On the effect of turbulence on bubbles: Experiments and numerical simulations of bubbles in wall-bounded flows

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On the effect of turbulence on bubbles

Experiments and numerical simulations of bubbles in wall-bounded flows

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 2 juli 2012 om 15:00 uur

door

Marcus Johannes Wilhelmus HARLEMAN natuurkundig ingenieur

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Prof. dr. ir. J. Westerweel Prof. dr. ir. T.J.C. van Terwisga

Samenstelling promotiecommissie:

Rector Magnificus, Technische Universiteit Delft, voorzitter Prof. dr. ir. J. Westerweel, Technische Universiteit Delft, promotor Prof. dr. ir. T.J.C. van Terwisga, Technische Universiteit Delft, promotor Prof. dr. rer. nat. D. Lohse Universiteit Twente,

Prof. dr. ir. W.S.J. Uijttewaal Technische Universiteit Delft, Prof. dr. ir. J.J.H. Brouwers Technische Universiteit Eindhoven, Prof. dr. E.K. Longmire University of Minnesota,

Dr. R. Delfos Technische Universiteit Delft,

Prof. dr. ir. G. Ooms Technische Universiteit Delft, reservelid.

Dr. R. Delfos heeft als begeleider in belangrijke mate aan de totstandkoming van dit proefschrift bijgedragen.

This research is supported by the Dutch Technology Foundation STW, which is part of the Netherlands Organisation for Scientific Research (NWO) and partly funded by the Ministry of Economic Affairs, Agriculture and Innovation (project number 07781).

ISBN 978-90-8891-437-9

Printed by: Proefschriftmaken.nl_{|| Uitgeverij BOXPress} Published by: Uitgeverij BOXPress, ’s-Hertogenbosch

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Summary

It has been shown in the literature that the addition of gas bubbles to a turbulent boundary layer in water can result in a local reduction of the skin-friction drag of up to 80% (Madavan

et al., 1984; McCormic & Bhattacharyya, 1973; Sanders et al., 2006). An application to ships

seems promising, since for long low-speed vessels up to 60% of the total drag results from skin-friction. Therefore, the drag of such vessels might be significantly reduced. Unfortunately, tests with full-scale ships in Japan (Kodama et al., 2002) and the Netherlands (Foeth et al., 2010) showed no significant drag reduction. More knowledge about the drag reduction mechanism is required to understand whether the location and method of bubble generation can be optimised or whether unfavourable scale effects limit a successful application of bubble drag reduction.

Previous research has shown that the maximum amount of drag reduction correlates well with the volume fraction of added gas in the inner region of the turbulent boundary layer (z+ . 300). This maximum is obtained directly after the gas injection location and the amount of drag reduction quickly decreases downstream of this point. This limits the effectiveness of bubble drag reduction. It is not clear if this limited drag reduction persistence results from the dispersion of bubbles away from the wall, the formation of a bubble-free layer or from the method of bubble generation. In this thesis, the dispersion of bubbles in fully developed turbulent channel flow is studied by experiments and by a direct numerical simulation (DNS). It is shown that for small ‘passive’ bubbles (St _{1, with the Stokes number, St, defined as the} ratio between bubble response time and turbulence time scale) dispersion can be modelled by a gradient-diffusion hypothesis with a dispersion coefficient that is proportional to the turbulent (eddy) viscosity. For fully developed channel flow an analytical expression for the equilibrium wall-normal bubble concentration profile can be obtained, which, in analogy to the dispersion of sediment in rivers (Rouse, 1937), is characterised by a Rouse number, Prouse, i.e the ratio

between bubble rise velocity and turbulent friction velocity. In developing boundary layer flows the same gradient-diffusion hypothesis applies, but no analytical expression is available, since advection and non-zero wall-normal fluxes need to be taken into account. In addition, it is shown that when _gνu3τ = Fr2_Re_{1, bubbles are driven away from the wall towards a region}

with high wall-normal velocity fluctuations. This turbophoresis effect may cause the formation of a bubble-free layer. It has been speculated that the formation of such a bubble-free layer is responsible for the limited drag reduction persistence (Madavan et al., 1985a; Sanders et al., 2006).

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There is an ongoing debate about the effect of the bubble size on the drag reduction effec-tiveness. In the majority of experiments reported in literature, bubbles are large relative to the viscous lengthscale of the flow,δ_ν, (d+= d/δ_ν_{∼ 10}2_{) and the maximum amount of drag}

reduc-tion is about 1 to 3 times the local gas volume fracreduc-tion in the flow. This suggests that the main mechanism for the observed drag reduction is a modification of the local density and viscosity, possibly in combination with a local positive effect of the disturbance of the boundary layer flow by the bubble injection. There are some reports, however, of experiments and simulations with much smaller bubbles (d+_{∼ 1) with a 10 to 10}3 times larger ratio of the amount of drag reduction to gas void fraction (Ferrante & Elghobashi, 2004; Hara et al., 2011; Jacob et al., 2010). This high efficiency implies that an additional drag reduction mechanism exists that involves interaction between bubbles and fluid.

It is known that bubbles can cluster in regions with high vorticity as a result of centrifugal forces. This occurs when the Stokes number is about unity and the bubble rise velocity is smaller than the typical fluctuations in fluid velocity. An estimation of the Stokes numbers for the simulations and experiments with the very high drag reduction efficiency suggests that

St_{1. Therefore, this type of bubble clustering is not relevant for bubble drag reduction. The}

simulations described in this thesis indicate that also bubbles with St _{1 show preferential} concentration, namely in downward moving fluid regions. As a result, the average Reynolds stress and streamwise fluid velocity, conditionally averaged at the bubble locations, differs from that of the fluid. These differences are largest when the Rouse number is large. This type of preferential concentration can occur in the reported work with high drag reduction by small bubbles and may therefore be relevant for the mechanism of this very efficient drag reduction. The effect of the wall-normal concentration distribution and preferential concentration of bub-bles on the skin-friction drag is not studied in the simulations, since the effect of the bubbub-bles on the flow is not included in the DNS (one-way coupling).

