POLITECNICO DI MILANO
Scuola di Ingegneria Industriale e dell’Informazione
Corso di Laurea Magistrale in Ingegneria Aeronautica
A Lagrangian 3D Particle
Tracking Solver for in-Flight Ice
Accretion
Relatore:
Prof.
ALBERTO GUARDONE
Correlatori:
Dott. MARTA ZOCCA
Ing.
GIANLUCA PARMA
Tesi di Laurea Magistrale di:
TOMMASO BELLOSTA
Matricola 874927
gas o liquidi al cui interno si trovano particelle in una fase differente (solida, liquida o gassosa). La diffusione di inquinanti nell’atmosfera e il moto di gocce d’acqua in una corrente d’aria sono solo alcuni esempi di questo tipo di correnti. Il secondo problema è tipico della modellazione dell’accrescimento di giaccio in volo, che avviene quando un aereo vola attraverso una nuvola contente gocce d’acqua in uno stato metastabile detto sottoraffreddato. Le gocce di acqua solidificano all’impatto o in istanti successivi causando la formazione di ghiaccio sul velivolo, che pone importanti problemi di sicurezza e può limitare l’inviluppo di volo.
In questa tesi l’accrescimento di ghiaccio è simulato attraverso il codice PoliMIce sviluppato al Politecnico di Milano. Il calcolo della massa d’acqua che viene raccolta dalle superfici aerodinamiche di un aereo è ottenuta dalla soluzione del moto delle gocce d’acqua immerse nella corrente d’aria attraverso un solutore Euleriano-Lagrangiano. La soluzione del moto delle particelle, che prende il nome di particle tracking, rappre-senta la parte Lagrangiana del solutore, ed è affrontata da questo lavoro.
Nell’ambito di questa tesi il solutore Lagrangiano è stato completamente riscritto per migliorarne l’accuratezza. Sono stati valutati gli effetti della procedura di interpolazio-ne usata per calcolare le forze agenti sulle particelle, insieme a quelli dati dallo schema numerico usato per integrare le traiettorie delle gocce. Viene presentato un algoritmo per localizzare le particelle all’interno di una griglia di calcolo non strutturata facendo uso di aritmetica a virgola mobile con precisione arbitraria per rendere robusta la procedura di calcolo in prossimità dei contorni degli elementi della griglia. Il nuovo codice per il particle tracking è stato usato per replicare alcune misure sperimentali sia in due che tre dimensioni. I risultati ottenuti sono in buon accordo con i dati ottenuti sperimentalmente, confermando la bontè dell’approccio seguito. Viene inoltre presen-tata un’implementazione parallela del codice per architetture a memoria distribuita ottenuta tramite lo standard MPI.
problems in science and engineering; for example pollution dispersion in the atmosphere or tracking of water droplets in the air near a flying object. The latter is a problem arising in the simulation of in-flight ice accretion which occurs if the aircraft flies through a cloud of supercooled droplets. In the present thesis, in-flight icing is simulated using the software suite PoliMIce developed at Politecnico di Milano. The computation of the water mass impinging the aerodynamic surfaces of a plane is obtained via the solution of the flow of water droplets in air by means of an Eulerian-Lagrangian solver. The particle tracking accounts for the Eulerian-Lagrangian part of the solver and it is addressed in this thesis. The accuracy of the particle solver is the main focus of this work. The effects of the interpolation strategy used to compute the forces acting on the particles, together with the numerical scheme for integrating the particles trajectories are evaluated. A particle tracking scheme for unstructured meshes is presented, making use of multiple precision arithmetics to address some robustness issues. A particle tracking code was developed implementing the algorithms devised and it was used to replicate experimental data in both two and three dimensional cases. Numerical simulations compare fairly well with available experimental data, thus confirming the correctness of the present approach. A parallel implementation using MPI is also provided for distributed memory architectures.
Al termine di questa esperiemza voglio ringraziare tutti coloro che hanno reso possibile il mio percorso al Politecnico.
In primis ci tengo a ringraziare chi mi ha guidato nella conclusione di questo percorso. Ringrazio il Prof. Alberto Guardone per avermi dato la possiblità di confrontarmi con un lavoro stimolante, per la sua guida e la fiducia dimostratami. Sono grato a Gianluca e Marta per i loro preziosi consigli che mi hanno aiutato nello svolgimento di questa tesi.
Un grazie anche alla mia famiglia, in particolare ai miei genitori che mi hanno supportato (anche) in questo percorso universitario, hanno appoggiato le mie scelte e non mi hanno mai fatto mancare niente. A loro dedico questo lavoro.
Un grazie infine a tutti gli amici che ho incontrato in questi anni. A Simo, Albe, Cri, Tom e Matte con i quali ho condiviso questi ultimi mesi di lavoro in aula studio. A Onto, Paolo e Lollo che mi hanno accolto al mio arrivo in Bovisa sei anni fa. Ringrazio anche tutti i coinquilini che mi hanno sopportato in questi anni in appartamento, in particolare Zoff e Macchio per questi ultimi due anni.
1 Introduction 1
1.1 Particle Laden Flows . . . 5
1.1.1 Phase Coupling . . . 8
1.2 Numerical Simulation of Particle Laden Flows . . . 11
1.2.1 Eulerian Description . . . 13
1.2.2 Lagrangian Description . . . 16
1.3 One-way Coupled Lagrangian Particle Tracking . . . 23
1.4 Thesis goals and structure . . . 23
2 Geometry and Mesh Structures 27 2.1 PoliGeom . . . 28
2.2 PoliMesh . . . 30
3 Solution of the particle tracking equations 35 3.1 Trajectory computation . . . 36
3.1.1 Testing the time integration . . . 40
3.2 Interpolation . . . 43
3.2.1 Testing the interpolation. . . 47
4 Tracking particles 55 4.1 Tracking algorithm . . . 59
4.1.1 Effects of floating point arithmetics. . . 62
4.1.2 Modified tracking algorithm . . . 64
4.2 Tree-based tracking . . . 68
4.3 Algorithm performance . . . 70
5 PoliDrop 75 5.1 Setting up the Simulation . . . 75
5.2 Data structures . . . 77
5.2.1 Particle . . . 77
5.2.2 Cloud . . . 78
5.2.3 CollectionEfficiency . . . 83
5.3 Main loop . . . 83
5.4 Parallelization of the particle tracking code . . . 84
5.4.1 MPI implementation . . . 86
6 Results 93 6.1 NACA23012 profile with simulated ice shapes . . . 93
6.1.1 Clean profile . . . 94
6.1.2 Iced geometries. . . 98
6.2 Straight wing in wind tunnel . . . 107
Appendix A Configuration File 113 Appendix B Wall interaction models 117
1.1 Example of icing on a wing, wind turbine, engine and a sensor. . . 2
1.2 Glaze (left) and rime ice accretion on a wind tunnel model and wing leading edge [2]. . . 3
1.3 Contributions to the energy and mass balance in the ice accretion computation. . . 5
1.4 Two particle system used to define inter-particle spacing. . . 7
1.5 Coupling zones as defined by Elghobashi [11]. . . 10
1.6 Cell-centered (hashed) and node-centered (shaded) finite volumes from reference [36]. . . 12
1.7 Ratio of length scale to particle diameter for a cube containing 50 particles. . . 14
1.8 CDof a sphere ad different Rp. Experimental data and analytical models. 20 1.9 Solution of a single time step of an Eulerian-Lagrangian particle tracking solver for a two-way coupled (left) and one-way coupled regimes. . . . 22
2.1 Graphical representation of the PoliGeom data structure. . . 29
2.2 Type of elements considered in PoliMesh. . . 33
2.3 PoliMesh data structure. . . 34
3.1 Contours of the norm of the carrier fluid velocity. . . 41
