Experiments in particle-laden turbulence
simultaneous particle/fluid measurements in grid-generated turbulence using particle image velocimetry
Experiments in particle-laden turbulence
simultaneous particle/fluid measurements in grid-generated turbulence using particle image velocimetry
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op dinsdag 5 oktober 2004 om 13:00 uur
door
Christian POELMA scheikundig ingenieur geboren te Spijkenisse.
Prof. dr. ir. J. Westerweel
Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof. dr. ir. G. Ooms, Technische Universiteit Delft, promotor Prof. dr. ir. J. Westerweel, Technische Universiteit Delft, promotor Prof. dr. ir. F.T.M. Nieuwstadt, Technische Universiteit Delft
Prof. dr. ir. G.S. Stelling, Technische Universiteit Delft Prof. dr. ir. A.A. van Steenhoven, Technische Universiteit Eindhoven Prof. dr. ir. L. van Wijngaarden, Universiteit Twente
Prof. dr. K.T. Kiger, University of Maryland
The work presented in this thesis was supported financially by the Dutch Technology Foundation STW (project DSF.4996). Copyright c 2004 by C. Poelma
All rights reserved. ISBN 90-9018560-7
voor Govertine
If I lose my way, or step out of line Let me hear you say, I’ll be fine LUKABLOOM
Contents
Summary xi
Samenvatting xiii
1 Introduction 1
1.1 Motivation . . . 1
1.2 Classification of two-phase flows . . . 3
1.3 Physical background . . . 4
1.3.1 Relevant parameters . . . 4
1.3.2 A single particle suspended in a flow . . . 8
1.4 Aim of this work . . . 11
1.5 Outline of this thesis . . . 13
2 Particle-Turbulence Interaction 15 2.1 Introduction . . . 16
2.2 Relevant parameters . . . 17
2.3 Direct numerical simulations . . . 18
2.3.1 Effect of particles on turbulence spectrum . . . 18
2.3.2 Physical mechanisms . . . 22
2.3.3 Effect of finite particle size . . . 24
2.3.4 Discussion of DNS results . . . 24
2.4 Theoretical models . . . 26
2.4.1 Introduction . . . 26
2.4.2 Physical mechanisms . . . 26
2.4.3 Discussion of results from theoretical models . . . 31
2.5 Experimental Work . . . 32
2.5.1 Introduction . . . 32
2.5.2 Grid-generated turbulence . . . 33
2.5.3 Overview of Experiments . . . 33
2.5.4 Influence of the particles on the carrier phase spectrum . . . 34
2.5.5 Discussion of Experimental Work . . . 40
2.6 Conclusion . . . 41
2.7 Acknowledgments . . . 42 vii
3 Experimental Facility and Measurement Equipment 43
3.1 Introduction . . . 43
3.2 Design motivation . . . 43
3.2.1 criterium 1: high intensity turbulence . . . 43
3.2.2 criterium 2: homogeneity . . . 45
3.2.3 criterium 3: isotropy . . . 45
3.2.4 criterium 4: optical considerations . . . 46
3.2.5 criterium 5: particle trapping . . . 46
3.3 Description of the VerMeer facility . . . 47
3.4 Particle load measurement . . . 49
3.5 Laser Doppler Anemometry . . . 51
3.5.1 Introduction . . . 51
3.5.2 LDA equipment . . . 52
3.5.3 Single-phase LDA measurements . . . 53
3.6 Particle Image Velocimetry . . . 54
3.6.1 Introduction . . . 54
3.6.2 Image processing . . . 56
3.6.3 PIV equipment . . . 57
3.6.4 Discussion of PIV system . . . 58
3.6.5 Single-phase PIV measurements . . . 63
4 Data processing 67 4.1 Introduction . . . 67
4.2 The slotting method . . . 68
4.3 Validation of the slotting method . . . 70
4.3.1 Synthetic data . . . 70
4.3.2 Implementation of the methods . . . 71
4.3.3 Results for the complete signal . . . 72
4.3.4 Results for the intermittent signal . . . 72
4.3.5 Noise filtering and data consistency . . . 77
4.4 Accuracy of spectral estimate . . . 79
4.5 Extension of the slotting method to two-dimensional data . . . 80
4.6 Turbulence statistics . . . 83
4.6.1 Taylor’s hypothesis . . . 83
4.6.2 Turbulence statistics . . . 84
4.6.3 Structure functions . . . 85
5 Single-phase characterization 87 5.1 Laser Doppler Anemometry results . . . 87
5.1.1 Boundary layer suction . . . 88
5.1.2 Turbulence statistics results . . . 89
5.2 Particle Image Velocimetry results . . . 97
Contents ix
5.3 Discussion of the single-phase results . . . 103
6 Two-Phase Measurements 107 6.1 PIV in two-phase flows . . . 107
6.1.1 Index-matching . . . 108
6.2 Phase discrimination . . . 109
6.2.1 Phase discrimination by post-processing . . . 109
6.2.2 Phase separation by optical methods . . . 111
6.3 Combined PIV/PTV set-up . . . 113
6.3.1 System alignment and calibration . . . 114
6.4 Preliminary two-phase tests . . . 117
6.5 Fluid phase measurements: PIV . . . 121
6.6 Particle phase measurements: PTV . . . 122
6.6.1 Particle recognition . . . 123
6.6.2 Particle-pair matching . . . 126
6.6.3 Algorithm performance comparison . . . 127
6.6.4 PTV measurements in the VerMeer facility . . . 130
6.6.5 Combined PIV/PTV measurements in the VerMeer facility . . . 131
7 Two-Phase Results 133 7.1 Introduction . . . 133
7.1.1 Particle details . . . 134
7.1.2 Cases . . . 135
7.2 Particle phase results . . . 135
7.3 Global fluid statistics . . . 138
7.3.1 Virtual origin correction . . . 140
7.3.2 Decay dynamics . . . 142
7.4 Flow structure . . . 144
7.4.1 Preferential concentration effects . . . 148
7.5 Comparison with model . . . 156
8 Discussion and Conclusions 161 8.1 Review of earlier work . . . 161
8.2 VerMeer facility . . . 162
8.3 Preliminary two-phase results . . . 162
8.4 Future work . . . 165
A Comparison DNS results and theoretical model 167 A.1 Introduction . . . 167
A.2 Theoretical model . . . 169
A.3 DNS-results for a decaying turbulent suspension . . . 173
A.4 Theoretical predictions . . . 176
B Equipment Overview 185
C Particle Size Distributions 187
D A model for decaying, particle-laden turbulence 189
D.1 Single-phase . . . 189 D.2 Two-phase flow . . . 191
Bibliography 196
List of Publications 209
Acknowledgments 211
Summary
Experiments in particle-laden turbulence - C. Poelma
Turbulent dispersed two-phase flows are abundant in both industry and nature. The ability to predict the behaviour of this type of flow - either using numerical or theoretical models - will be benificial to a wide range of disciplines. However, a lack of consistent experimental data currently makes validation of both numerical and theoretical results difficult. The main aim of this thesis is to generate experimental data that could fill this void. In order to do so, a facility is built to generate isotropic, homogeneous turbulence. This is done using a grid in a vertical water channel. This new facility is validated and documented for the particle-free case using Laser Doppler Anemometry (LDA) and Particle Image Velocimetry (PIV). The results of both measurement techniques are in good quantitative agreement, and the flow is a close approximation of decaying homogeneous, isotropic turbulence.