Simultaneous particle image velocimetry (PIV) and bubble shadowgraphy experiments have been performed to measure fluid velocity fields together with bubble locations and velocities. Bubbles are generated with sizes between 30 and 150 µm in fully developed turbulent channel flow with bulk velocities between 0.1 and 0.6 m/s (360< Reτ< 1655). Although the bubble distribution does not reach an equilibrium in the test facility, the experiments confirm the valid-ity of the Rouse profiles and the occurrence of preferential concentration, as predicted by the DNS. Bubble rise velocities are determined for a range of bubble sizes (0.03 < Reb< 3) from

the velocity difference between bubbles and surrounding fluid. Surprisingly, the measured rise velocities are only about 75% of the theoretical bubble rise velocities. Although the measured velocity differences are very small and the measurement uncertainty is significant, the low bub-ble rise velocity is consistently observed in several measurement series and has been previously observed for particles in turbulent channel flow (Kiger & Pan, 2002).

Finally, the drag reducing effect of electrolysis bubbles is measured in turbulent channel flow (0.1 < Ubulk< 0.3 m/s). Ideally, the local skin-friction force should be measured by a direct

force sensor, but at these low velocities the forces are so small that no commercial sensors are available. Therefore, a highly sensitive force sensor is developed that measures forces as

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small as 0.35 mN (τw= 0.02 N/m2). The skin-friction can also be obtained indirectly from the

velocity profile in the viscous or the logarithmic region of the boundary layer. The direct and indirect methods are compared in single-phase developing boundary layer flow (0.1 < U0< 0.8

m/s) and show good agreement. Since the direct force sensor does not work in two-phase flow, the velocity profiles measured with PIV are used to access the drag reducing effect of bubbles. At 0.12 and 0.17 m/s a gas void fraction of 0.05% or less results in a significant change in the velocity in the top half of the horizontal channel, which can be interpreted as a drag reduction. This local drag reduction, however, is caused by a reduction in bulk fluid velocity as a result of a global drag increase from the upstream roughness of a stagnant bubble layer. At 0.28 m/s no stagnant bubble layer is formed and no significant changes in bulk flow rate or velocity profile are observed. These measurements indicate that low concentrations of small bubbles (d+_{∼ 1)} do not create a significant global or local drag reduction. In addition, it shows the importance of both local and global drag measurements for the interpretation of the cause of a local drag reduction.

With the experience obtained from the simulations and experiments described in this thesis, the results of previous drag reduction experiments can be better understood. The apparent scale effect between the high drag reduction in laboratory experiments and negligible drag reduction in full-scale applications is primarily caused by a comparison between local friction reductions in the former case with global drag reductions in the latter case. In addition, high local drag reduction are typically measured directly downstream of the bubble injection location and might therefore be partly caused by the flow disturbances from the injection method itself. For an economically feasible application of drag reduction the required amount of injected gas should be minimised. Therefore, future work should focus on a better understanding of lift force and turbophoresis effects that force bubbles away from the wall.

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Samenvatting

Het is aangetoond in de literatuur dat het toevoegen van gas bellen aan een turbulente grenslaag in water kan leiden tot weerstandsreducties van wel 80% (Madavan et al., 1984; McCormic & Bhattacharyya, 1973; Sanders et al., 2006). Een toepassing bij schepen lijkt veelbelovend aangezien voor lange schepen met een lage snelheid to 60% van de totale weerstand bestaat uit wrijvingsweerstand. Hierdoor kan de weerstand van zulke schepen wellicht significant worden verminderd. Helaas toonden testen met schepen op ware grootte in Japan (Kodama et al., 2002) en Nederland (Foeth et al., 2010) geen significante weerstandsreductie. Meer kennis van het weerstand reducerende mechanisme is nodig om te begrijpen of de locatie en de manier van bellen productie kan worden geoptimaliseerd, of dat ongunstige schaaleffecten een succesvolle toepassing van weerstandsreductie met bellen verhinderen.

Eerder onderzoek heeft aangetoond dat de maximale hoeveelheid weerstandsvermindering even-redig is met de gas volumefractie in het binnenste deel van de turbulente grenslaag (z+. 300). Dit maximum wordt bereikt direct achter de locatie van gasinjectie en de mate van reductie neemt stroomafwaarts van dit punt snel af. Dit beperkt de effectiviteit van weerstands-reductie door bellen. Het is niet duidelijk of deze afnemende weerstandsweerstands-reductie het gevolg is van de dispersie van bellen weg van de wand, het vormen van een belvrije laag, of het gevolg is van de methode van bellen productie. In dit proefschrift wordt de dispersie van bellen in volledig ontwikkelde turbulente kanaalstroming bestudeerd met experimenten en directe nu-merieke simulaties (DNS). Het is aangetoond dat de dispersie van kleine ‘passieve’ bellen (St _{1, waarbij het Stokes getal, St, is gedefinieerd als de verhouding tussen de reactietijd} van de bellen en de turbulente tijdschaal) kan worden gemodelleerd met een gradiënt-diffusie hypothese met een dispersiecoëfficiënt die evenredig is met de turbulente viscositeit. Voor volledig ontwikkelde turbulente kanaalstroming kan een analytische uitdrukking voor de wand-normale evenwichtsprofiel van de bellenconcentratie worden bepaald, welke in analogie met de dispersie van zand in rivieren (Rouse, 1937), wordt gekarakteriseerd door het Rouse getal,

Prouse, i.e. de verhouding tussen de stijgsnelheid van de bel en de turbulente wrijvingssnelheid.