3.2 100 µm particle trajectory computed with first order schemes for differ-ent time steps. . . 44
3.3 Trajectory computed with the forward euler scheme with h = 1e − 4. The black x represents the exact position of the particle after 50 seconds. 45 3.4 Trajectory computed with the adaptive RK45 scheme. The black x represents the exact position of the particle after 50 seconds. . . 46
3.5 Simulation elapsed time for RK45 at different droplet’s diameter. . . . 47
3.6 Collection efficiency β computed for a three dimensional and two di-mensional case from reference [34]. . . 49
3.7 Cloud front computed with different interpolations. From the right: piecewise constant, distance weighted, linear. The green particles represent the reference solution. . . 50
3.8 Coarse and fine meshes used in the evaluation of the interpolation strategy. 50 3.9 Collection efficiency on a cylinder for droplets of diameter 5e − 5m. . . 51
3.10 Collection efficiency on a cylinder for droplets of diameter 10e − 5m. . 51
3.11 Collection efficiency on a cylinder for droplets of diameter 25e − 5m. . 52
3.12 Effect of the initial particle velocity recovered with the distance weighted interpolation on the collection efficiency. MVD 250µm . . . 52
3.13 Error computed for the interpolation schemes tested. In blue linear, yellow distance-weighted and red constant. . . 53
4.1 Shaded cells are checked when a particle is found to be in the central rectangle of the background mesh (bold lines) [52]. . . 56
4.2 Virtual gap forming between cells due to different decompositions and proposed solution [48]. . . 59 4.3 T2L test in 2D and 3D. In three dimension the test must be applied to
every edge of the face [44]. . . 59 4.4 Pseudo code of the algorithm used to assign the new particle position
to a cell. . . 60 4.5 Computation of intersection between faces and trajectory. . . 62 4.6 Actual cell area that would return true to a in-cell query in the modified
version of the robust algorithm. . . 63 4.7 Tracking of a particle near a node presenting errors due to floating point
finite precision. . . 64 4.8 Tracking of a particle near a node with different versions of the algorithm.
Only the presented algorithm making use of MPFR number types to compute the intersection is able to map the particle to cell 6. . . 65 4.9 Fictional trajectories tracked in order to extend the known vicinity
algorithm to the initial positioning of particles. . . 69 4.10 Kd-tree of a set of bidimensional points [58]. . . 69 4.11 Particle search in the neighbourhood of the closest centroid. . . 70 4.12 Cost of the presented algorithms changing the number of cells crossed
at each step. . . 72 4.13 Cost of the presented algorithms changing the number of elements in
the mesh; the number of cells crossed at each step are kept constant. . 73 4.14 Three dimensional tetrahedral mesh used in the performance evaluation
of the algorithm. . . 73 5.1 Dependencies of the particle tracking code. . . 76 5.2 Discontinuous droplet front after splashing from the discretized boundary. 82 5.3 Droplets front splashing upon impact with the surface of a naca 23012
airfoil computed using a discrete representation of the boundary. . . . 82 5.4 Droplets front splashing upon impact with the surface of a naca 23012
airfoil computed using a continuous representation of the boundary. . 83 5.5 Flowchart of the polidrop code. Shaded blocks are called only by the
parallel implementation. . . 85 5.6 CPU clock speed vs time. Data from [61] . . . 87 5.7 Prallelization strategy implemented in the particle tracking code. . . . 89 5.8 Different ways to divide the particles between available processors. . . 90 5.9 Percentage distance from a perfectly balanced load on 5 processors
running a 1000 droplet simulation (Figure 5.8). The alternated division of the cloud leads to a better distribution of the workload. . . 90
5.10 Scaling for a simulation of one million droplets. . . 91
6.1 NACA 23012 profile in the test section of the IRC wind tunnel, reference [59]. . . 94 6.2 Measured pressure distribution from reference [59] and SU2 values at
α = 2.5◦. . . 96
6.3 Values for the collection efficiency on a NACA 23012 profile. Circles experimental data, dashed line LEWICE, continuous line PoliDrop. . . 97 6.4 Values for the collection efficiency on a NACA 23012 profile. Circles
experimental data, dashed line LEWICE, continuous lines PoliDrop with splash models. . . 99 6.5 Values for the collection efficiency on a NACA 23012 profile after an
6.6 Values for the collection efficiency on a NACA 23012 profile after an exposure time of 10 minutes. . . 102 6.7 Values for the collection efficiency on a NACA 23012 profile after an
exposure time of 15 minutes. . . 103 6.8 Values for the collection efficiency on a NACA 23012 profile after an
exposure time of 22.5 minutes. . . 104 6.9 Values for the collection efficiency on a NACA 23012 profile after an
exposure time of 45 minutes. . . 105 6.10 Collection efficiency computed considering the re impinging mass for
the 45 min. iced geometry and MVD of 236µm. On the left the single parcel discretization is compared to the multiple parcel representation.
On the right multiple values for np were tested. . . 106
6.11 Wind Tunnel mesh used in the simulation of the collection efficiency on the GLC-305 model. . . 107 6.12 Experimental and computed pressure coefficient at mid-span of the
GLC-305 model. . . 108 6.13 Experimental and computed collection efficiency at mid-span of the
GLC-305 model. The simulated results are reported for different num-bers of particles used. . . 109 6.14 Collection efficiency for the wind tunnel GLC-305 model near the model
1.1 Idealized cloud division and related icing threat. . . 3
1.2 LWC values from reference [9] . . . 3
3.1 Errors and elapsed times of forward Euler (FWE) and ONERA 3D scheme 3.18 (ONE) compared to the Runge-Kutta (RK45) time inte-gration. The RK45 scheme used for comparison yielded an error of
2.79 · 10−7. Droplet diameter 100 µm, simulation time 0.7 s. . . 43
3.2 Error due to the interpolation used. . . 49 5.1 Results from time profiling a particle tracking simulation on a 40k
elements mesh with two different cloud sizes. Results are time percentage of total simulation time. . . 87
Introduction
In-flight ice accretion is a relevant issue in aircraft design and aviation safety. If an aircraft flies through cold wet air, the water particles from clouds can freeze after impacting upon its surfaces, possibly creating ice build-ups in the most exposed zones. Ice formations have a detrimental effect on aircraft performance as they modify the aerodynamic shape of wings and blades, and can also affect aircraft stability as they modify the weight distribution. Moreover ice build up on control surfaces and sensors can prevent the pilot or the control system from correctly manouvering the plane, which has led in the past to many accidents, some of which led to casualties. There are also reports of damages to engines due to ice formation [1]. Non uniform ice accumulation on the blades of the fan can lead to unbalanced loads on the system. Moreover, due to the high centrifugal force, ice build-ups can detach and hit the first stage of the compressor, possibly damaging the blades. All those effects are to be avoided or at least mitigated in order to comply with safety regulations. Of the total number of weather related accidents, icing accounts for 12% of them. A thorough understanding of the dynamics of ice accretion could help preventing that share of accidents.