Due to the presence of the dispersed phase, there is a relatively high percentage of data drop-out in the PIV results. This increase occurs due to the deteriorated image quality of the two-phase measurements. To process the PIV data, the slotting method is used. This method, originally developed to process non-equidistant LDA data, is able to process the PIV data with-out the need for interpolation of the missing data. The autocovariance function of a signal is reconstructed, from which important quantities can be extracted (i.e. variance, length scales, power spectrum). The advantage of this method is a better performance compared to classical methods (i.e. based on interpolation and Fast Fourier Transforms of the data).
In order to measure both the fluid phase and the particle phase simultaneously, a two-camera measurement set-up is developed. This is chosen over a one-two-camera set-up to be able to optimize the settings and thus image quality for each camera. Using a mirror and a semi-transparent mirror, the two cameras have the same field of view and there is no need for perspec-tive correction. The tracer particles for the fluid phase contain a fluorescent dye, so that they emit light at a higher wavelength than the light forming the PIV light sheet. Phase discrimination can therefore be achieved using color filtering: one camera captures the images of the (fluorescent) tracer particles, while the scattered light of the dispersed phase is blocked. The other camera captures both the tracer particle images and the dispersed phase images. These dispersed phase particles are orders of magnitude brighter than the tracer particles; in practice, the latter can not be distinguished from the background noise. In order to extract velocity data from the dispersed-phase images, an implementation of the so-called Particle Tracking Velocimetry (PTV) technique is implemented. It has been found that at the working conditions in the water channel (i.e. laser
path length, optical depth, dispersed particle size), it is feasible to measure up to a volume load of approximately 0.5%.
Two-phase measurements have been performed succesfully using 5 particle types: glass particles in two size classes, ceramic glass particles in two size classes and neutrally buoyant particles. Measurements have been done at different volume loads (with a maximum of 0.42%, equivalent to a mass load of 1.05%). Even though these loads are very low compared to the loads used in the majority of numerical calculations (often of order unity), significant effects in the decay rates of the turbulent kinetic energy can be observed.
Two significant effects can be observed in all measurements: the ‘origin’ of the turbulence apparently moves to a more upstream location. It is expected that this shift is a result of the change in the turbulence production mechanisms at the grid (i.e. jet instability), but it is not possible to study this in detail in this facility. In order to be able to compare the two-phase results with the single-phase results, the ‘virtual isotropic origin’ is introduced. This enabled a translation along the axial axis, so that the initial conditions of the particle-free and particle-laden cases are more or less identical.
The second effect that is found is the (significantly) lower apparent decay rate for the axial component of the turbulent kinetic energy. With increasing mass load and increasing particle Stokes time this decay rate becomes lower. The horizontal component shows a slightly faster decay. This results in a flow that becomes more anisotropic as it develops.
To explain these phenomena, the production of turbulence by the particles is studied. It is found that even particles that are relatively light (i.e. St 1Rep1) are capable of generating turbulence, due to the alignment of the mean flow direction with gravity. This is in contradiction with observations and rules-of-thumb given in literature. The particle-induced turbulence can account for the slower decay in the axial direction. The faster decay in the horizontal direction can be attributed to the observed decrease in macroscopic (integral) length scale. An alternative explanation is the fact that the production is anisotropic (i.e. mainly in the axial direction), while the dissipation is anisotropic. Since production and dissipation must balance, this leads to the observed higher dissipation. Based on these observations, a one-dimensional Reynolds stress model is developed. This seems to describe the development of the turbulent kinetic energy relatively well.
A final observation is the occurrence of ‘streaky’ structures far downstream. There ap-pears to be a mechanism that generates energy at very large scales (larger than the field-of-view of the current measurement system). The significance of preferential concentration effects is stud-ied, but there seem to be no inhomogeneities in the distribution of the particles strong enough to cause the streaks. Their origin is found to be related to either the wake-like disturbances of the particles or hydrodynamic interaction between the particles. Both of these effects only become visible when the ambient turbulence level has sufficiently decayed.
Samenvatting
Experimenten in een turbulente stroming met deeltjes - C. Poelma
Turbulente gedispergeerde twee-fasen stromingen zijn alomtegenwoordig in zowel industrie als de natuur. Wanneer het gedrag van dit type stromingen voorspeld kan worden - door numerieke of theoretische modellen - dan zal zit van groot belang zijn voor vele vakgebieden. Door een gebrek aan experimentele data kunnen de numerieke en theoretische resultaten echter moeilijk gevalideerd worden. Het belangrijkste doel van dit werk is het genereren van experimentele data om dit hiaat te vullen. Hiervoor is een opstelling gebouwd om een isotrope, homogene turbulente stroming te maken. Dit wordt gedaan door een rooster te plaatsen in een vertikaal wa-ter kanaal. De stroming zonder deeltjes in de nieuwe opstelling is gedocumenteerd met behulp van Laser Doppler Anemometry (LDA) en Particle Image Velocimetry (PIV). De resultaten van beide meettechnieken komen kwantitatief goed overeen en tonen aan dat de stroming een goede benadering is van vervallende homogene, isotrope turbulentie.
Door de gedispergeerde fase is er relatief veel uitval in de PIV data door de verminderde beeldkwaliteit van de metingen. Om deze data te verwerken, wordt de ‘slotting’ methode ge-bruikt. Deze methode, oorspronkelijk ontwikkeld om niet-equidistante LDA data te verwerken, kan de PIV data verwerken zonder dat de ontbrekende data door middel van interpolatie opgevuld hoeft te worden. De autocovariantie functie van het signaal wordt gereconstrueerd, waaruit een aantal belangrijke grootheden afgeleid kan worden (variantie, lengteschalen, vermogensspec-trum). De slotting methode geeft hiervoor betere resultaten dan de klassieke methode (gebaseerd op interpolatie en Fourier Transformaties van de data).
Om zowel de vloeistof- als deeltjesfase gelijktijdig te meten, is er een meetopstelling met twee cameras ontwikkeld. Dit is verkozen boven een opstelling met ´e´en camera omdat nu de instellingen - en dus beeldkwaliteit - van beide cameras onafhankelijk geoptimaliseerd kunnen worden. Door middel van een spiegel en een half-doorlatende spiegel zien beide cameras hetzelfde meetgebied zonder de noodzaak voor perspectief-correctie. De tracer deeltjes voor de vloeistoffase bevatten een fluorescerende kleurstof, zodat ze licht op een hogere golflengte uitzenden dan het licht van het PIV lichtvlak. Er kan dus onderscheid gemaakt worden tussen de twee fasen door middel van kleurenfilters: ´e´en camera ziet de beelden van de (fluorescerende) tracer deeltjes, terwijl het verstrooide licht van de gedispergeerde fase wordt geblokkeerd. De andere camera ziet zowel de tracer deeltjes als de gedispergeerde fase. De intensiteit van deze laatste is echter enkele ordes groter dan die van de tracer deeltjes; in praktijk zijn deze tracer deeltjes niet van de achtergrond ruis te onderscheiden. Om de beelden van de gedispergeerde
fase om te zetten in snelheidsmetingen wordt Particle Tracking Velocimetry (PTV) toegepast. Bij de huidige meetomstandigheden (laser pad lengte, optische diepte, grootte van de gedispergeerde fase) blijkt het mogelijk te zijn om te meten tot een volumefractie van circa 0.5 %.