Dezelfde gradi¨ent-diffusie hypothese is van toepassing in een ontwikkelende grenslaag, maar een analytische uitdrukking is niet beschikbaar aangezien advectie en een wand-normale flux ongelijk aan nul in beschouwing moet worden genomen. Verder is geconstateerd dat wanneer

f racu3_τgν= Fr2_Re_{1, bellen van de wand weg worden gedreven naar een gebied met grote}

snelheidsfluctuaties in de wand-normale richting. Dit turbophorese-effect kan verantwoordelijk zijn voor de vorming van een bellenvrije laag. Het is gespeculeerd dat de vorming van zo’n

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lenvrije laag verantwoordelijk is voor de stroomafwaartse afname van de weerstandsreductie door bellen (Madavan et al., 1985a; Sanders et al., 2006).

Er is een debat gaande over het effect van de bellengrootte op de effectiviteit van weerstands-reductie met bellen. In de meerderheid van de in de literatuur gerapporteerde experimenten zijn bellen groot ten opzichte van de viskeuze lengteschaal van de stroming,δ_ν, (d+= d/δ_ν_{∼ 10}2₎

en is de maximale weerstandsreductie ongeveer 1 `a 3 keer de lokale gas volumefractie in de stroming. Dit suggereert dat het belangrijkste mechanisme voor de waargenomen weerstands-reductie een aanpassing van de lokale dichtheid en viscositeit is, mogelijk in combinatie met een lokaal positief effect van de verstoring van de grenslaagstroming door de belleninjectie. Er zijn echter resultaten gepubliceerd van experimenten en simulaties met veel kleinere bellen (d+_{∼ 1)} met een 10 tot 103 keer grotere verhouding van de hoeveelheid weerstandsreductie tot de gas volumefractie (Ferrante & Elghobashi, 2004; Hara et al., 2011; Jacob et al., 2010). Deze hoge effectiviteit impliceert dat er een aanvullend weerstandsreducerend mechanisme moet bestaan, waarbij bel-vloeistof interactie een rol speelt.

Het is bekend dat bellen kunnen clusteren in gebieden met een grote vorticiteit als gevolg van centrifugaal krachten. Dit gebeurt wanneer het Stokes getal ongeveer één is en de stijgsnelheid van de bellen kleiner is dan de typische fluctuaties in vloeistofsnelheid. Een afschatting van de Stokes getallen voor de simulaties en experimenten met de zeer hoge weerstandsreductie suggereert dat St _{1. Daarom is dit type van bellenclustering niet relevant voor} weerstands-reductie door bellen. De simulaties die worden beschreven in dit proefschrift tonen aan dat ook bellen met St _{1 een voorkeursconcentratie laten zien en wel in naar beneden stromende} vloeistofgebieden. Als gevolg hiervan wijken de gemiddelde Reynolds spanning en vloeistof-snelheid in de stromingsrichting, conditioneel gemiddeld over de locaties van de bellen, af van die van de vloeistof. Deze verschillen zijn het grootst als het Rouse getal groot is. Dit type voorkeursconcentratie kan ook voorkomen in het gepubliceerde werk met hoge weerstands-reductie door kleine bellen en kan daarom relevant zijn voor het mechanisme van deze zeer efficiënte weerstandsreductie. Het effect van de wand-normale concentratieverdeling en de voorkeursconcentratie van bellen op de wrijvingsweerstand is niet bestudeerd in de simulaties omdat het effect van de bellen op de stroming niet is opgenomen in de DNS (één-weg koppe-ling).

Experimenten met gelijktijdige ‘particle image velocimetry’ (PIV) en bellenschaduwgrafie zijn uitgevoerd om gelijktijdig het snelheidsveld van de vloeistof en de locaties en snelheden van de bellen te meten. Er zijn bellen gegenereerd met afmetingen tussen de 30 en 150 µm in een volledig ontwikkelde kanaalstroming met bulk snelheden tussen 0.1 en 0.6 m/s (360 <

Re_τ< 1655). Hoewel de bellenconcentratieverdeling geen evenwicht bereikt in de test faciliteit, bevestigen de experimenten de geldigheid van de Rouse profielen en het voorkomen van pref-erente concentratie, zoals voorspelt door de DNS. Voor een range aan belafmetingen (0.03 <

Reb< 3) is de belstijgsnelheid bepaald op basis van het snelheidsverschil tussen de bellen en

de omringende vloeistof. Verrassend genoeg zijn de gemeten stijgsnelheden slechts 75% van the theoretische waarden. Hoewel de gemeten snelheidsverschillen klein zijn en de meetonzek-erheid significant is, zijn de lage stijgsnelheden van de bellen consistent waargenomen in

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schillende meetseries en zijn ze eerder waargenomen voor deeltjes in turbulente kanaalstroming (Kiger & Pan, 2002).

Tenslotte is het weerstandsreducerende effect van elektrolysebellen gemeten in turbulente kanaal-stroming (0.1 < Ubulk< 0.3 m/s). Idealiter zou de locale wrijvingskracht met een directe kracht

sensor moeten worden gemeten, maar bij deze lage snelheden zijn de krachten zo klein dat er geen commerci¨ele sensoren beschikbaar zijn. Daarom is een zeer gevoelige krachtsensor ont-worpen die krachten meet vanaf 0.35 mN (τw= 0.02 N/m2). De wrijvingskracht kan ook

indi-rect worden bepaald met behulp van het snelheidsprofiel in het viskeuze of logaritmische gebied van de grenslaag. De directe en indirecte methoden zijn vergeleken in een ´e´en-fase, ontwikke-lende grenslaagstroming (0.1 < U0< 0.8 m/s) en komen goed met elkaar overeen. Aangezien