Icing occurs when a plane flies through a region in the atmosphere containing liquid water. These regions are called clouds: they consist in small water droplets and ice crystals suspended in air. Remarkably, icing on aircraft is not caused by actual ice crystals, but by water droplets in a super-cooled state. Ice crystals tend to rebound on the aircraft surface, while water droplets stick to it by freezing upon impact or forming a liquid film. Table 1.1 reports an idealised subdivision for clouds based on their composition (ice crystals or water droplets) with the related risk of ice formation. Droplets are formed in a process called nucleation. The atmosphere contains significant concentrations of particles of size of less than a micron, for example dust, which have an affinity for water, and therefore act as condensation nuclei. The water vapour present in supersaturated humid air condensates around those nuclei forming the droplets. Due to their small dimension, droplets formed in such a way exist in liquid
form well below 0◦ in an unstable equilibrium state called supercooled state. These
Figure 1.1 Example of icing on a wing, wind turbine, engine and a sensor.
Transition to solid state can also happen if the equilibrium is perturbed, for example by impacting upon the surface of an aircraft flying through a cloud. Two parameters used to describe clouds are the droplet size and their concentration. The latter is expressed through the Liquid Water Content (LWC), namely the measure of the mass of water in a cloud in a specified amount of dry air. It is usually measured in grams per volume of air or mass of air. The LWC of a cloud varies significantly depending
on the type of clouds in the atmosphere: typical values range from 0.002 to 3 g/m3
. Typical liquid water content values for different type of clouds are reported in table 1.2. The other important parameter for the characterisation of clouds is the size of the droplets. Particles composing a cloud do not have all the same size but their diameter can be described by a probability density function [76]. A representative value for the size of droplets in a cloud is the Median Volume Diameter (MVD) or alternatively the
Mean Effective Diameter (MED). They are defined as the median value of the drop
diameter in the drop size distribution, averaged by the liquid water content. Half the water volume will be in smaller drops and half the volume in larger drops, as such droplets with diameters smaller than the MVD (or MED) represent the majority of the cloud particles. The difference between the two is that the MED is calculated based on an assumed drop size distribution, while the MVD is obtained by actual drop size measurements [10]. In the case of an actual measurement the MVD is computed
Figure 1.2 Glaze (left) and rime ice accretion on a wind tunnel model and wing leading edge [2].
Cloud Type T (◦C) icing threat
Ice cloud < −40◦ none
mixed cloud −40◦<T< −15◦ moderate
supercooled liquid cloud −15◦ <T< −0◦ high
liquid water cloud > 0◦ none
Table 1.1 Idealized cloud division and related icing threat.
as: PK i niDi3 Pnbins i niDi3 = 0.5 (1.1)
where ni is the number of droplets whose diameters are closest to Di, and K is the
bin containing the MVD. Typical values of the MVD are between 15 and 40 µm and droplets with higher values are called Supercooled Large Droplets (SLD).
Depending on the weather conditions and the exposure time, two types of ice can form on a flying airplane: rime and glaze. Rime ice formation occurs at very low temperatures and low LWC. It presents an opaque texture and lower density. It is formed when water droplets freeze as they impact the plane surface. Air can remain trapped in this kind of formations hence the low density and texture of rime ice. Glaze ice instead is characterised by a higher density and clear color. Water particles do not freeze upon impact, instead they form a liquid film which can later freeze creating the ice formation reported in figure 1.2.
Cloud Type LWC (g/m3) Cirrus 0.002-0.3 Fog 0.06 Stratus 0.28-0.3 Cumulus 0.26-0.3 Stratocumulus 0.44 Cumulonimbus 1.0-3.0
In order to avoid ice formation, manufacturers have devised with different measures. Anti-ice systems are used to avoid any ice from forming on a plane. Examples of such systems consist in heating the wing surface with electrical resistances or hot air from the compressor. Another alternative is to spray the airplane with anti-icing chemicals before departure. These substances have the function of lowering the freezing point of water or creating a layer on which ice slips away. A different approach is followed by de-icing system, where ice formations are allowed and then removed when they grow above a given limit. Pneumatic boots are installed on the aerodynamic surfaces and are inflated in order to break-up ice formations causing them to detach.
The reasons highlighted earlier led to a great research effort in understanding the physics of ice-accretion, which brought to the development of models and computer software able to predict the ice formation. Ice accretion softwares are nowadays a fundamental tool in aircraft design. They are used to predict ice shapes, to investigate performances degradation of the lifting surfaces and to aid the design of anti or de-icing systems. Examples are LEWICE [3] and GlennICE [4] developed at NASA, FENSAP-ICE [5], the two dimensional (IGLOO2D) and three dimensional (ONFENSAP-ICE3D) codes from ONERA [6] and MULTI-ICE [7], the latter developed by the Italian Aerospace Research Center (CIRA).
The simulation of in-flight ice accretion requires the computation of the multiphase flow surrounding the aircraft. The flow is composed by a gaseous phase, the air, carrying a liquid phase made of water particles, the droplets. From the solution of the flow field, the mass of water impinging on the surfaces is extracted. Ice accretion is then computed by solving the energy and mass balance on the surface of the plane. Figure 1.3 reports the main heat flux contribution to the ice accretion problem.
The term multiphase flow is used to represent the flowing of a mixture composed by elements with different phases, dust particles in air, fluidized beds, clouds, or immiscible fluids such as oil and water. The importance of this type of flows in industrial and scientific applications led to the development of many models and simulation tools capable of dealing with the unique characteristics of multiphase flows. A part from being able to describe the coupling of mass, momentum and energy the models must be able to represent the interface between the different phases through which the flow variables may present very abrupt changes (i.e. in the flow of air bubbles in molten steel the density ratio is about 10000). Also the interface between phases may present itself in very complicated geometries and undergo deformations and topology changes due to instabilities. The phenomena involved in such process insist on broad range of scales and thus require a modelling able to capture all the important features.
A class of multiphase flows is particle laden flows, in which a carrier phase, liquid or gas, is mixed with a particulate of another phase. The carrier phase is continuously connected while the other is made of small, immiscible particles. In this set of problems the interface between phases is no longer an issue since the particles are supposed to be separated and instead their characteristic length and concentration become important parameters in describing such flows. The problem of an airplane flying through a cloud of supercooled water droplets can be modelled in this framework. In this chapter the main properties of particle laden flows and their simulation techniques will be briefly
Figure 1.3 Contributions to the energy and mass balance in the ice accretion computation.
described highlighting the assumptions which specialise the general problem to the one of interest.
1.1
Particle Laden Flows
Before starting with a classification of the different regimes of particle laden flows a few definitions and hypothesis are in order. The dispersed phase is composed by particles, which can be solid (dust) or fluid (droplets for a dispersed liquid phase in a gaseous media and bubbles for gas in liquid). Properties related to a single particle will be associated to the subscript p while the ones regarding the carrier fluid with f. The subscript m will refer to properties of the mixture of dispersed and carrier phase. The fluid is considered as a continuum; therefore in order to satisfy this condition the particle diameter must be much greater than the mean free path of the molecules of the carrier fluid. This assumption can be expressed via the dimensionless Knudsen number, namely the ratio of mean free path to a reference length, being much less
than unity. By taking the particle diameter dp as reference length and the mean free
path as twice the ratio of the carrier fluid dynamic viscosity µf to density ρf times
speed of sound cf this condition reads
µf
dpcfρf
In the case of interest water droplets size is described via the Median Value Diameter (MVD), the median value of the probability distribution of the diameters of the droplets. MVD is usually around 15 − 40 µm which at standard temperature and
pressure conditions gives values of the Knudsen number in the order of 10−3. For a
continuum density at a certain point can be defined as
ρ(¯x) = lim δV →0 δM δV x=¯x
where δM is the mass contained in a volume δV around point ¯x. This volume however cannot be taken arbitrary small as no real element is actually a continuum, but must contain enough molecules to yield a stationary average, meaning that if the volume was slightly changed the average would remain the same. This volume can be considered a point if its dimension is much smaller than the characteristic length of the flow; this is just another way to say the system can be considered as a continuum. This concept is extended to the dispersed phase and it is used to defined some important properties.