Twee-fasen metingen zijn uitgevoerd met 5 deeltjestypen: glazen deeltjes in twee groot-teklassen, keramische deeltjes in twee grootteklassen en deeltjes met ongeveer de dichtheid van water. Metingen zijn verricht bij verschillende beladingen (met een maximum van 0.42 vol.%, overeenkomstig met 1.05 massa %). Hoewel deze beladingen relatief laag zijn in vergelijking numerieke werk (vaak van orde 1), worden er significante verschillen gevonden in de vervalsnel-heden van de turbulente kinetische energy.
Twee effecten kunnen worden opgemerkt in de resultaten: allereerst verschuift de ‘oor-sprong’ van de turbulentie naar een meer stroomopwaartse positie. Waarschijnlijk wordt deze verschuiving veroorzaakt door veranderingen in het productieproces van de turbulentie bij het rooster (m.a.w. de instabiliteiten van de stralen uit het rooster). Dit kan echter in de huidige op-stelling niet nader bestudeerd worden. Om de twee-fasen resultaten met de ´e´en-fase metingen te kunnen vergelijken wordt de ‘virtuele isotrope oorsprong’ gedefineerd. Door de twee-fase data over de axiale as te verschuiven naar dit punt worden de beginconditities van de beide gevallen vergelijkbaar. Het tweede effect dat kan worden waargenomen is de (aanzienlijke) lagere ver-valsnelheid van de axiale component van de turbulente kinetische energie. Met toenemende belading en Stokes tijd van de deeltjes neemt deze vervalsnelheid af. De horizontale compo-nent laat een iets sneller verval zien. Het resultaat van deze verschillen in vervalsnelheid is een toenemende anisotropie van de stroming.
Om bovengenoemde fenomenen te verklaren is de productie van turbulentie door deeltjes bestudeerd. Zelfs deeltjes die relatief licht zijn (i.e. St 1Rep 1) blijken turbulentie te kunnen genereren, doordat de gemiddelde snelheid in de opstelling samenvalt met de richting van de zwaartekracht. Dit is in tegenspraak met eerdere resultaten in de literatuur. De door de deeltjes geproduceerde turbulentie kan een verklaring zijn voor de gemeten lagere vervalsnelheid in de axiale richting. Het snellere verval in de horizontale richting kan verklaard worden door de kleinere integrale lengteschaal van de stroming. Een alternatieve verklaring is de anisotrope productie van turbulentie, terwijl de dissipatie isotroop is. Aangezien productie en dissipatie in evenwicht moeten zijn, zal er een hogere dissipatie in horizontale richting waargenomen worden. Op basis van deze verklaring kan er een ´e´en-dimensionaal Reynoldsspanning model opgesteld worden. Dit model beschrijft de ontwikkeling van de turbulente kinetische energie redelijk goed. Tot slot wordt opgemerkt dat er langwerpige structuren in de axiale richting verschijnen in de twee-fasen stroming. Er lijkt een mechanisme actief dat energie genereert op (zeer) grote schalen, groter dan het huidige meetvolume. Het belang van inhomogeniteiten in de deeltjescon-centratie is onderzocht als mogelijke verklaring. De verdeling van de deeltjes over de stroming lijkt echter geen voorkeuren te hebben, zodat deze verklaring onwaarschijnlijk is. De werkelijke verklaring wordt daarom gezocht in het feit dat de ‘omgevingsturbulentie’ stroomafwaarts zeer laag wordt, zodat de zog-gebieden van de deeltjes of hydrodynamische interacties waarneembaar worden.
Chapter 1
Introduction
This chapter describes the importance of dispersed two-phase flows in both industry and na-ture. A classification is introduced based on the volumetric particle load. The main problem in predicting these flows correctly is currently the lack of computing power. To overcome this problem, closure relationships are needed. These relationships require a deeper understanding of the underlying physics of particle-fluid interaction. Some theoretical background of dispersed two-phase flows and turbulence-particle interaction is reviewed. Finally, the first research aim of this project is described: generating experimental data of dispersed two-phase turbulent flows. Grid-generated turbulence is chosen as a base flow, since this is a close approximation to ho-mogeneous, isotropic turbulence. The advantages of using combined whole-field measurement techniques (PIV+PTV) are described. The second aim of this project is a physical interpretation of the results. Three questions are posed to help find this interpretation. Additionally, the results are used to validate predictions made by a newly developed theoretical model.
1.1 Motivation
The occurrence of dispersed two-phase flows in nature and industry is so abundant, that it is surprising how little is understood about the fundamentals of these flows today. Whether it is a turbulent atmospheric boundary layer bringing heavy rain, the dispersion of pollutants in an urban environment or the fluidized catalytic cracking of carbohydrates, the same basic principles govern these flows: all consist of a turbulent continuous phase - most commonly water or air - laden with small particles in suspension. The density of these particles can be similar to the carrier phase or orders of magnitude higher. When their size is small enough, even air bubbles can be considered to be part of this family of flows. Table 1.1 gives a (far from exhausting) overview of some turbulent two-phase suspensions. Evidently, a better understanding of this kind of flow would be a great benefit to predicting their behavior. Practical use of this knowledge may be used in egchemical engineering science, to be able to scale-up process equipment or to improve mixing efficiencies. On a geophysical level, prediction of sediment transport is a con-crete application that could benefit from a better understanding of particle-turbulence interaction.
Table 1.1: Examples of dispersed two-phase flows
Solid-Liquid Liquid-Liquid
fluid catalytic cracking oil/water separation sediment transport milk, mayonnaise
blood flow reactant mixing
Solid-Gas Liquid-Gas
smoke/soot dispersion inkjet printing erosion, sand storms, dune formation cloud dynamics
spray drying aerosol transport snow flake formation beer, champagne
The lack of understanding of the phenomena under consideration is not due to lack of interest: research in this field can be traced back to egStevin (1605) and Newton (1687) and has been continuing ever since. The lack of understanding can be attributed to a number of causes: first of all, there is the problem of turbulence itself. Even though the Navier-Stokes equations have been known for over a century, the research into the behavior of these highly non-linear equations is still extensive. The non-linearity of the equations leads to chaotic behavior, making a generally applicable solution virtually impossible. In order to do any sensible predictions, either a large number of realizations have to be calculated or parts of the equations have to be modeled (eg by looking at averages). This modeling eventually reaches a stage at which the so-called ‘closure problem’ is encountered: as a result of averaging an equation or term, one or more extra terms appear. These extra terms have to be modeled too, so no analytical average end solution can be found.
While numerical simulations have become an invaluable tool to study single-phase flows, today’s computational capabilities are not sufficient to do exact simulations of complex flows in real-life geometries. The main reason for this is evident: a direct numerical simulation (DNS) divides up the physical space in cells, after which the numerical integration of the Navier-Stokes equa-tions yields one realization of the flow. The number of cells needed to fully resolve these flows depends on the Reynolds number of the flow, since a higher Reynolds number implies a large separation of large and small scales. Today’s supercomputers are capable of higher and higher Reynolds number simulations, yet for engineering applications, simplifications still have to be made (Pope 2000).
When particles are added to the flow, the only way to exactly describe the system, including the interaction of the phases, is to fully resolve the particle surfaces. This is needed because in order to calculate the force on a particle exerted by the fluid, one needs to integrate the stress tensor over the surface (and the volume integral over the body forces). This increases the already significant number of required cells (and thus memory and processing speed) so dramatically, that at the moment only very simple geometries with limited numbers of particles can be studied this way (Ten Cate 2002). Most numerical work therefore uses simplifications, eg assuming
1.2. Classification of two-phase flows 3 the particle to be a point-force and choosing a model for the particle equation of motion (eg Stokes’ drag law). Alternatively, one can choose to model ensembles of particles instead. As a limiting case, the two-fluid model considers the dispersed phase as an interpenetrating fluid. It is beyond the scope of this thesis to elaborate on all approaches that have been introduced to tackle two-phase problems. For more details, one is referred to egthe book by Sommerfeld et al. (1997). In Chapter 2.3, an overview of important numerical work on two-phase flows is given.
