the central part, from −15◦ to 15◦, the
angles are lower tham the analytic one. For these reason the Porous curves have an higher values in the central part of the plot.
(a) Forces in x direction
(b) Forces in y direction
(c) Polar curves
Chapter 3
The three-dimensional case
In this chapter will be treated the 3D implementation of the screen's modeling. In the previously chapters are explained the physics and there is a simpli cation of the real case. In fact the screen has three dimensions, the considered case of the only section (the 2D case) was the starter point.
The fact that the values obtain through the modeling are a good approximation of the real screen suggests that the linear modeling could be used also in the 3D case.
But, as the reader could expect, the 3D implementation adds some troubles. First of all the physical 3D model, since it isn't possible to use the simpli ed section cause of the fact the third component could be a fundamental role in the model or perhaps not, but a priory it is needed the 3D model.
When this problem is solved, the mesh around the 3D model will be generated and the model will be tested and a validation will be done.
Then the OpenFoam model will be constructed, in a very similar way of the 2D case, and will be shown a comparison between results as in the previous chapter.
In the last section a real building will be tested with the Openfoam modeling around the structure.
3.1 The 3D model of a screen
The scope of this section is to describe the procedure used to generate the stl le. In the following gures 3.1 are shown some view of the screen that will be drawn using the software Inventor.
(a) front view of the real screen
(b) length of the screen's passage
(c) height of the screen's passage
3.1. The 3D model of a screen Observing the gures 3.1, it is noted that the two characteristic dimensions are the length and the height of the screen's passages.
In the treated case the two dimensions are : • length around 25cm
• height around 5cm
At now it will be described the principal steps used to draw the screen.
First of all it is drawn the pro le of the same dimensions of the real screen. The resultant pro le is shown in the gure 3.2 , as it is possible to see the dimensions are around the same of real length and height. The thickness of the pro le is around 2mm.
Figure 3.2: pro le of the CAD screen
Then the pro le is mirrored in order to obtain a series of pro les next to each other, with the same geometry, as is shown in gure 3.3.
One of the most important thing to remind is to constrain the initial sketch because when the sketch will be extruded it is signi cant to preserve the initial shape.
After these simple steps, the following one is to create a work plane inclined around 45◦, for the treated screen, and to move all the sketches on that plane.
Once all the sketch are moved, the simple geometries are extruded in the direction perpendicular to the work plane. The extrusion is around 40mm, that is the thickness of the real screen.
All the described procedure is remade exactly the same, but the pro le at this time will be symmetric to the previous one.
At now the core of the screen is done, simply applying a pattern to the core of the screen is mirrored in the two perpendicular directions (the original ones).The lateral view of the obtained model is shown in the gure 3.4.
Finally a very similar screen is created, the front view is illustrated in the gure 3.5.
Figure 3.3: mirroring of the initial sketch
Figure 3.4: lateral view
In the gure 3.5 it is important to note that the blocks are all aligned, that is possible considering the correct distance between the patterns. In fact for reproduc-ing the eect of expanded metal sheet it is necessary that the two symmetric series of pro les are slightly overlying.
When the screen is realized will be cut a square of desired dimensions. In this case 600 × 600mm2.
3.1. The 3D model of a screen
Figure 3.5: front view
3.1.1 Setup of SnappyHexMesh : mesh generation around the model One of the most critical steps is the generation of the mesh around the a complex stl le.
There are two principal issues that could be veri ed after this step: • The mesh could be open, or very rough approximate the original shape • The computation cost, in terms of time and requested memory of the calculator In the treated case the screen is very thin, therefore it will be needed a large number of cells to obtain a good approximation of the screen. In fact it is expected that there will be at least three cells in the thickness.
After a lot of attempts the best setup found for the snappy and castellated control is reported in the algorithm 3.1.
The calculation are all performed on a cluster, since the computation cost is too high in terms of requested memory to be supported from a personal computer.