de directe krachtsensor niet werkt in twee-fase stroming zijn de met PIV gemeten snelheids-profielen gebruikt om het weerstandsreducerende effect van bellen te bepalen. Bij 0.12 en 0.17 m/s resulteert een gas volumefractie van 0.05% of minder in een significante verandering van het snelheidsprofiel in de bovenste helft van het horizontale kanaal, hetgeen kan worden ge¨ınterpreteerd als weerstandsreductie. Deze lokale weerstandsreductie is echter veroorzaakt door een afname van de bulk vloeistofsnelheid, wat het gevolg is van een globale weerstands-toename door de stroomopwaartse ruwheid van een stationaire bellenlaag. Bij 0.28 m/s wordt geen stationaire bellenlaag gevormd en is er geen significante verandering in vloeistof debiet of snelheidsprofiel waargenomen. Deze metingen geven aan dat lage concentraties kleine bellen (d+ _{∼ 1) geen significante lokale of globale weerstandsreductie veroorzaken. Verder tonen ze} het belang aan van zowel lokale als globale weerstandsmetingen voor de interpretatie van de oorzaak van lokale weerstandsreducties.

Met de ervaring van de in dit proefschrift beschreven simulaties en experimenten kunnen de re-sultaten van eerder uitgevoerde experimenten met weerstandsreductie beter worden begrepen. Het ogenschijnlijke schaaleffect tussen de grote weerstandsreducties in laboratorium expe-rimenten en de verwaarloosbare weerstandsreductie bij ware-grootte toepassingen is vooral veroorzaakt door een vergelijking van lokale weerstandsreducties in het eerste en globale weer-standsreducties in het laatste type experiment. Daarnaast zijn hoge weerweer-standsreducties di-rect stroomafwaarts van de bellen injectielocatie mogelijk ten dele veroorzaakt door verstoring van de grenslaagstroming door de injectiemethode zelf. Het weerstandsreducerende mecha-nisme lijkt met name de verandering van de lokale dichtheid en viscositeit van het gas-vloeistof mengsel te zijn en is vooral effectief als bellen zich in het binnenste gebied van de grenslaag bevinden (z+ . 300). Voor een economisch rendabele toepassing van weerstandsreductie door bellen moet de benodigde gas volumefractie worden geminimaliseerd. Daarom moet toekom-stig werk focussen op een beter begrip van lift krachten en turbophorese effecten die bellen van de wand weg drijven.

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Nomenclature

Roman symbols

a bubble radius [m]

B width [m]

B constant in the law of the wall [-]

C volume averaged bubble concentration or void fraction [-]

Cd drag coeffient [-]

Cf friction coeffient [-]

Cl lift coefficient [-]

Cm added mass coefficient [-]

c local bubble concentration [-]

c local bubble concentration averaged over t, z or d. [-]

c circle of confusion [m]

ca reference concentration for a Rouse profile [-]

cf local friction coeffient [-]

D aperture diameter [m]

Di j deviation matrix for PTV [m]

d bubble diameter [m]

dt time between two laser pulses [s]

Eo E¨otv¨os number [-]

F force [kgms−1]

F Faradays constant [Asmol−1]

Fl lift force [kgms−1]

Fr Froude number [-]

f_/# aperture or f-stop [-]

G velocity gradient over a bubble surface [s−1]

g gravitational acceleration [ms−2]

H channel height [m]

H shape factor [-]

I current [A]

i image distance [m]

I intensity (grey-value count) [#]

J lift force correction factor [-]

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k roughness height [m]

Kb bubble disperion coefficient [-]

l development length [m] M optical magnification [-] M displacement vector [m] m mass [kg] N number of ... [#] o object distance [m]

Prouse Rouse number [-]

P pressure [kgm−1s−2]

p pressure devided by density [m2s−2]

Qa gas flow rate [m3s−1]

Qw water flow rate [m3s−1]

R relative difference between a bubble and Poisson distribution [-]

R ideal gas constant [kg

m2s−2K−1mol−1]

Rebulk Reynolds number based Ubulk, H andν(channel flow) [-] Red Reynolds number based Wrise, d andν(bubbles) [-] Rel Reynolds number based on U0, l andν(boundary layer flow) [-] Re_θ Reynolds number based on U0,θandν(boundary layer flow) [-] Re_τ Reynolds number based on u_τ, H andν(channel flow) [-]

S reference area [m2]

Si j mean rate of strain tensor [s−1]

si j fluctuating rate of strain tensor [s−1]

Sc turbulent Schmidt number [-]

Sr shear rate [-]

St bubble Stokes number [-]

T temperature [K]

t time [s]

ta effective air layer thickness [m]

U velocity vector [ms−1]

U streamwise component of the velocity [ms−1]

U0 free stream velocity [ms−1]

Ubulk bulk velocity [ms−1]

u_τ friction velocity [ms−1]

V spanwise velocity [ms−1]

V velocity scale in the Weber number [ms−1]

Vb bubble volume [m3]

W wall-normal velocity [ms−1]

Wrise bubble rise velocity [ms−1]

We Weber number [-]

x streamwise distance [m]

x displacement vector [m]

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y spanwise distance [m]

z wall-normal distance [m]

za reference location for a Rouse profile [m]

Greek symbols

α gas void fraction [-]

α proportionality constant between the bubble dispersion coeffi-cient and the turbulent viscosity

[-]

β bubble trapping coefficient [-]

δ boundary layer thickness [m]

δ99 boundary layer thickness [m]

δ∗ _{displacement thickness} _[m]

δν viscous lengthscale [m]

ε dissipation per unit mass [m2s−3]

η Kolmogorov lengthscale [m]

ηDR drag reduction efficiency [-]

θ momentum thickness [m]

κ von K´arm´an constant [-]

λ number of bubbles [#]

λ wavelength [m]

µ dynamic viscosity [kgm−1s−1]

ν kinematic viscosity [m2s−1]

νT turbulent or eddy viscosity [m2s−1]

ρ density [kgm−3]

σ surface tension [kgs−2]

σT turbulent Prandtl number [-]