The particle number density np is an indication of the number of particles per unit
volume and is defined as
np(¯x) = lim δV →δV0 δN δVm x=¯x
where δN is the number of particles in δV0 which is the smallest volume yielding a
stationary average. Clearly the limiting volume for the mixture will be much larger than the one of the only carrier phase. The volume fraction α is defined as the ratio between the volume occupied by one phase to the total volume of the mixture
αi(¯x) = lim δV →δV0 δVi δVm x=¯x i = p, m
The bulk density of a specie is its mass per unit volume of mixture, and can be expressed as
¯
ρi= αiρi
while the mass concentration Cp of the dispersed phase is defined as
Cp=
¯
ρp
¯
ρf
Similarly the mass loading z is defined as the ratio of the dispersed phase mass flux to the carrier phase
z = ρ¯pup
¯
ρfuf
Dilute and Dense phase
Concentration and spacing between particles play an important role in the dispersed phase as they dictate interference between particles. A dispersed phase in which particles can be taken as isolated entities without considering the presence of other particles is called a dilute phase; otherwise it is said to be dense and particle interactions must be modelled. This interference can be two-fold: particles can collide with each other, or they can modify the fluid flow around them, thus changing the fluid dynamic forces. The number of collisions per unit time is assumed to be proportional to the number density times the section area of a particle and a measure of the relative
d
pL
Figure 1.4 Two particle system used to define inter-particle spacing.
velocity between particles. Particle-particle collisions can be assumed to be negligible
if the number of collisions during the characteristic time of the flow field is small1. By
taking the terminal velocity Vterm of a particle in quiescent flow as a measure of inter
particle relative speed a criterion for negligible collisions is
Vtermnpπd2L
VR
1
where VR and L are the reference velocity and length of the system.
Interaction via the flow usually gives more restrictive criteria. Considering a simple system composed by two particles like the one in figure 1.4, the volume fraction of the dispersed phase is
αp=
πd3
6l3
where the particles are considered as three dimensional, and thus the spacing between
particles can be written as a function of αp
l
d =
π
6αp
13
As a criterion for negligible particle-particle fluid dynamic interaction we can take
α1/3p 1. In clouds the parameter related to the concentration of the dispersed phase
is the Liquid Water Content (LWC) that provides the mass of water in a unit volume of air; the volume fraction of the dispersed phase is obtained by dividing the LWC by the density of water. As reported in [9] typical values of LWC in in-flight ice
problems range from 0.002 to 3 g/m3 and therefore this type of interaction can be
neglected.
1 In fact, the criterion is to confront the characteristic time with the longest between the flow
characteristic time and the particle response time which will be defined later in (1.2). For Stokes number (1.5) around unity no error is made by considering only the former.
Kinetic Equilibrium
An important parameter used to characterize particle laden flows is the response time
τpof a particle subjected to drag forces to a step change of the carrier fluid velocity,
defined as
τp=
ρpd2
18µff
(1.2) The above formula is derived for a sphere in a stokesian flow and the coefficient f takes into account the effects of non-stokesians particles. It is defined as the ratio of the actual drag coefficient to the Stokes drag coefficient. The parameter governing the drag coefficient is the particle Reynolds number defined as
Rep=
ρf||up− uf||dp
µf
(1.3)
where ||up− uf|| is the relative velocity between a particle and the surrounding fluid.
The Stokes drag coefficient used in obtaining the expression (1.2) is defined as CDStokes =
24
Rep (1.4)
A particle is said to be in kinetic equilibrium with the surrounding flow if it readily adapts its velocity to the one of the carrier phase. This can be expressed with the particle Stokes number
Stp=
τp
τL
(1.5) namely, the ratio of the particle response time to the characteristic time of the flow
τL. For very low Stokes numbers the particles response time is much lower than the
characteristic time of the flow τp τL and thus it adapts almost instantaneously
to the external velocity changes, the particles subject only to drag forces act as a tracer and their trajectories follow very closely the flow streamlines. As the Stokes number increases, particles take more time to adjust to the external flow velocity and
for Stp → ∞their trajectory are independent from the carrier fluid motion. For the
case of water droplets in high speed flows Stokes regimes vary in a range around unity: for small droplets of 10 µm diameter flowing at 200 m/s diameter, the particle Stokes
number is of the order of 10−2 while for larger particles, such as Supercooled Large
Droplets (SLD), it can exceede unity.
1.1.1
Phase Coupling
Volume fraction and mass concentration are key parameters in determining coupling between phases in particle laden flows. If the dispersed phase does not alter the dynamics of the continuous fluid the flow is said to be one way coupled; otherwise it is two way coupled. Coupling can happen through mass, momentum and energy transfer between the two phases. Mass can be exchanged through evaporation and condensation; momentum coupling is given by the fluid dynamic forces exchanged at the interface of the two phases and energy coupling occurs through heat transfer. In the following, water droplets in a cloud are considered to be initially in thermal equilibrium with the surrounding air. As the fluid properties change close to solid boundaries from their far field values, differences in temperature may arise. Due to the
high speeds of the flows usually encountered in ice accretion problems, particles tend to transit in these zones of non-equilibrium quickly thus changes in their temperature are negligible given the difference between the heat capacity of the two phases. Therefore particles temperature is considered constant at all times. Given the low volume fraction of water droplets this difference in temperature is not enough to change the properties of the carrier fluid and therefore energy coupling won’t be considered. Evaporation and condensation are negligible before interaction with the solid walls and therefore also mass coupling will be disregarded.
Momentum Coupling
Coupling of momentum between the two phases obviously depends on the concentration of the dispersed phase: for high concentrations the forces resulting from a change in momentum of a great number of particles will produce a sensible change to the net momentum of the fluid while if the concentrations are lower the change will be negligible. A momentum coupling parameter can be obtained by confronting the momentum flux of the two phases. The dynamics of the particles in high speed flows is mainly governed by the drag forces therefore we can define a momentum coupling parameter as
Π = Dp
Mf
where Dp is the drag force due to the particles in a reference volume and Mf is the
momentum flux through he volume of the carrier phase. By considering the particles as spheres subject to a drag force obtained for a Stokes flow this parameter becomes [16]
Π = Cp
1 + Stp
Therefore for cases in which the particle Stokes number is close to unity negligible momentum coupling is equivalent to having a mass concentration (or loading) much
lower than one; Elghobashi [11] suggests to use z < 10−3.