A second route to understanding turbulent suspensions is by means of theoretical work. As men-tioned earlier, due to the mathematical obstacles associated with turbulence, i.e. chaotic behavior, no ‘ultimate’ solution exist. Attempts at quantitative predictions based on analytical descriptions have been made by eg L’vov et al. (2003), but these are limited to idealized flows. Modeling seems to be the next best thing, but even more than single-phase flows, these models rely on em-pirical constants, egin closure relationships and above all the effective drag coefficient. This coefficient relates the drag force to the ideal Stokes drag and is a function of, among others, the particle Reynolds number, the free stream turbulence level, and particle volume load. No general solution has been found that describes this complex behavior. An overview of some theoretical models is given in Chapter 2.4.
The final way of gaining insight into turbulent suspensions is by experimental investigation. The presence of particles strongly limits the applicability of the standard single-phase techniques: either due to mechanical problems as with the fragile Hot Wire Anemometers or due to optical problems as with Particle Image Velocimetry or Laser Doppler Anemometry (opacity of the fluid). A review of the experimental work that has been done in two-phase flows is given in Chapter 2.5. A major problem is the fact that it is difficult to compare the experiments directly, given the large number of parameters involved. Even if the experiments can be compared, there are contradictory results. Examples of this can be found in Chapter 2.5.
1.2 Classification of two-phase flows
In order to understand the way particles and fluid interact, a classification can be made, based on their volume or mass load (Elghobashi 1994):
one-way coupling: in these flows, the effect of the particles on the fluid phase is negligi-ble. It suffices to calculate the fluid phase behavior and subsequently integrate the particle equations of motion. The particles behave as passive (non-ideal) tracers, following all, a part or none of the fluid motions. Rules-of-thumb found in literature state that this regime ends at a volume load of 10 6(Elghobashi 1994) to 10 5(Elghobashi and Truesdell 1993).
Due to the fact that particles may not follow all fluid motions, so-called preferential con-centration (or ‘clustering’) can occur. This is especially the case when the Stokes number,
St1, is of the order of unity (Eaton and Fessler 1994).
two-way coupling: with a higher load (or with heavier particles), the particle phase starts to influence the fluid phase. In practice, the turbulence level can be attenuated or aug-mented, depending on the particle characteristics. The fluid phase and particle phase equa-tions should be solved simultaneously. Boundaries for the two-way coupling regime are often stated at volume loads of 10 5and 10 2.
four-way coupling: if the particle load is increased even further, particle-particle inter-actions should be taken into account (viz collisions, hydrodynamic interactions). In this regime, the distribution of the particles can become significantly non-random, with large regions devoid of particles. These large-scale structures can for instance be observed in fluidized bed reactors.
In this project, the focus will be on the two-way coupling regime. As mentioned, the particle characteristics determine whether the fluid turbulence is damped or enhanced. Despite numerous suggestions, there is no single parameter that seems capable of prediction this behavior. El-ghobashi (1994) suggests the use of the particle Stokes number as a distinguishing parameter: for St100, turbulence is enhanced, while for smaller values turbulence is dissipated. The idea behind this is the fact that a less responsive particle will exhibit a greater instantaneous slip ve-locity (due to eg‘overshooting’ in a vortex). If this slip velocity is large enough, it may generate turbulence. Another often-cited rule (Gore and Crowe 1989) suggest that the ratio of particle size and fluid length scale is the discriminating parameter (i.e. dpΛ 01 means turbulence damp-ening, otherwise augmentation). This criterium was based on a large survey of experimental data. A final criterium, suggested by Hetsroni (1989), is based on the particle Reynolds number: for Rep400, particle are supposed to generate turbulence by means of vortex shedding. How-ever, there is not sufficient experimental evidence to verify these claims. Also, contradictory experimental results are found (see Chapter 2). This project hopes to shed some light on this subject.
1.3 Physical background
At this point, some basic concepts of dispersed two-phase flows are introduced. This is done using two approaches: first of all, a summary is given of all relevant physical properties. Using these, a number of dimensionless parameters can be introduced to define different regimes. The second approach starts at the behavior of a single-particle suspended in a flow.
1.3.1 Relevant parameters
It is no simple task to give an overview of the parameters involved in dispersed two-phase flows. Even for idealized, single-phase turbulent flows there is a range of physical properties that
de-1The Stokes number is defined as the ratio of the response time of the particle and a representative time scale of
1.3. Physical background 5 scribe the flow. A reason for this is that there is separation of scales, inherent to turbulent flows. This separation of scales originates in the so-called cascade process. Energy is fed into the flow at large scales (due to ega pressure gradient). These large structures (or eddies) are unstable and break-up into smaller eddies. This process continuous until the fluctuating motions are so small that viscous dissipation is dominant and the remainder of the fluctuations is converted into heat. For a more details on turbulence, one is referred to the classic text books (Hinze 1959, Tennekes and Lumley 1972, Pope 2000). The large-scale structures can be characterized by a length scale Λ, a velocity scale u and a time scale T . In idealized turbulence, i.e. homogeneous isotropic turbulence, it is then sufficient to specify a dissipation rate (ε) and a viscosity (ν) to describe the flow. Based on dimensional arguments, the small scales in the flow (λk, ukandτk) can be derived
from the dissipation rate and the viscosity. These scales are often referred to as the Kolmogorov scales.
In addition to the scales involved in single-phase flows, the presence of particles introduces three new parameters: the particle densityρp, the particle size dpand the volume loadΦv. This volume
load is obviously connected to the mass load (ΦmorΦ) via the densities2
Φm Φvρp
1ΦvρvΦvρp
(1.1) Often, the particle Reynolds number Repand the terminal velocity UTV are introduced as
param-eters to describe suspensions. The former characterizes the local flow around the particle, while the latter is a measure of the maximum settling velocity a particle can attain before drag balances gravitational forces. These two parameters will be further discussed in Section 1.3.2.
Given the particle diameter dp and mass volume loadΦv, the number density np can be
calcu-lated:
np πΦv
6d3p (1.2)
In Table 1.2, all fundamental parameters to describe the overall behavior of a turbulent two-phase suspension are summarized3.
Dimensionless groups Using the scales introduced above, it is possible to create a large
num-ber of dimensionless quantities. The most significant of these are listed in Table 1.3. For the fluid phase, two well-known parameters are the turbulence intensity and the Reynolds number. The turbulence intensity I is defined as the ratio of the root-mean-square value of the fluctuations and
2There can be some confusion over the definition of volume and mass load: often, the mass load is defined as
the mass of dispersed phase per unit of mass of carrier phase. Here, it is defined as the mass of the dispersed phase per unit of mass of suspension. For solid-gas system (i.e. high density ratios), these definitions give the same result.