τ shear stress [kgms−1]

τb bubble timescale [s−1]

τκ Kolmogorov timescale [s−1]

τw wall shear stress [kgms−1]

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Sub- and superscripts

+ dimensionless variable based on viscous scaling

0 _{fluctuation relative to a time-average}

∗ dimensionless variable 0 in the absence of gas

b related to bubbles

f related to fluid

f, b related to fluid at the location of a bubble

g related to gas

w related to water

Abbreviations and acronyms

CCD charge-coupled device

Ch channel flow

DDPIV defocusing digital particle image velocimetry

DNS direct numerical simulation

DOF depth of field

DR drag reduction

Exp experiment

FW HM full width half mean

GPS glare point sizing

ILIDS interferometric laser imaging for droplet sizing

LDA laser Doppler velocimetry

LED light emitting diode

MEMS micro-electro-mechanical system

OD optical density

PDA phase Doppler anemometry

PDF probability density function

PID proportional-integral-derivative

PIV particle image velocimetry

PMMA poly(methyl methacrylate)

PTV particle tracking velocimetry

T BL turbulent boundary layer

TC Taylor-Couette flow

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Chapter 1 On the effect of turbulence on bubbles

This chapter gives an extensive overview of the motivation, research goals, applied methods, results, and conclusions of this thesis. As such, it gives a complete description of the work performed in this PhD-research. The full details of this work are described in the subsequent chapters.

1.1 The potential of bubble drag reduction for ships

There is a continuous effort to develop ships with improved fuel efficiency through better propulsion systems and lower drag. Drag reduction saves fuel and installed engine power and is therefore economically desirable. In addition, it decreases pollution and stimulates a shift from cargo transport by road to transport by waterways. The three most significant contributions to the drag of a ship are the skin-friction drag, the wave making drag, and form drag. Wave making drag results from the generation of free-surface waves, while form drag results from pressure forces and is influenced by flow separation and the shape of a ships wake. Wave making and form drag are highly minimized nowadays, but the major drag contribution to a ship is usually the skin-friction drag. For long low-speed vessels, it can account for up to 60% of the total drag. As a consequence, it should be realised that an increase in wall roughness, for example, by biological fouling, can add significantly to the total drag of a ship. On the other hand, one can argue that the potential gain of techniques that reduce the skin-friction drag are very large for many types of ships. Naturally, there has been a substantial interest in the development of drag-reducing techniques like riblets, the creation of air films, cavities or air filled chambers, and the addition of fibers, polymers or gas bubbles.

Riblets are elongated grooves that protrude from the wall into the flow, which can achieve a reduction of the drag by 5 to 10% (Bechert et al., 1997). They need to be aligned with the flow direction and their spacing and size is related to the smallest flow structures in a

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lent boundary layer. For a typical ship the groove spacing and extention should be around 0.1 mm, which makes these structures hard to manufacture and even harder to maintain in a ma-rine environment. Although riblets have been succesfully applied to airplanes and yachts, the application of riblets to commercial ships has so far not been feasible.

Polymers and fibres are known to generate significant drag reductions already at very low con-centrations (Virk, 1971; Warholic et al., 1999). A famous application is the addition of polymers to the crude oil that is transported in the Trans-Alaska pipeline system. The polymers have to be carefully mixed with a fluid, after which this concentrated mixture can be added to the pipe flow in order to reduce the drag. The obvious disadvantages of these techniques for applications to ships is that the additives consumed during a journey need to be taken along with the ship, while methods that rely on the injection of gas can take air from the atmosphere.

As a consequence, the application of air chambers, films, cavities, and air bubbles seem most feasible for application to ships and are the focus of the current research project ’Drag reduction by air lubrication’. This joint research project, which is funded by the Dutch Technology Foun-dation (STW), is a cooperation between the ’Physics of Fluids’ group at Twente University, the ’Ship Hydromechanics’ group at the Delft University of Technology and the ’Laboratory for Aero & Hydrodynamics’ at the Delft University of Technology. The groups focus on the drag reducing ability of bubbles in Taylor-Couette flow and the drag reduction potential and stability of air films and cavities under a flat plate. The work presented in this thesis is focused on the dispersion and drag reducing ability of micrometre-sized bubbles in turbulent channel flow.

1.2 An overview of bubble drag reduction

The potentially beneficial effect of surrounding a ship with an air layer has long been specu-lated, but one of the first documented investigations is by McCormic & Bhattacharyya (1973). The drag of an axisymmetric submerged body that was towed through a water channel was re-duced by bubbles that where generated by electrolysis. In the 1980’s a systematic investigation of bubble drag reduction on flat plates in a water tunnel was performed at the Pennsylvania State University’s Applied Research Laboratory (Madavan et al., 1984, 1985a; Pal et al., 1989, 1988). The researchers varied fluid velocities between 4.6 and 16.7 m/s, varied the plate ori-entation, the method and amount of gas injection and observed local drag reduction up to 80% in the first 254 mm downstream of the bubble injection. It should be realised however, that for these high drag reductions a large amount of air is injected, which corresponds to a gas concentration in the boundary layer of up to 40%.

In the nineties the focus of bubble drag reduction shifted from the USA to Japan and from appli-cation to high speed (military) vessels to an appliappli-cation on large commercial ships with a much lower speed. A series of turbulent channel flow experiments was performed at Japanese Uni-versities (Guin et al., 1996; Kato et al., 1999; Moriguchi & Kato, 2002; Murai et al., 2006) and at the National Maritime Research Institute (NMRI) in Japan (Kitagawa et al., 2005; Kodama

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1.2. An overview of bubble drag reduction 3

et al., 2000; Nagaya et al., 2002). In these 10 to 15 mm high channels with velocities between 5

and 10 m/s, air is typically injected by porous or perforated plates, giving bubble sizes from 0.5 to several millimetres. Drag reductions up to 40% are reported with gas concentrations in the channel below 15%, which are comparable to those previously observed in turbulent boundary layers.