A more thorough analysis of the different coupling regimes in particle laden flows is found in [11], were interaction with turbulence is also considered. The presence of turbulence doesn’t override the criteria stated earlier but it introduces new ones regarding the interaction of particles with the smaller scales of turbulence. As an example in the case of the criterion for momentum coupling the one given reduces to negligible coupling on the mean flow, while if using the Kolmogorov scale to derive the same expression a criterion is obtained for negligible coupling on any part of the turbulent spectrum. In this work the effect of turbulence on the dispersed phase will not be taken into account. For high Stokes numbers particles trajectories are almost independent of the carrier flow and therefore also of its turbulent state; for very low Stokes number instead they tend to follow even the smallest turbulent structures. For the intermediate Stokes numbers of interest there the effect of turbulence depend on
the ratio of particle density to carrier fluid density. Buoyant particles (ρp/ρf < 1)
are more prone to have their trajectories disturbed by the low pressure core of the
turbulent eddies whilst heavy particles (ρp/ρf > 1) are less affected. Therefore in
the case considered turbulence effects are mitigated by the higher density of water compared to that of air. Another observation that can help justify the choice of
10-7 10-6 10-5 10-4 p 10-4 10-2 100 102 p / L
2
3
4
1
Figure 1.5 Coupling zones as defined by Elghobashi [11].
neglecting the effects of turbulence is that the cases considered consist of external high speed flows past aerodynamic bodies in which effect of turbulence are mainly confined inside the boundary layer and the wake. For dispersed phases in boundary layers and internal flows the effect of turbulence should be taken into account. A thorough description of those effects can be found in the work of Loth [12].
Referring to figure 1.5, Elghobashi identifies four different zones each relative to a different coupling regime. These zones are bounded by values of volume fraction of the dispersed phase and particle Stokes number. Zone number four represents particle laden flows with a dense discrete phase characterized by a full coupling between phases and non negligible particle-particle interaction; these flows are said to present a four-way coupling. Fluidized beds present this type of coupling. The other zones represent a dilute dispersed phase. The first contains flows regimes in which the carrier fluid dynamics doesn’t depend on the particulate, the coupling is only one-way; pollutant dispersion in the atmosphere can be modelled in this context. Flows in zone two and three are characterized by a two-way coupling regime in which the momentum, mass or energy transfer from the particles is enough to influence the turbulent structures. For a given volume fraction at lower Stokes numbers this
interaction enhances turbulent dissipation, region three, while increasing Stp results
in higher particle Reynolds numbers which in turn results in vortex shedding taking place thus increasing turbulent production. An example of two way coupled particle laden flows can be found in combustion processes where the dispersed phase made
of atomized fuel ignites and the energy transfer from the dispersed to the continuous phase must be accounted for.
Given the low volume fractions of droplets in clouds and the limited effects of turbulence discussed above, this work will insist on particle laden flows that present one-way coupling. In the next section simulation techniques for particle laden flows are analysed, highlighting their pros and cons, and the most suited for the case at hand will be selected.
1.2
Numerical Simulation of Particle Laden Flows
Description of particle laden flows is usually classified by the treatment of the dispersed phase, which can be cast either in an Eulerian or Lagrangian framework. In the Eulerian framework the characteristics of the particulate are defined as continuous functions of space and time, and can be computed in each point of a computational grid which can be the same used for the computation of the carrier phase solution. The Lagrangian treatment consist on tracking the particles following their trajectories. The continuous fluid is instead always described in an Eulerian framework and the two approaches are therefore called Euler-Euler and Euler-Lagrange.
In an Eulerian perspective the fluid properties at a fixed location in space are sampled in time, thus providing the following representation:
u = u(x, t) (1.6)
where u is the fluid velocity at the point x and time t. In this representation both space and time are independent variables. In the Lagrangian framework single elements of the fluid, or material points, are followed as they move through the flow, and thus the time is the only independent variable. The position of each element is expressed as:
r = r(r0, t) (1.7)
where r0 denotes the position of the material point at a reference time t = t0. The
Lagrangian velocity is defined as the time derivative of the position:
v = dr
dt(r0, t) (1.8)
At a given position and time, the fluid velocity expressed in the two frameworks must be the same, therefore:
u ˜x, ˜t |˜x=r(r
0,˜t) = v r0, ˜t
(1.9) which is dubbed in [25] as the fundamental principle of kinematics, stating that the fluid velocity at a fixed position, the x on the left side, is the velocity of the fluid element that passes for that position at that instant. Also the rate of change of fluid properties cast in the two frameworks can be related to one another via the use of the material (or Lagrangian) derivative D/Dt:
df dt(r0, t) = Df Dt = ∂f ∂t + u · ∇f (1.10)
Figure 1.6 Cell-centered (hashed) and node-centered (shaded) finite volumes from reference [36].
By taking the material derivative of the fluid velocity, the acceleration of a particle of fluid is obtained for a velocity field expressed in an Eulerian framework.
Most popular solution strategies developed for partial differential equations, such as the ones governing the motion of fluids are based on an Eulerian framework; the most common approaches are the Finite Volume Method (FVM) and the Finite Element
Method (FEM). Computational methods based on a Lagrangian frame were also
developed, in particular the Smoothed-particle Hydrodynamics method [23], which was developed to solve problems related to astrophysics, but was later also used in fluid dynamics applications. FVM is today’s standard for fluid dynamics simulations and it is the method implemented in most commercial and free CFD codes. In the Finite Volume Method the discretized domain is represented by a mesh, and the integral form of the conservation equations is solved on a set of so called finite volumes. Those volumes are non-overlapping and sum up to the discrete domain. If the finite volumes are coincident with the mesh elements, the FV discretization is said to be cell-centered, whereas if they are built around the nodes, it is called node-centered. In figure 1.6 the volumes related to both discretizations are shown.
For the exemplary PDE
∂u
the integral form on the volume Vi reads: ∂ ∂t Z Vi u (x, t) = Z ∂Vi n · f (1.12)
For each volume Vi the the equation is solved for the following unknown
ui(t) = 1 Vi Z Vi u (x, t) ∀x ∈ Vi (1.13)
The discrete form of the PDE under the finite volume approximation is obtained by using 1.13 in the integral form 1.12 and finding an appropriate way to discretize the flux on the rhs of the latter equation.
1.2.1
Eulerian Description
Within the Euler description, therefore, the properties of the dispersed phase are considered as continua; control volumes must contain a large number of droplets, so that local properties such as number density or volume fraction can be defined in the proper limit. This hypothesis allows one to use the same discretization and numerical schemes used for the carrier phase. This allows for a straightforward description of two-way coupled regimes as the set of equations for both phases can be solved simultaneously given a proper modelling of interphase exchange terms. Euler-Euler approach fails for vanishing small volume fraction of the dispersed phase as the particulate can no longer be considered as a continuum. A continuum description of a set of particles is justified when it is possible to define a geometrical scale which is both much smaller than the scale of variation of the flow parameters and large enough to contain a significant number of particles. To obtain an estimate of such scale Varaksin [17] considers a cubic element of mixed flow with edge l that must
contain Np particles. The particle volume fraction of such element can be written
considering the dispersed phase as composed of spheres of diameter dp:
αp= π 6 dp l 3 Np
and used to obtain an expression for the ratio of the cube edge to particle diame-ter: l dp = 3 s Np 6 Np αp
In figure 1.7 is reported the ratio of length scale of a volume containing 50 particles to particle diameter as a function of volume fraction. It shows that the scale increases with decreasing volume fraction and with increasing particle size. The smaller l with respect to the scale of variation of the flow the more particles are present in the system. Therefore as the concentration increases and particle inertia decreases the continuum approximation used for the Euler formulation for the dispersed phase becomes more and more valid.