3this analysis is limited to the case of spherical particles for simplicity, details about the general case can be
Table 1.2: Fundamental and derived parameters in turbulent suspensions particle phase densityρp[kgm 3], size d p[m] fluid phase densityρf [kgm 3], viscosityν [m2 s], dissipation rateε [m2 s 3]
macro length scaleΛ [m], micro length scale λk[m]
macro time scale T [s], micro time scaleτk [s],
macro velocity scale u [ms], micro velocity scale uk [ms]
suspension
volume loadΦv[-]
number density np[1m
3]
mass loadΦm[-]
particle response (‘Stokes’) timeτp[s]
terminal velocity UTV [ms]
the mean convective velocity (u
U). The Reynolds number Re expresses the ratio of inertial over viscous forces. Several definitions are used in the literature. Here, the definition based on the macroscopic length scales is used (ReΛu
Λ
ν). Similarly, one can define the particle Reynolds number (Rep), based on the terminal velocity and the particle size (RepdpuTVν). Using this number, it is possible to characterize the flow behavior around the particle, egboundary layer separation, vortex shedding; for some visualizations of the different flow regimes, one is referred to egthe book by Van Dyke (1982).
In a random distribution, the ratio of the mean distance between two particles and their radius only depends on the volume fraction. The order of this value can easily be estimated from a particle distribution on a regular grid4:
δ dp π 6Φv 1 3 1; (1.3)
In Figure 1.1, the mean spacing is plotted as a function of the volume load. This spacing can be used as an alternative classification of two-phase flows, since the importance of particle-particle interactions can easily be estimated. If the particles are not evenly distributed, due to eg pref-erential concentration effects, additional parameters (i.e. a length scale) have to be introduced. More details about this can be found in Chapter 2.
To compare the response time of a particle (τp ρρpf d 2
p
18ν, see next section) and a typical flow time
scale, the Stokes number is introduced (StτpT ). In the literature both the macroscopic time
4Note that for randomly distributed points, the solution is given by Chandrasekhar (1943, p.87): δ
0 53396Φ 1 3. The difference, apart from the constant, is the fact that the Chandrasekhar’s equation describes
1.3. Physical background 7 10−6 10−4 10−2 100 101 Φv δ/d p
Figure 1.1: The dimensionless mean distance between particles as function of the volume fraction.
scale as well as the microscopic time scale of the flow are used. A very small Stokes number indicates a particle that follows all fluid motions. A very large Stokes number indicates a non-responsive particle. For Stokes numbers of the order of unity, the particle is susceptible for large scale motions, yet unresponsive to small scale motions5.
To check if gravitational effects are dominant, the dimensionless parameterβuTVu
is intro-duced. This parameter compares the terminal velocity of a particle to a typical fluid fluctuation (i.e. the root-mean-square value of the fluctuating velocity component). Forβ1, gravity will be dominant and there will be little time for the particle to interact with fluid fluctuations, i.e. the particle is relatively unresponsive. Forβ1, gravitational effects can be neglected.
Depending on the terminal velocity, two regimes exist: for an almost neutrally buoyant particle, the fluid fluctuations that are observed by the particle are caused by the temporal changes in the fluid. Therefore, the response time of a particle should be compared to the typical ‘life time’ of an eddy, which scales as Lu
. For a particle with a high terminal velocity, the particle can be considered to drop through ‘frozen’ turbulence and it will observe only spatial changes. Here, the response time of the particle should be compared to the ‘residence time’ of a particle in a typical eddy. A new parameter can thus be identified: µτpuTVu
Λ (see also Spelt (1996) for a similar analysis). This parameter obviously is of the same form as the Stokes number. Yet by taking the slip velocity into account, it should provide a better means of describing the particle-fluid interaction. Similar to the Stokes number, a very large µ indicates a non-responsive
5Alternatively, one can compare the particle diameter and the eddy size, St
L dpΛ, which describes the spatial
Table 1.3: Dimensionless groups in turbulent suspensions
fluid phase
u
U I turbulence intensity
Λu
ν ReΛ Reynolds number based on the turbulence macro scales
suspension
ρpρf ratio of particle and fluid densities
δdp dimensionless mean inter-particle distance dpuTVν Rep particle Reynolds number
τpT St particle Stokes number based on the turbulent macro scale uTVu
β gravitational versus turbulence effects τpuTVu
Λ µ particle response number particle, etc
1.3.2 A single particle suspended in a flow
The second approach to understanding the physical phenomena in two-phase flows is by consid-ering the behavior of a single sphere in a flow. Two cases are considered: a particle in a steady creeping flow (the limiting case for Repapproaching zero) and an approximation for a particle in
an unsteady flow.
Stokes flow The behavior of a single suspended, rigid sphere in a flow has attracted the interest
of scientists for centuries. For an overview, see egthe book by Clift et al. (1978). Milestones in this field include the famous analytical solution for the creeping flow case by Stokes (1851), Oseen’s suggestion for an improvement of this solution to take into account the inertial effects in the far field and Lamb’s (1911) approximation for this improved (but unsolvable) solution. All these results were obtained for flows in which the Reynolds number of the particle, Rep, was
sufficiently low. Still, the results can be useful for the understanding of the behavior of flows in which inertia does play a role. For example, the well-know Stokes’s drag law
FD 3πηdpup (1.4)
(in which FDis the force exerted on the particle,η the dynamic viscosity, dpthe particle diameter
and upthe relative velocity of particle), can be used to derive the so-called Stokes time scale or
relaxation time, i.e. the time a particle at rest needs to reach 1e of the velocity of the surrounding fluid: τp ρρp f d2 p 18ν (1.5)
If the Stokes drag force is equated with the gravitational force (minus buoyancy force), the so-called terminal velocity (UTV) can be defined:
1.3. Physical background 9 π 6d 3 pρpρfg 3πηdpup UTV dpρpρfg 18η (1.6)
This velocity is the theoretical limit a particle can reach, falling in a quiescent fluid. Of course, once the particle starts moving, its Reynolds number will increase and also fluid acceleration effects occur. In practice, Equation 1.4 only holds for Rep 05 (Lamb 1945).
Unsteady flow The results obtained in the previous section for the creeping flow case are the
limit of what is still analytically solvable. In‘real’ flows, so with non-zero Reynolds number and non-stationary in time, no exact governing equation (let alone solution) is available. The most used approximation is the so-called Basset-Boussinesq-Oseen (BBO) equation. The original BBO equation described the motion of a particle in a stationary fluid. Tchen (1947) extended the equation to include non-stationary effects. Over the years, several publications have appeared, pointing out inconsistencies and errors in his work. Maxey and Riley (1983) derived the BBO equations in its ‘conclusive’ form:
mpdvdti 1 mpmfgi 2 mf Dui Dt Y t 3 1 2mf d dtvituiYt 1 10a2∇2ui Y t 4 6πaηvituiYtt 1 6a 2∇2u i Y t 5 (1.7) 6πa 2η t 0 dτ d dτvituiYτ 1 6a2∇2ui Y t πνtτ 6
in which uiis the velocity of the particle in the i-direction, at location Yt, mpthe particle mass, mf the displaced fluid mass and a the particle radius. This equation is valid under the
assump-tion that the Reynolds number (based on slip-velocity and particle radius) is smaller than unity. Furthermore, it is assumed that the particle is smaller than the smallest structures of the flow, i.e. the particle is a point force. Note that the mathematical complexity of the equation arises from the fact it uses the ‘memory’ of the flow in the last term and that both Eulerian (ddt) and Lagrangian (DDt) derivatives are used. The labeled terms can be interpreted as follows:
1. the total force exerted on the particle by the fluid 2. buoyancy or gravitational contribution
4. ’added mass’ contribution, due to the fact that a part of the surrounding fluid has to be accelerated along with the particle, i.e. non-viscous unsteady drag. The volume of this added mass is usually assumed to be half that of the displaced fluid.