In this speci c case, considering only the snappy and castellated time, the run time is around 4 hours using 1 node with 8 processors.
Remark the fact that the given data are referred to a screen 600 × 600mm2 that
is very small in the real applications.
But it is possible to imagine that in this case the problem isn't the time, but the requested memory.
For understanding the order of magnitude of the cells used in the snappy and castellated process in this case it is used around eight millions of cells.
Algorithm 3.1 optimized snappyHexMeshDict: snappy and castellated controls.
Observing the snappyHexMesh some comments could be done.
First of all the level 6 requested, obviously it is a high level of re nement, but in a case in which the nal shape must be as much as possible coherent to the real screen, it is a good choice.
Further, one of the parameter that has an important role in the snappy is the smoothness that adapts the shape of the cells, it is an important parameter in this case cause of the presence of overlying surface that are better approximated with a value around 3.
Finally the mesh result is shown in the gure 3.6, as it is noted the screen is correctly approximated from the generated mesh.
After the checkMesh the mesh results close, then the performed screen could be considered a good approximation of the real one.
3.2. Validation of the screen
Figure 3.6: Screen obtained after snappy and castellated
3.2 Validation of the screen
In the chapter 1 is introduced the wind tunnel experimental setup.
The scope of this experiment, besides to measure the pressure drops at some ow speed which characterized the behavior of the screen, it is to have a comparison between the real experiments and the simulations.
In this case the values obtained by the real measurement into wind tunnel are directly compared with the data achieved from the simulations.
First of all the setup of the simulation will be discussed.
It is necessary that the simulation is very similar to the wind tunnel setup, then it is important to be careful about the B.C. and to extract the forces from the simulation.
In order to simulate it is created a bounding box with the same dimensions of the real wind tunnel . Reminding that the wind tunnel section dimensions are 1,5×2m2.
The length of the bounding box is 6m and the screen is positioned at 2 meters from the inlet.
The procedure to create the screen are just described in the previously section, in this case the screen is cut of dimensions 1450 × 1900mm2 ( that are almost he same
of the wind tunnel sections. Obviously the screen isn't cut of the same dimensions of the section in order to avoid problems on the boundaries.
In the bounding box the number of cells in which is subdivided the domain is around 4 millions.
The B.C in this case aren't a problem, in fact the four lateral patch are obviously de ned as wall while the inlet and the outlet are exactly the same of the ones used in the 2D cases.
In particular for the walls it is used the following boundary conditions: pwall: zero gradient
Uwall: noSlip
The RANS model used is κ − and to model the ow near the wall are used wall functions.
As for the previously 2D treatment, the choice of the model hasn't a fundamental role for obtaining the correct result. In fact the same setup was run with a κ − ω RANS model, but the results are around the same.
In this way the real experiment is reproduced through the CFD simulations in the most faithful way to reality.
Now it is described the procedure used to extract the forces, reminding that in the wind tunnel experiment the forces are measured with four balances, as explained in the chapter 1. But these four balances measure only the forces on the central part of the screen, in particular the central square of dimensions 600 × 600mm2.
In order to measure the same forces with the simulation the idea is to use a topoSetDict that extracts only the forces on the central part of the screen.
The topoSet works in the following way:
1. select a box of cells of the desired dimensions (in this case 660 × 600mm2)
2. the selected box and the screen are intersected 3. the result of the intersection is the desired faceSet
After all the procedure is described and the case is run the results are reported in the gure 3.7, as is possible to see the gures are exactly the same reported in the chapter 1, but at now also the forces calculated from the simulations are added.
It is implied that the simulations are run at the same velocity of the the experi-mental test as it is noted in the gure 3.7. Further also the conventions are the same of the chapter 1.
3.2. Validation of the screen
(a) Fx
(b) Fy
(c) Fz
Figure 3.7: Results from the validation The curves are very similar if compared one which the others.
This means that the modeling screen is a very good approximation of the real screen, then the screen model could be considered validated.