10-6 10-5 10-4 10-3 10-2 p 0 50 100 150 200 250 300 l/d p
Figure 1.7 Ratio of length scale to particle diameter for a cube containing 50 particles.
Another shortfall of this approach comes from the intrinsic nature of a continuum description: representation of the dispersed phase through an Eulerian framework doesn’t provide information on the positions and velocities of the single particles. This results in difficulties in modelling particle-wall interactions such as a particle sticking to or rebounding from a surface which is of paramount importance in the in-flight ice accretion framework of interest here. This can also be an issue if each particle-particle collision must be resolved, as the continuous description allows to treat this interaction only in a stochastic manner. Moreover, if the particle properties are not homogeneous, only a Lagrangian description of the dispersed phase yields all the data needed to resolve each collision. The Eulerian description can be further divided into mixed-fluid and separated-fluid approach.
Mixed-fluid
The the mixed-fluid representation is based on the Local Homogeneous Flow (LHF) approximation presented in [15]. The two phases are supposed to be in local kinetic and thermal equilibrium therefore differences in velocity and temperature of the two phases are considered to be zero. As such the contribution of particles inertia forces is negligible as relative acceleration is zero. The analysis is equivalent to considering a single phase fluid whose properties vary due to the variations in concentration of the dispersed phase. This method only computes the volume fraction of the two phases in addition to the properties of the mixed-fluid. In addition it does not require the knowledge of the size and shape of the dispersed phase and therefore it is the simplest
way to represent some class of particle laden flows. When particle dimension and
time scale are small compared to that of the carrier flow (Stp 1and dp L) this
approximation can be acceptable as the relative velocity between phases is negligible. The mixed fluid approximation uses the same set of equations used for a single phase fluid with modified bulk properties such as density, viscosity, speed of sound etc. For example the density of the mixed fluid is written as the sum of the bulk densities of the two phases
ρm= ¯ρf+ ¯ρp= (1 − αp)ρf+ αpρp
Also viscosity, specific heat ratio, specific energy can be computed from the values of the single phases using the volume fraction or the mass loading. In the case of an inviscid flow the equations can be written as a function of mixed fluid density, velocity and energy: ∂A ∂t + ∂Bj ∂xj = S
where A is the vector of the transported quantities, B the fluxes and S the source terms. A = [ρm ρmumi ρmem] T B = [ρmumi ρmumjumi ρmumiem]T S = [0 Smom Sener] T Smom= (1 − αp) ∂p ∂xi + ρm ∂ ∂xj (µm(∂umi/∂xj+ ∂umj/∂xi)) Senerg= (1 − αp)p ∂umi ∂xi + ρm ∂ ∂xj (µmumj(∂umi/∂xj+ ∂umj/∂xi))
Pressure p is obtained from the equation of state, i.e. in the case of a constant specific
heat (polytropic) ideal gas p = ρmRmTm, and the volume fraction can be computed
as
αp=
ρm− ρf
ρp− ρf
where the carrier fluid density is obtained from its equation of state and the hypothesis
of thermal equilibrium between phases, for example ρf = p/(RfTm).
The mixed-fluid model provides a simple way of simulating particle laden flows characterized by very low particle Stokes number presenting one or two-way coupling. When interphase differences are not negligible this model yields large errors [15], therefore other Eulerian representations were developed to take those difference into account.
Separated-fluid
This second group of Euler-Euler methods is called separated-fluid approximation and assumes that both phases are two separated and intermixed continua, and therefore it is also called two-fluid approach. Modelling two different fluids this approach can take into account the difference in velocity and temperature between phases that were neglected in the LHF approximation. The equations for the dispersed phase read ∂Ap ∂t + ∂Bpj ∂xj = Sp
where the A, B, S are function of particle velocity, density, density number and volume fraction Ap= [αp αpupi αpep] T Bp= [αpupj αpupiupj αpupjep] T Sp= [0 npFpi npupiFpi] T (1.14) The terms appearing in the above equations are averaged on a given computational
volume and do not refer to a single particle. In the source term S np and Fpi are the
number of particles and average force component acting on particles in a given volume and need modeling. Equation (1.14) contains three underlining hypothesis: the first
is negligible heat transfer between phases (Qpi = 0) in the source term relative to
the energy equation; the second is no exchange of mass is considered as evaporation
nor condensation are contemplated (Sp= 0in the mass conservation law). The third
hypothesis is considering the dispersed particles as incompressible. This is justified in the case of water droplets in air due to the huge difference in densities. In the case of gas bubbles the incompressible approximation doesn’t hold and the above equations
are solved substituting the volume fraction with the particle spatial density σ = αpρp
and an additional equation of state is needed.
The advantage of the two-fluid approach is the use of similar equations for the description of the continuous and dispersed phases. This allows to use the many years of experience in simulation of single-phase flows and apply the same numerical methods to the entire set of equations. The disadvantages of such models include some loss of information about the motion of individual particles, as well as the difficulties in the formulation of boundary conditions for the dispersed phase on surfaces which bound the flow.
1.2.2
Lagrangian Description
Another class of methods for the simulation of particle laden flows is called Euler-Lagrange description. This time the dispersed phase in no longer considered to be a continua but as a set of discrete particles which move at their own independent velocity and whose properties are computed in a Lagrangian reference frame following each particle along its trajectory. The set of equations to be resolved for the evolution of the dispersed phase consists only of ordinary differential equations, which can be solved with the standard time-step methods available in literature. In flow regimes where the particle volume fraction is too small to consider the dispersed phase as a continuum, Euler-Lagrange methods offer a much more physical description than the eulerian approach described earlier. At higher concentrations the increasing number of particles per unit volume of fluid makes the momentum transfer between the dispersed phase and the carrier fluid not negligible and therefore the properties of the latter depend on the particulate. In this case, two-way coupled regime, Euler-Lagrange method tend to be less efficient than full Euler methods as at each time step the Euler solution of the carrier flow and the Lagrangian step for the particles must be iterated to catch the full behaviour of the mixture. Euler-Euler methods are usually less computationally expensive than Euler-Lagrange ones, as the dispersed phase dynamics
can be described by a number of nodes that is usually smaller than the number of actual particles required by the Lagrange description. Anyway, Lagrangian methods are the best choice for very low concentration regimes and when particle-boundary interaction is of interest. To lower the computational cost of the Lagrangian simulation, a smaller number of particles can be tracked. Each of these particles is the statistical representation of a neighbouring group of droplets having similar characteristics and it is named computational particle or parcel.
As pointed out in [12], the Lagrangian description con be subdivided in two classes: point-volume representation and resolved-volume representation. In the point-volume approximation all of the particle properties and forces are applied to a point, so that the carrier fluid domain and discretization are not altered by the presence of particles. This allows for a faster computation of the carrier fluid properties since the flow around each particle is not resolved. On the other hand the interphase forces have to be modelled with the help of some empiric formulation given that it is not possible to obtain them by integration of the surface stress. In the resolved-volume method the volume of each particle is actually represented and the flow around particles resolved. Formulations of this method were developed to tackle both rigid and deformable particles, and used with both Chimera grids or unstructured solution-adaptive grids. Often this second class of Lagrangian methods is unfeasible due to the large number of particles needed in the simulation of problems of interest in engineering but they are still useful for validating the models for particle forces used in the more common point-volume representation, or to trace for example large ice chunks shedding from the surface of the aircraft.