5. Stokes drag contribution, i.e. steady-state viscous drag.
6. Basset history contribution, arising from viscous unsteady drag. Acting as a memory inte-gral over the relative acceleration of the particle, it accounts for the diffusion of vorticity away from the particle surface (Clift et al. 1978).
Note that this equation is still an approximation (Stokes flow, Rep 1). Therefore, no lift force
can be present due to the symmetry of the flow around the particle. For bubbles (or light parti-cles), it may be necessary to add this term. Saffman (1965) derived the following expression for the lift force:
Fl 646a
2η
upv Gν
1 2 (1.8)
This equation is only valid if the Reynolds number based on the local shear rate (G) is small, i.e. ReG a
2G
ν 1 (1.9)
If also the particle rotation is taken into account, the resulting Magnus effect acts as an extra lift force in the direction of the side where the difference between particle surface velocity and fluid velocity is minimal.
Fortunately some simplifications can be made, based on scaling considerations: when the parti-cles are small compared to the fluid length scales (i.e. aλk 1), the particles can be considered to be a point force. This implies that the flow around a particle is more or less uniform, so that the Faxen terms (present as Laplacian operators of the fluid velocity field) can all be neglected (van Haarlem 2000).
Furthermore, for heavy particles (i.e.ρpρf 1), the steady Stokes drag (term 5) and the grav-itational effects (term 2) are dominant, so the equation is considerably simplified and a similar expression as the earlier result equation 1.6 is retrieved:
mpdvi
dt mpmfgi6πaηvituiYtt (1.10) This equation is the starting point for most numerical simulations. In this project, the continuous phase is water, so ρpρf 12. It this thus a priori difficult to predict which terms will be significant.
1.4. Aim of this work 11
Two-way coupling Once the particle equation of motion has been found, there needs to be
some sort of feedback to the fluid: the force that is exerted on the particle by the fluid - egin the form of Equation 1.7 or in a simplified form - also works in opposite direction on the fluid via Newton’s third law of motion. Thus in the equations of motion for the fluid, these forces have to be incorporated. This is done by adding an extra forcing term to the original Navier-Stokes equations, see egFerrante and Elghobashi (2003, Eq.1-4). Again, this is mostly done as a point-force at the location of the particle.
In addition, one can also account for the presence of other particles. Even if particle collisions do no play a role, particles can interact through so-called hydrodynamic interactions. Recent studies (Poelma and Ooms 2002, Ooms, Gunning, Poelma and Westerweel 2002) indicate that these do not play a role in a highly turbulent flow with moderate loading (up to a few volume percent), unlike egin ‘hindered settling’ of suspensions (Batchelor 1972).
1.4 Aim of this work
There is a lack of consistent data to support the numerous theories for turbulence-particle inter-action that have been developed in recent years. The main aim of this project is to generate a systematic series data sets, describing both fluid and particle properties. Some choices have to be made to limit the scope of the work. First of all, only rigid, approximately spherical particles are considered. This excludes bubbles, since these often have oscillating surfaces. Also, no phase transitions (i.e. evaporation, condensation) are considered. In order to study the fundamentals of the coupling mechanisms, the flow geometry is chosen to be as simple as possible: in the ideal case, homogeneous, isotropic turbulence is preferred. As a close approximation, grid-generated turbulence is chosen, since this can be realized relatively easy in a laboratory. Next to that, a sig-nificant amount of single-phase experimental data is present for comparison (see Figure 1.2 for an example of this type of flow). The particles are distributed as evenly as possible in the flow, so no mean concentration gradients are present. In practice, they will be recirculating in the flow fa-cility, fully suspended by a high mean velocity. A final choice for the geometry is the orientation: either horizontal or vertical. In this investigation, it was chosen to place the channel vertically, so the flow remains axisymmetric. Also, transversal concentration gradients are avoided this way. Measurements will be done with optical whole-field techniques: particle image velocimetry (PIV) and particle tracking velocimetry (PTV); for a detailed description of the methods, one is referred to Chapter 3. PIV was chosen because this yields invaluable information about the spatial structure of the suspension, whereas laser Doppler anemometry and hot-film anemometry only yield single-point information. PIV has the drawback that it is far from straightforward to obtain turbulence spectra from the data, since little work has been done in this field as compared to the single-point methods. Special care will therefore be given to this topic. The single-phase measurement techniques developed for PIV have to be extended to cope with two-phase flows. Additionally, a similar system is added to study the particle-phase simultaneously. The main part of this thesis deals with developing this combined PIV/PTV set-up. Using the data obtained by
Figure 1.2: A visualization of grid-generated turbulence. Photograph by T. Corke and H. Nagib, reproduced from Van Dyke (1982).
this set-up, an attempt can be made to gain more insight into the two-coupling effects in dispersed phase flows.
Due to the limited time available, only some preliminary results are described in this thesis. This analysis is by no means meant to be a conclusive study of dispersed two-phase flows. It is mainly meant as an initial classification of the generated data. The focus will be on three main questions, starting on a macroscopic level and zooming in to the microscopic or particle level:
How does the overall fluid phase behavior change ?
What is the spatial distribution of the particles and how does this relate to the structure of the fluid flow ?
Can anything be said on a qualitative and/or quantitative basis about the interaction of the two phases ?
In order to answer these questions, experiments of the two-phase flows are compared to their single-phase equivalent. Several statistics are available for the first question: turbulence level, decay rates, length scales, autocorrelation functions and power spectra. To answer the second and third question, it is possible to use the unique ability of PIV to measure whole-fields: using conditional sampling, the structure of the flow can be examined in detail. The final question can be answered by studying the effect of different particles and mass loads.
Recently, a theoretical model for predicting the effect of two-way coupling in a turbulent suspen-sion has been published (Ooms and Poelma 2004). This model is described in detail in Appendix
1.5. Outline of this thesis 13 A. Qualitatively, the model agrees well with numerical work by Ferrante and Elghobashi (2003). To see if the model also agrees with experimental work, a comparison with the initial results is made.
1.5 Outline of this thesis
The general lay-out of the thesis is as follows: first, an overview of previous numerical, theoret-ical and experimental work in the field of dispersed two-phase flows available in the literature is given (Chapter 2). The focus of this review is on homogeneous, isotropic turbulence and the power spectrum of the fluid is used as the main indicator of two-way coupling effects. In Chapter 3 a description of the new facility is given, along with the (single-phase) measurement techniques that are used to characterize it. The data processing technique for the single-phase data, the slotting method, is explained in Chapter 4. Additionally, this method is extended to be able to process two-dimensional data. Using the tools described in this chapter, the results for the single-phase flow are reported in Chapter 5. Subsequently, Chapter 6 deals with the extension of the single-phase measurement set-up, so it is capable of measuring both the fluid and particle phase simultaneously. Preliminary results of the two-phase measurements are given in Chapter 7. Finally, in Chapter 8, a discussion and the main conclusions of this project can be found, as well as some recommendations for future work.