In this period the research gradually focussed on detailed measurements of bubble concentration and fluid velocity profiles. Besides friction measurements with local sensors, bubble concentra-tion profiles are measured with succoncentra-tion tubes and optical fibres (Guin et al., 1996; Kodama et al., 2000). Velocity fluctuations close to the wall can be measured with laser-Doppler anemometry (LDA), as shown by Kato et al. (1999). At low bubble void fractions (< 8%), optical techniqes, such as combined shadowgraphy and particle image velocimetry (PIV) with fluorescent tracers, can be used to measure both bubble and fluid velocities simultaneously (Kitagawa et al., 2005; Murai et al., 2006). These detailed measurement techniques enabled the observation of bubble induced changes in the fluid flow and the study of possible bubble-fluid interaction.

Even more details about two-phase flows can be obtained from numerical simulations that re-solve all fluid scales and include bubble-fluid interaction. The first reported drag reduction in a numerical simulation is for deformable bubbles in Couette flow by Kanai & Miyata (2001). Drag reduction was also obtained for deformable bubbles in channel flow by Lu et al. (2005), while Kawamura & Kodama (2002) observed a drag increase. In addition, drag reduction is ob-tained for large spherical bubbles in channel flow (Xu et al., 2002) and for small point-bubbles in developing boundary layer flow (Ferrante & Elghobashi, 2004, 2005).

In order to investigate the downstream persistence of bubble drag reduction, Watanabe et al. (1998) measured the skin friction at several locations on 20 and 40 metre long ship models that where towed with 7 m/s through a towing tank. The amount of local drag reduction decreased from about 50% directly after bubble injection to about 5% after 40 metres. Researchers at the University of Michigan studied drag reduction persistance at high Reynolds numbers by air injection under a 13-metre long plate at velocities up to 20 m/s in the massive W.B. Morgan Large Cavitation Channel (Elbing et al., 2008; Sanders et al., 2006). They also observed a large decrease in drag reduction efficiency over their 13 metre plate when the air in the boundary layer consisted of individual bubbles. At their lower velocities (but still 6.7 m/s or more) and with very large gas injection rates, a stable air film was formed that resulted in drag reductions near 100% over the full length of the plate.

Unfortunately, to date not many full-scale tests have been performed, and the outcome of these tests is often neither public, nor very detailed. The earliest reported full-scale sea trial was performed in 2001 with a 116-metre long Japanese training ship SEIUN-MARU, which was modified to contain 6 air injection locations near the bow of the ship. At one velocity a power reduction of 2% was obtained, but for the other cases an increase in the required engine power was reported, possibly by a reduced propeller efficiency as a result of bubble entrainment (partly described by Kodama et al. (2002)). Full-scale and model-scale measurements with a modified Dutch river ship also showed no measurable drag reduction with bubble injection (Foeth et al.,

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2010).

This brief overview of bubble-induced drag reduction research makes clear that the topic has been studied by many researchers on different scales and with different techniques. Neverthe-less, our knowledge of the dominant phenomena is still not sufficient for a successful appli-cation to ships. The following subsections focus in more detail on remaining questions about bubble drag reduction. First of all, it is concluded from the existing literature that there is no universal scaling of bubble drag reduction. In addition, it is still not clear whether the bubble size is of importance for the drag reduction efficiency. Finally, some proposed bubble drag reduction mechanisms are discussed.

1.2.1 Universal scaling of bubble drag reduction

The amount of drag reduction in turbulent boundary layers is commonly expressed as a change of the friction coefficient

Cf = τw 1 2ρU 2 0 , (1.1)

with a fluid density ρ, wall shear stress τw, and free-stream velocity U0. Typically, the free

stream velocity does not change with gas injection, so that the amount of drag reduction (DR) can be expressed by a change in either a friction coefficient, a wall shear stress, or a wall friction velocity (τw=ρu2_τ): DR=Cf,0−Cf Cf,0 =τw,0−τw τw,0 =u 2 τ,0− u2τ u2_τ_,0 . (1.2)

The indices with a zero indicate the situation without air injection. In channel flow the bulk velocity is not fixed, since usually a constant liquid flow rate is used (i.e. from a constant pump frequency or pressure head). In those cases the bulk velocity increases proportional to the gas injection rate as a result of the increased volume flux. If the bubbles generate some (local) drag reduction, the bulk flow velocity might increase even further. Therefore, the reference friction coefficient Cf,0 needs to be corrected for the appropriate velocity. This can be done

with an empirical formula for the friction coefficient, or by interpolation of friction coefficients measured at different velocities (Kodama et al., 2000).

Many researchers have attempted to find a universal relation between the amount of drag reduc-tion, water flow velocity and gas volume fraction. It is generally observed that the amount of drag reduction increases with the volumetric amount of air that is added to a turbulent boundary layer, decreases with the free stream velocity U0, and decreases with downstream distance of

the bubble injection location. The first two trends suggest that drag reduction scales with the ratio of gas fraction to velocity. Madavan et al. (1984) studied bubble drag reduction in a tur-bulent boundary layer and measured the shear force by a 102_{×254 mm}2force balance directly after the porous plate through which the bubbles were injected underneath the plate. Drag re-ductions obtained with different gas flow rates, Qa, and for different velocities (4.2 < U0< 17.4

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m/s) collapse on a single curve when plotted against the non-dimensional parameter

Qa SU0

, (1.3)

with S the area of the porous plate injector. Data measured with different plate orientations i.e. mounted on either top, bottom or side of a water tunnel, and data measured at different down-stream locations, do not collapse on the same curve (Madavan et al., 1984, 1985a). Comparable results are obtained when the reduction in skin friction coefficient is plotted against

ta= Qa BU0

, (1.4)

with B the width of the test-section (Elbing et al., 2008; Fukuda et al., 2000; Kodama et al., 2002). This parameter ta, with a dimension of length, can be interpreted as the thickness of an

air-layer with volume flux Qa that moves with a velocity U0. Consequently, scaling with ta is

used at high air concentrations, where bubbles coalesce into an air layer. Again, this parameter does not describe the downstream variation in drag reduction effectiveness. A variation of the previous scaling parameters that includes a downstream dependence is

Qa BU0X

= ta

X, (1.5)

with X the distance between gas injector and measurement location (Deutsch et al., 2004). It suggests that drag reduction reduces linearly with downstream distance. Deutsch et al. find a reasonable collapse of all data in their data set, but it should be noted that they did not vary the measurement location X . Therefore, their scaling is identical to that of equation 1.4.