In the case of negligible effects of turbulence the criterion for point-volume approxima-tion depends on the particles diameter and the discretizaapproxima-tion used for the continuum phase. If the particles are smaller than a characteristic length of the discretization, for example the smallest edge of the numerical grid, point-volume description for the dispersed phase can be used:
dp
∆x < 1
In order to use the point-volume technique the surface-averaged mass, momentum and energy equations must be determined. The focus of this work is on flows of very low concentration water droplets in (almost everywhere) thermal equilibrium with the surrounding air and the particles temperature is considered fixed. Therefore mass and energy equations reduce to:
mp(t) = mp(0)
Tp(t) = Tp(0)
hence only the momentum equation requires substantial modelling, which takes the form:
mp
dup
dt = Fp (1.15)
The term on the right end side of the equation represents all the forces acting on the particle. These forces take into account the effects of drag, lift, gravity, history terms and wall interaction forces. In general drag and gravity forces have the largest impact on most particle laden-fluids, at least when particle-particle interactions can be neglected. In the following possible modelling for the forces are briefly presented.
Gravity and pressure gradient
A local pressure gradient of the carrier flow creates a force on the particle in the direction of the gradient. The pressure force can be written as:
Fp= Z ∂V −P n dS = Z V −∇P dV
where V and ∂V are the volume and surface of the particle, and n the surface normal. By assuming the pressure gradient constant over the integration volume, consistent
with the point-volume hypothesis of dp L
Fp= −∇pVp
The pressure force due to the hydrostatic pressure then reads:
Fg= −ρfgVp
This force can be summed to the gravity term thus obtaining a net force accounting for buoyancy and gravity:
Fg= Vpg(ρp− ρf)
If ρp= ρf the particle is in hydrostatic equilibrium with the still flow and its terminal
velocity is zero. The importance of the pressure force terms can be extimated by confronting them with inertia force of a particle. Pressure gradient can be taken to be
in the order of the continuous fluid acceleration (∼ ρfDuf/Dt), thus obtaining the
ratio: Vp∂p∂t mp dup dt ∼ ρf ρp Duf/Dt dup/dt
If the accelerations of the two phases are of the same order then the ratio of pressure
forces to inertia forces is proportional to ρf/ρp and therefore negligible in the case of
water droplets in air.
Virtual Mass and Stress Gradient effect
The viscous nature of the carrier fluid is responsible for the so called virtual mass effect. The virtual mass accounts for the extra mass of the fluid that is put in motion due to the particle acceleration. The stress gradient effect take into account the opposite process, that is the force on a particle due to a local acceleration in the nearby fluid. The virtual mass effect can be taken into account by augmenting the mass of the particle with the added mass of the fluid that is accelerated. The effective mass results:
me= mp+ ma = ρpVp+ ρfCMVp= ρp(1 + CM/Ψ)Vp
where CM is a coefficient that yields the volume of fluid that is accelerated by a
particle of volume Vp. Values for CM are obtained from approximated theories for
a spherical particle and correction expressions are available for other shapes. CM is
around 0.5 [12] and can grow up to unity for larger bubbles and droplets. The term Ψ in the above expression is the ratio of particle density to carrier fluid density. In the
mass effect can be neglected. Similarly the effect of fluid acceleration can be modelled as [13]:
Fstress= ρfVp(1 + CM)
Duf
Dt
Also in this case, after normalizing the dynamic equation by the particle mass, the stress gradient force is proportional to the density ratio, hence negligible.
Drag
The steady-state drag force for a solid particle can be expressed via the usual non-dimensional drag coefficient, obtained after dividing the force by the particle cross-section area and the square of the relative velocity
Fd= 1 2ρfπ dp 2 2 CD|ur|ur
The drag coefficient is dependent on the particle Reynolds number and many models were obtained starting from the drag on a rigid sphere in a Stokes flow. In this regime,
equivalent to Rep 1
CD=
24
Rep
At higher Reynolds numbers many empirical expressions are available to fit experiments data. The measured drag coefficient differs from the one obtained in Stokes flows
for Rep> 1and presents a drag drop at around Rep= 105 due to the transition of
the particle boundary layer to a turbulent regime. The expressions reported in figure 1.8 are due to Gent [20] and Morrison [25] and are confronted with the Stokes drag and the experimental data from Achenbach, Liebster and Wieselberger reported in [19].
When dealing with droplets and bubbles instead of solid particles, two complications arise: internal circulation and deformability. For particles with large density ratios (Ψ 1) internal circulation was found not to be important and the drag coefficient for a solid sphere can be used without introducing sensible errors. Drag coefficient is also strongly dependent on the particle geometry; for larger droplets, such as the SLD, deformation from the initial spherical shape are common. This deformation is governed by the breakup Weber number:
W eb=
ρp||uf − up||2dp
σ (1.16)
where σ is the surface tension of the droplet. As the Weber number increases the droplets undergo a continuous deformation process that modifies the steady state shape from that of a sphere to a disk. As for the sphere many empirical relations are available for the drag of a disk and the drag coefficient can be corrected to follow such deformations.
The above drag models were obtained for a particle in a uniform free stream velocity flow. To account for the effects of a non-uniform flow fields the drag force can be
10-4 10-2 100 102 104 106 Rep 10-6 10-4 10-2 100 102 104 106 C D Gent Stokes Experiment Morris
Figure 1.8 CDof a sphere ad different Rp. Experimental data and analytical models. extended by the addition of the Faxen force, that for the case of a Stokesian particle takes the form:
Ffaxen= µfπ d3 p 8 ∇ 2u f
The ratio of the Faxen force to uniform drag scales as the square of the ratio of the particle diameter to the characteristic length of the carrier flow field, and therefore becomes important for larger particles.
Lift
Lift forces on particles are due to two separate mechanisms: flow shear and particle rotation. These forces are usually referred to as Saffman lift (shear induced) and Magnus lift (spin induced). They were initially derived for low particle Reynolds
numbers and depend mostly on Rep and a non-dimensional parameter r obtained
normalizing the fluid vorticity or the particle rate of rotation by the time of convection past a particle. The expressions derived by Saffman, and Rubinow and Keller (for the Magnus force) con be written as
Fs= 1.61 r r s Rep ρfd2p||uf − up||2 Fm= π 8ρfd 2 prm||uf − up||2
These expressions where obtained for low Reynolds flows and values of r below unity.
[14]. In order to incorporate the Magnus lift into the point-volume model of the particle, the particle spin must be tracked by solving the angular momentum equation:
Idωp
dt = T
in addition to the momentum one presented in 1.15.
Both lift phenomena have negligible effect for low values of r [16]. Also lift forces depend linearly on the carrier fluid density; as for the case of virtual mass effect these forces can be considered negligible given the much greater density of particles.
History terms
The Basset history force takes into account the unsteady behaviour of the relative velocity which reflects on the forces acting on the particle through the development of the wake. In the case of a rigid sphere at low Reynolds numbers it takes the form: Fbasset= 3 2d 2 p √ πρfµf Z t 0 D(up− uf)/Dt t − t0 dt 0
where it is evident the hereditary nature of this term as it depends on the time history of the difference in fluid and particle accelerations. The considerations about the importance of the virtual mass effect done earlier apply also for the Basset force [16] and therefore history effects will be neglected in the rest of this work.