Chapter 2
Particle-Turbulence Interaction
1
A review is given of numerical, analytical and experimental research regarding the two-way coupling effect between particles and fluid turbulence in a homogeneous, isotropic turbulent sus-pension. The emphasis of this review is on the effect of the suspended particles on the spectrum of the carrier fluid, in order to explain the physical mechanisms that are involved. An important result of numerical simulations and analytical models (neglecting the effect of gravity) is, that for a homogeneous and isotropic suspension with particles with a response time much larger than the Kolmogorov time scale the main effect of the particles is suppression of the energy of eddies of all sizes. However for a suspension with particles with a response time comparable to or smaller than the Kolmogorov time, the Kolmogorov length scale will decrease and the turbulence energy of (nearly) all eddy sizes increases. For a suspension with particles with a response time in between the two limiting cases mentioned above the energy of the larger eddies is suppressed, whereas the energy of the smaller ones is enhanced. Attention is paid to several physical mechanisms, that were suggested in the literature to explain this influence of the parti-cles on the turbulence. In some of the experimental studies certain results from simulations and models have indeed been confirmed. However, in other experiments these results were not found. This is attributed to the role of gravity, which leads to turbulence production by the particles. Additional research effort is needed to fully understand the physical mechanisms causing the two-way coupling effect in a homogeneous, isotropic and turbulently flowing suspension. This review contains 47 references.
1A modified version of this chapter has been accepted for publication in Applied Mechanics Reviews:
‘Particles-turbulence interaction in a homogeneous, isotropic turbulent suspension’, Poelma and Ooms (2004). There is some overlap with the previous chapters in the first section, yet for the sake of clarity the article has been reproduced in its entirety. N.B.: there are minor differences in notation compared to the rest of this thesis.
2.1 Introduction
The occurrence of dispersed two-phase flows in nature and industrial applications is abundant. Despite a significant research interest, there are still several open questions in this topic, limiting a good quantitative prediction of these flows. In this publication, a review is given of recent progress relating to one of the more fundamental problems: the so-called two-way coupling be-tween the dispersed phase and turbulence. Two-way coupling refers to the effects of the fluid on the particles and vice versa. The review is limited to rigid particles in homogeneous and isotropic turbulence, in order to avoid further complications. For the same reason, no work on shear flows is incorporated. Even though inclusion of these publications would greatly increase the amount of available data, it would also significantly complicate the analysis, since - among other phenomena - shear-induced turbulence production and particle migration might obscure other effects.
Due to the growth of computer power detailed direct numerical simulations (DNS) of the behav-ior of particles in the turbulent fluid velocity field and of the two-way coupling effect became possible. Also theoretical models were developed, that try to capture the main physical mech-anisms occurring in turbulent suspensions. Moreover, because of the progress in experimental techniques new and accurate measurement results became available.
We will give a review of publications about the two-way coupling effect in turbulent suspensions that appeared in the literature during the last decade. It is, of course, impossible to discuss all important publications. However, we hope that the review will give a clear picture of the progress in understanding of turbulent suspensions. Attention will be paid to numerical simulations, theo-retical models and experiments. Emphasis will be placed on the influence of the particles on the turbulent kinetic energy spectrum of the carrier fluid. We know, that turbulence spectra are not sufficient to characterize turbulence completely. However, in the literature particular attention was given to the influence of particles on the turbulence spectrum in order to derive a physical understanding of the two-way coupling effect.
In the first part of this review we will integrate the knowledge from numerical simulations with respect to the dependence of the turbulence spectrum of the carrier fluid on the presence of the particles in the fluid. In the second part such an integration will be given with respect to the-oretical explanations given in the literature on the two-way coupling effect in general and the dependence of the turbulence spectrum on the particles in particular. The third part is devoted to a comparison between experimental data and results from numerical simulations and theoretical models.
Several good reviews about particle-laden turbulent flows were published during the last decade, see for instance Hetsroni (1989), Elghobashi (1994), Crowe et al. (1996) and Mashayek and Pandya (2003). Our review is different, in our opinion, in the sense, that we pay particular attention to the physical mechanisms that play an important role in the interaction between par-ticles and fluid turbulence in a turbulently flowing suspension. In recent years some interesting
2.2. Relevant parameters 17 publications have been written in which detailed information is given about these mechanisms.
2.2 Relevant parameters
We define a number of parameters that are of importance for turbulently flowing suspensions and that will be used in this review. The volume fraction of the dispersed phase (the particles) is defined as
ψ δVd
δV (2.1)
where δVd is the volume of the dispersed phase in the volumeδV of the suspension (assumed
large enough to ensure averaging). The ratio of the average particle spacing (dspacing) and the
particle diameter (dp) is related to the volume fraction by
dspacing dp π 6ψ 1 3 (2.2)
The particle mass loading is given by
φ ψρρp
f
(2.3)
in whichρpandρf are the densities of the particles and the carrier fluid respectively. The particle
response time to changes in the surrounding fluid is expressed by the Stokes time τp 2ρp
d2
p
9ρfν
(2.4)
whereν is fluid kinematic viscosity. For the fluid phase the Kolmogorov time is defined by
τk η2ν (2.5)
withη the Kolmogorov length scale. The integral time scale is given by τΛ Λu
(2.6)
in which Λ is the integral length scale of turbulence and u
the r.m.s.-value of the turbulent velocity fluctuations. The Stokes number is in this review defined as the ratio of the particle time scale and the Kolmogorov time scale
St τpτk (2.7)
For St 0, the particle behaves as a fluid tracer. For St ∞, the particle is unresponsive to the fluctuations of the flow. Obviously, the physically most interesting situation occurs when it approaches unity: particles follow the large scale fluid motions.
Finally, the flow regime around the particles can be characterized by means of a Reynolds num-ber:
Rep uTVνdp (2.8)
In this equation, uTV represents the terminal velocity (the velocity a particle attains falling in a
quiescent medium). Alternatively, it can be defined using eg
the root-mean-square value of the free stream turbulence. In has been postulated by Hetsroni (1989) that Rep can be used to determine whether particles attenuate (Rep 100) or enhance
(Rep400) the fluid turbulence level.
2.3 Direct numerical simulations
2.3.1 Effect of particles on turbulence spectrum
An early indication about the influence of the particles on the turbulence spectrum was given by Squires and Eaton (1990) and Squires and Eaton (1994). They used direct numerical sim-ulation to study statistically stationary, homogeneous, isotropic turbulence. They considered particle motion in the Stokes regime. Gravitational settling was neglected. Computations were performed using both 323and 643 grid points. The particles were treated as point particles. To achieve the stationary flow, a steady non-uniform body force was added to the governing equa-tions. Particle sample sizes up to 106 were used in the simulations. Mass loadings of 0.1, 0.5
and 1.0 were considered. They used the following values for the ratio of the particle response timeτpand the Kolmogorov time scale of turbulenceτk:τpτk 0.75, 1.4, 1.5, 5.2, 7.5 and 15.0. Squires and Eaton calculated the effect of the mass loading on the spatial turbulent kinetic energy spectrum of the carrier fluid. The (dimensionless) spatial spectrum for τpτk 15 as function of the (dimensionless) wave number for different mass loadings is shown in Figure 2.1. Eκ is the turbulent energy spectrum as function of wave number. As mentioned earlier η is the Kolmogorov length and φ the mass loading. The spectra are normalized using q2, the total
turbulent kinetic energy for each particular case. It can be seen, that with increasing mass load-ing the energy at large wave numbers increases relative to the energy at small wave numbers. As Squires and Eaton found that the total turbulent energy decreases with increasing mass loading, it can be concluded that at small wave numbers (where most of the energy is located) the energy decreases with increasing mass loading compared to the particle-free case. Whether the energy at large wave numbers decreases or increases with respect to the particle-free case cannot directly be concluded from the results of Squires and Eaton. For that it would be necessary to multiply the spectra of 2.1 with the total energy q2. The relative increase of the distribution of energy at
large wave numbers with respect to the energy at small wave numbers was found for all particle response times used in the simulations.