An alternative scaling parameter for bubble drag reduction is the gas void fractionαdescribed as

α= Qa

Qa+ Qw

, (1.6)

with water flux Qw. In channel flow this water flux is taken as the total water flux through

the channel, but in turbulent boundary layer flow it is typically based on integral boundary layer scales. Madavan et al. (1985a) proposed to use Qw= BU0(δ−δ∗), with boundary layer

thickness δ and displacement thickness δ∗. Alternative expressions based on the momentum thickness θ are used by Deutsch et al. (2004) and Sanders et al. (2006). Again, a reasonable data collapse is observed for a given measurement location, but large data scatter is observed when data from different researchers and with different flow topologies are compared (figure 1.1).

The above findings make it clear that the trends in bubble drag reduction can not be fully char-acterised by global flow parameters. Especially the persistence or downstream development of bubble drag reduction is not fully understood. A possibly related issue is the effect of buoyancy on the local gas distribution in a channel or boundary layer. Finally, there is still an ongoing debate about the effects of bubble size and deformability on the effectiveness of the drag re-duction. These topics are covered in the subsequent sections before the possible mechanisms responsible for bubble drag reduction are discussed.

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Figure 1.1: Skin friction ratio as a function of the gas void fraction (equation 1.6), as measured by different investigators. It is clear from the scatter in the data that bubble drag reduction can not be characterised by the gas void fraction only. The figure is reproduced from Sanders et al. (2006)

Table 1.1: Overview of proposed scaling parameters for bubble drag reduction

Qa

SU0 [-] Madavan et al.

(1984)

TBL Single x-location: good data col-lapse, but not with different plate orientations.

Qa

Qa+BU0(δ−δ∗) [-] Madavan et al.

(1985a)

TBL Several x-locations: no data col-lapse.

Qa Qa+BU0θ0

u_τ,0,R

u_τ,0 [-] Deutsch et al. (2004) TBL Single x-location: good data

col-lapse for different roughness.

Qa

BU0X [-] Deutsch et al. (2004) TBL Single x-location: good data

col-lapse for different roughness.

Qa

Qa+BU0(θ0−θ0,in j) [-] Sanders et al. (2006) TBL Single x-location: good data

col-lapse

Qa

Qa+Qw [-] Guin et al. (1996) CH Single x-location: good data

col-lapse, but not with different injec-tion locainjec-tion

Qa

BU0 [m] Fukuda et al. (2000) CH Total force measurements: no data

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1.2.2 Wall-normal bubble distribution and drag reduction persistence

It is generally observed in turbulent boundary layers that the amount of drag reduction decreases with the downstream distance from the location of bubble injection. This is most obvious in large-scale experiments, like in the boundary layer experiments at Michigan University (Elbing

et al., 2008; Sanders et al., 2006), where one metre after gas injection up to 80% of drag

reduction was observed, which reduced to about 10% after 10 metres. Experiments with 40 and 50-metre long model ships (Kodama et al., 2002; Watanabe et al., 1998) show that drag reduction can persist over lengths of 40 metres but decreases from 50% to 5%. Sanders et al. (2006) observed that a bubble-free layer was formed over a distance of 0.5 to 1.5 mm from the wall, even though bubbles where injected underneath their plate i.e. buoyancy pushes the bubbles toward the plate. This thickness corresponds to about 300 viscous lengthscales δ_ν=

ν/uτ, with the friction velocity u_τdefined by the wall shear stressτw fromτw=ρu2τ. Sanders et al. argue that the lift force counteracts buoyancy and turbulent dispersion, and is responsible

for the bubble free layer, which limits drag reduction persistance. A comparable observation is made by Madavan et al. (1985a) and Pal et al. (1988), who injected gas above a flat plate so that buoyancy pushes bubbles away from the wall. The initial drag reduction of up to 80% completely disappears within a downstream distance of 200 mm as soon as a bubble free layer near the wall is formed of about 200δ_ν thick. Apparently, it is essential for the mechanism of bubble drag reduction that the bubbles are located in the inner boundary layer (z< 300δ_ν). The effect of the wall-normal bubble distribution on drag reduction can also be studied in a channel flow, but drag reduction persistence is expected to be different from that in boundary layer flow. In a boundary layer flow, bubbles can in principle leave the boundary layer by turbu-lent dispersion, and the boundary layer thickness increases because of entrainment. As a result, the gas void fraction in the boundary layer will decrease for increasing distance downstream of injection. In fully-developed channel flow, on the other hand, the flow reaches an equilibrium state, and the bubble distribution over the channel height might also reach an equilibrium distri-bution. This implies that when drag reduction can be obtained in this equilibrium state, it might be obtained over very long distances. It should be noted, however, that the absolute pressure decreases downstream, which results in a (small) expansion of the bubbles. Since it is shown by Shen et al. (2006) that the bubble volume fraction (and not its mass) is important for bubble drag reduction, this might be relevant at high velocities and small channel heights, when the pressure gradient is large (for example: the pressure drop over a channel section of 10 mm high and 1 m long is 0.06 bar at a velocity of 5 m/s.). As a consequence of the development of an equilibrium, it is only possible to manipulate the wall-normal bubble distribution, for example by the way the bubbles are generated, in the channel section where the two-phase flow is still under development. The advantage is however, that if drag reduction is obtained for equilib-rium two-phase channel flow, it is actually the result of the bubbles and not for example by the disturbance of the bubble generation (see section 1.2.4).