The Euler-Lagrange model for particle laden flows can tackle both uncoupled and coupled regimes. Now that the main forces governing the motion of a point-volume particle in a fluid were introduced, the governing equations and a possible solution scheme is presented. The equations for the carrier fluid are simply those of a fluid continuum augmented with a source term accounting for the coupling with the dispersed phase. Taking for example the Navier-Stokes equations:
∂ρf
∂t + ∇(ρfuf) = 0
∂(ρfuf)
∂t + ∇ · (ρfuf ⊗ uf) = −∇p + ∇ · τ + S
where the energy balance was not reported as energy coupling is not considered in this work. The term S accounts for the momentum coupling and represents the force per unit volume exerted by the particles on the fluid. It is equivalent to (minus) the forces acting on the dispersed phase:
S = −
Np
X
i=1
fpi(xip)δ(x − xip)
where Np is the number of particles and xipthe position of the i-th particle and fpi the
forces acting upon it. Numerically the Lagrangian solution of the dispersed phase needs to be linked to the continuous flow computational grid in order to obtain a suitable expression for the coupling term. That can be obtained by properly interpolating the
∆tn
Solve carrier flow
Compute par-ticles forces Update parti-cles positions Compute cou-pling terms Convergence? ∆tn+1 yes no tn Load carrier fluid solution Map particles to owners Interpolate solution and compute forces on particles Evolve parti-cles positions t < tn+1 tn+1 no yes
Figure 1.9 Solution of a single time step of an Eulerian-Lagrangian particle tracking solver for a two-way coupled (left) and one-way coupled regimes.
particle properties known at positions that are not necessarily coincident with the grid nodes. The interphase momentum transfer at the m-th grid node can be obtained as: Sm= 1 Vm Np X i=1 fpiWiK(xip, xm) were K(xi
p, xm)is a kernel with compact support that determined the influence of a
particle at xpon the node xmand W is the weighting which depends on the type of
interpolation used. Possible strategies to carry on such interpolation procedure are highlighted in [21]. The carrier flow variables obtained by solving the above equations are used to obtain the forces on the right end side of the particle equation of motion by interpolating the values known at the cell nodes to the particle location, and the new particles positions are computed. In the case of two-way coupling regimes the solution of the carrier flow and the computation of the particles trajectories must be iterated till convergence. The scheme for each time step is illustrated in 1.9. In the case of negligible interphase momentum coupling there’s no need to iterate the Lagrangian solution for the dispersed phase with the eulerian one for the carrier fluid; the continuous solution is independent and particles are free to evolve up the the next time step at which the carrier fluid solution is known.
1.3
One-way Coupled Lagrangian Particle Tracking
In the previous sections an overview of the possible realizations of particle laden flows was given together with some common numerical methods for the solution of such flows. Each method has its own strengths and flaws and behaves best in a certain flow regime. The choice of the model and numerical solution are driven by the flow physics one want to resolve together with the flow regime one want to simulate. For the problem of an object flying through a cloud of supercooled water droplets the physics we are interested in catching is the particle-wall interaction, as it determines the amount of mass that sticks to the walls which is of great importance in computing ice accretion. This interaction is hard to model in an Euler-Euler framework as the position and trajectory of every particle is not resolved. A Lagrange model for the dispersed phase instead allows for a straightforward implementation of models for droplets impinging on walls. Also a continuum representation of the dispersed phase is not justified for very low LWC and volume fractions of droplets in some clouds, hence the choice of an Euler-Lagrange model to represent the problem at hand. Moreover, the fraction volumes considered allow for the momentum coupling to be neglected, further simplifying the problem. Given that also mass and energy transfer between phases are not considered, the problem of the carrier flow becomes completely independent on that of the dispersed phase. That allows to first solve the continuous fluid equations and then, as a post processing, evolve the particles trajectories. This task is often referred to as particle tracking. The hypotheses adopted make the problem simpler than more general two phase flows, leaving only a series of technical difficulties. Given the carrier fluid solution on a computational grid the most time consuming task in particle tracking is the mapping of each particle to the mesh cell that in which they are contained. The cell that contains a particle is called the particle owner. The mapping of a particle to its cell owner is needed to interpolate the carrier fluid solution to the particles positions, to compute the forces acting on the particle itself. Also if a particle owner is a boundary cell, occurrences of the particle hitting a boundary face must be checked in order to apply the right wall interaction model.
In the following, a particle tracking code able to deal with one-way coupled particle laden flows is developed. The solution scheme that will be followed is reported in figure 1.9 for the case of an unsteady carrier flow.
1.4
Thesis goals and structure
The work presented in this thesis focuses on the solution of the multiphase flows encountered by a plane flying through air with droplets. The goal of this thesis is to develop a particle tracking code to solve one-way coupled particle laden flows. This work builds on the previous experience acquired at Politecnico di Milano in developing an integrated framework for the simulation of in-flight ice accretion (PoliMIce). In this project the available implementation of a Lagrangian particle tracking was assessed to compute the amount of water collected by the aerodynamic surfaces subject to the ice-accretion computation. Some limitations emerged from the use of this code
namely problems in the tracking algorithm and a non satisfactory behaviour of droplets rebounding on the walls.
The problems of the tracking algorithm are due to the floating-point arithmetics used in the mapping of particles to their owner cells. Algorithms that make decisions based on geometric tests can fail if these tests give wrong answers because of errors introduced by finite precision operations. A common problem is testing two floating-point numbers for equality, or determining the sign of their difference. If the two values are close, the test is possibly to be dominated by round-off errors and therefore may not return the answer one would expect if doing the same computation by hand on a piece of paper. Geometric tests are vulnerable to this limitation: examples of queries that can be affected by this problem are assessing the collinearity of three points, determining the belonging of a point in some subspace of the plane, the computation of intersections or even determine if two points are coincident. A tracking algorithm aims at finding the mesh cell that contains the particle at a given time. The fastest and most robust way to do so is by following the particle along its trajectory and computing the intersections with the cell faces it encounters along the way. The face-cell connectivity of the mesh can then be used to get the owner cell. Such an algorithm makes use of all the geometric tests pointed out above and is therefore prone to experience failures due to finite-precision computations. A possible solution to this problem is investigated, making use of multiple precision arithmetics to compute the geometric tests needed.
As water droplets hit the aerodynamic surfaces different scenarios can take place. They can either stay on the boundary by sticking or spreading or they can re-enter the domain in case of rebounding or splashing. In the latter case, only a fraction of the droplet mass is re-introduced in the domain. An important parameter in determining which interaction occurs is the normal to the surface in the point of impingement. As the whole computation is done on the discretized domain used for the carrier flow solution, surfaces are approximated by panels (2D) or patches (3D). In the case of coarse meshes the geometry of such surfaces can be grossly approximated especially if large curvatures are present. Therefore the normal computed on the approximated surface can be very different from the actual one thus possibly leading to the use of the wrong wall interaction. Even in the case of a finer discretization some problems persist. Once a droplet splashes or rebounds from a surfaces the angle at which it is re-introduced in the domain depends on the wall normal. In the computation this results in the front of the particles rebounding from the surface presenting a dashed distribution, which is of course not what one would expect. This behaviour should be avoided as it leads to computing wrong trajectories for the rebounding particles. Those particles can fall back to or near the point where they first impacted or be advected downstream. If their trajectories are wrong the amount and distribution of water caught by the boundaries can be distorted. To deal with this problem a continuous approximation of the boundaries was implemented in the particle tracking using splines or other curves.
In order to compute the particles trajectories two aspects are of paramount importance. The first is being able to compute the forces acting on each particle: to do so, it is necessary to interpolate the Eulerian solution known at the grid nodes to the particle