More details about the influence of particles on the spatial turbulence spectrum of the fluid be-came available via the work of Elghobashi and Truesdell (1993). In contrast to Squires and
2.3. Direct numerical simulations 19 10−1 100 10−5 10−4 10−3 10−2 10−1 100 kη kE(k)/<q 2 > Φ = 0 Φ = 0.1 Φ = 0.5 Φ = 1.0
Figure 2.1: Effect of mass loading on spatial energy spectra forτpτk 1 5. With increasing mass loading the
energy at large wave numbers increases relative to the energy at small wave numbers. Data taken from Squires and Eaton (1990).
Eaton, they examined turbulence modulation by particles in decaying isotropic turbulence. A point-particle approximation was again made. They used the particle equation of motion derived by Maxey and Riley (1983). Like Squires and Eaton they found that the coupling between the particles and fluid results in an increase in the turbulent energy at the large wave numbers relative to the energy at small wave numbers. Moreover, they concluded from their calculations that the large wave number energy for a flow with particles (with a sufficient small response time) is even larger than the large wave number energy for the particle-free case at the same time in the decay process.
Like Squires and Eaton also Boivin et al. (1998) made a detailed DNS study of the modulation of statistically stationary, homogeneous and isotropic turbulence by particles. Gravitational settling was neglected and the particle motion was assumed to be governed by drag. The ratio of the par-ticle response time to the Kolmogorov time scale had the following values: 1.26, 4.49 and 11.38, and the particle mass loading was equal to: 0.2, 0.5 and 1.0. The velocity field was made sta-tistically stationary by forcing the small wave numbers of the flow. Again the effect of particles on the turbulence was included by using a point-force approximation. Fluid turbulence spatial energy spectra, derived by Boivin, Squires and Eaton, for different mass loadings are shown in Figure 2.2 for τpτk 126 and in Figure 2.3 for τpτk 1138. Note that the figures as they are represented here are replotted with double-logarithmic axes, to facilitate comparison with the other figures. In their simulations the dimensionless maximum value of k is about equal to the number of grid points in each direction of their cubic simulation domain. So the region around k 1 represents the energy containing eddies and we can say that the wave number along the
101 10−4 10−3 10−2 10−1 100 k E(k) φ = 0 φ = 0.2 φ = 0.5 φ = 1.0
Figure 2.2: Turbulence kinetic energy spectrum forτpτk 1 26:φ 0 (solid line), 0.2 (dotted), 0.5 (dashed) and
1.0 (dash-dotted). With increasing mass loading the energy at large wave numbers is enhanced relative to the energy at small wave numbers. Data taken from Boivin, Simonin and Squires (1998).
horizontal axis is made dimensionless by means of the integral length scale. In general there is a similar damping effect on the smaller wave numbers of the fluid turbulence by both particles with a large response time and particles with a small response time. At the larger wave numbers the turbulence kinetic energy is attenuated by particles with a large response time, but increased by particles with a small response time. It should be noted that in Figures 2.1, 2.2 and 2.3 the highest wavenumbers appear to be contaminated with noise, since they show a distinct ‘plateau’ (2.1) or even increase (2.2,2.3) at these wavenumbers. Nevertheless, even if this part of the graph is ignored, the trends mentioned above remain.
A next step in achieving information about the dependence of the turbulence spectrum on the presence of particles in a turbulent suspension was provided by Sundaram and Collins (1999). Like Elghobashi and Truesdell they performed DNS simulations of particle-laden isotropic de-caying turbulence. The particle response time was in the range 16 τpτk 64. The ratio of the particle density and fluid density was of the order 103. The drag force on the particles was
de-scribed by Stokes law and the influence of the gravitational force was neglected. The DNS results showed again that the turbulent energy spectrum of the fluid is reduced at small wave-numbers and increased at large wave-numbers (compared to the particle-free case) by the two-way cou-pling effect. They also concluded that the location of the cross-over point (the wave number where the influence of the particles changes from a damping effect to a turbulence-enhancing one) is shifted toward larger wave numbers for larger values of the particle response timeτp.
2.3. Direct numerical simulations 21 100 101 10−4 10−2 100 k E(k) φ = 0 φ = 0.2 φ = 0.5 φ = 1.0
Figure 2.3: Turbulence kinetic energy spectrum forτpτk 11 38: φ 0 (solid line), 0.2 (dotted), 0.5 (dashed)
and 1.0 (dash-dotted). With increasing mass loading the energy at all wave numbers decreases. Data taken Boivin, Simonin and Squires (1998).
In their DNS calculations, Ferrante and Elghobashi (2003) fixed both the volume fractionψ 10 3and mass fractionφ 1for four different types of particles, classified by their ratio of the particle response time and the Kolmogorov time scale of turbulence. The ratioτpτkhad the values 01, 025, 10 and 50. The ratio of the particle density ρp and the fluid densityρf is 103. In Figure 2.4 the spatial turbulent energy spectrum Etκ for the carrier fluid in the suspension is given (for the case without gravity) at a certain moment during the decay process. κ is the wave number made dimensionless by means of the the integral length scale L. In the figure the result indicated by case A is for the particle-free flow, the results indicated by cases B, C, D and E are for the carrier fluid in the suspension with particles of increasing response time (τpτk 0102510 and 50) respectively. Microparticles (case B) increase Etκ relative to the particle-free flow (case A) at wave numbers κ 12, and reduce Etκrelative to case A for κ 12, such that Et
Etκdκ in case B is larger than in case A (the particle-free case). Also for the cases C, D and E the particles dampen the turbulence at small wavenumber compared to the particle-free flow and enhance the turbulence at large wave number. However the cross-over wave number increases with increasing particle response time. As can be seen from Figure 2.4 large particles (case E) contribute to a faster decay of the turbulent kinetic energy by reducing the energy content at almost all wave numbers, except for κ87, where a slight increase of Etκoccurs.
κ
32 64 96128 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 Case A Case B Case C Case D Case EE(k)
16Figure 2.4: Kinetic energy spectrum of the carrier fluid at t 5 0. The cross-over wave number increases with increasing particle response time. From Ferrante and Elghobashi (2003).
2.3.2 Physical mechanisms
In the publication of Squires and Eaton (1994) a first proposal is made to explain the physi-cal mechanism responsible for the non-uniform distortion of the turbulence energy spectrum by particles. In their opinion this non-uniform distortion is due to the preferential concentration of particles in the turbulent flow field. They showed that particles with a small response time (τpτΛ 1) exhibit significant effects of preferential concentration in regions of low vorticity and high strain rate. The effect of high concentrations of particles in these regions leads to an in-crease in small-scale turbulent velocity fluctuations. This production of small-scale fluctuations subsequently causes the viscous dissipation rate in the carrier fluid to be increased (for parti-cles with a small response time). Squires and Eaton also showed that preferential concentration causes a significant disruption of the balance between production and destruction of dissipation, again leading to a selective modification of the turbulence spectrum. More research is, in our opinion, needed to fully understand the contribution of preferential concentration to the two-way coupling effect.
Boivin et al. (1998) found that in a turbulently flowing suspension the cascade process (energy transport from the large to the small eddies) is influenced by the particles. They calculated the spectrum of the fluid-particle energy exchange rate. In the small wave number part of this spec-trum the turbulent fluid motion transfers energy to the particles, ie the particles act as a sink of kinetic energy. At larger wave numbers of the spectrum the energy exchange rate is positive,
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