The development length of two-phase channel flow and the effect of the wall-normal bubble distribution on bubble drag reduction can be obtained from a series of channel flow experi-ments performed in Japan. Kitagawa et al. (2005) measured the skin friction at 0.5 and 1.0 m

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after bubble injection (33 and 66 channel heights, H) and found only half the amount of drag reduction at 1.0 m, compared to 0.5 m. Unfortunately, the bubble distribution was measured at one location only, so it is not clear whether the lower downstream drag reduction efficiency is related to the bubble distribution. Moriguchi & Kato (2002) used a comparable channel and measured the drag at 0.75 and 1.25 m after bubble injection (75H and 125H). They found a comparable amount of drag reduction at both locations, so apparently fully-developed channel flow is again obtained after a distance of about 75H. Again, the bubble distribution was only reported at a single location. Kodama et al. (2000), on the other hand, measured the changes in skin friction and bubble distribution at 0.5, 1.0 and 1.5 m after bubble injection (33H, 66H and 100H, respectively). They observed identical amounts of drag reduction and identical bub-ble distributions at a flow velocity of 5 m/s, but observed a gradual diffusion of bubbub-bles away from the top wall at a flow velocity of 10 m/s. In that case also a lower drag reduction was obtained in the most downstream location. Apparently, the development length depends on the flow velocity and is longer at higher flow velocities. Guin et al. (1996) studied the effect of the wall-normal bubble distribution on drag reduction by injecting bubbles through porous plates in either the top or bottom of a 10 mm high channel. Both skin friction and bubble distribution are measured 0.67 m (67H) after bubble injection. The measured drag reduction was higher when bubbles were injected from the top wall, in which case the bubble concentration near the wall was highest. Based on this observation the authors suggest that the bubble concentration in the outer part of the boundary layer is of less importance for bubble drag reduction than the concentration in the inner part of the boundary layer. This is in agreement with observations by others in turbulent boundary layers (Madavan et al., 1985a; Pal et al., 1988; Sanders et al., 2006).

1.2.3 The effect of bubble size

A possible explanation for the inability to find a universal scaling for drag reduction is the intuitive hypothesis that the bubble size should be of some influence on the drag reduction mechanism and efficiency. Most bubble injectors used in experiments, like porous plates, have the problem that the bubble size is determined not only by the injector geometry, but also by the gas flux and the local shear and therefore by the flow velocity. It is therefore very difficult, if not impossible, to vary both the flow velocity, gas flux and bubble size independently of each other. Secondly, bubbles in a turbulent flow can split or coalesce depending on the local shear, so it can be argued that far downstream of injection the bubble size distribution will reach an equilibrium based on the local turbulence level.

Despite these difficulties, Moriguchi & Kato (2002) managed to temporarily vary the bubble size between 0.5 and 2.5 mm by placing the injector in either a converging or diverging inlet of their channel, but found no effect on the drag reduction. Shen et al. (2006) varied the bubble size from 44 to 476 µm by adding surfactants and salts to the water and also found no significant effect on drag reduction. Kawamura et al. (2004), on the other hand, found a higher drag reduction for 1.3 mm bubbles than for 0.3mm bubbles, but since the smaller bubbles were

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distributed over a larger wall-normal distance than the larger bubbles this more likely indicates the importance of the local bubble concentration than the direct effect of bubble size.

Although dedicated experiments show no significant change in drag reduction with varying bubble size, results from different research groups do show large differences in drag reduction. Given the lack of a universal scaling parameter (section 1.2.1) there is not a real objective way to compare drag reduction results from different researchers. In order to get at least a qualitative comparison a drag reduction efficiency,ηDR, is defined as

ηDR= DR α = 1 α Cf,0−Cf C f0 , (1.7)

with α the gas bubble void fraction (equation 1.6 with Qw = BU0(δ−δ∗) for boundary

lay-ers). Although the definition of α differs for channel flow and boundary layer flow, they are physically quite similar. The maximum values for the reported drag reduction (usually shortly after injection) from different studies are summarized in table 1.2 together with the bulk or free stream flow velocity and the bubble size. The bubble size is scaled with the relevant length scale in the inner boundary layer, the viscous length scaleδ_ν=ν/u_τ, with friction velocity u_τdefined by the wall shear stress τw fromτw =ρu2τ. In addition, a continuous air layer thickness ta is

given, which is defined as the thickness of the (imaginary) air layer with volume flux Qa, that

flows with a bulk or free stream velocity. Since channel heights and boundary layer thicknesses differ significantly between the reported experiments, ta gives a good intuitive impression of

the amount of air that is used. The data in table 1.2 is on purpose split into three parts. The top part contains the maximum amounts of drag reduction obtained in both towing tank, turbulent boundary layer and channel flow geometries from research groups all over the world. Drag reductions as high as 76% are reported, but in general only after the injection of a significant amount of air. As a result, the calculated drag reduction efficiencies are rather low and of the order one.

In the bottom part of table 1.2 it is shown that in the often cited numerical simulation of Ferrante & Elghobashi (2005) a drag reduction of around 20% is obtained with a bubble volume fraction of only 1%. Recent experiments with small electrolysis bubbles in a turbulent boundary layer (Jacob et al., 2010; Ortiz-Villafuerte & Hassan, 2006) and in turbulent channel flow (Hara

et al., 2011) also show drag reductions at a remarkably low bubble void fraction. As a result,

the drag reduction efficiencies are an order of magnitude larger than in the previously described experiments. The reason for this large difference in drag reduction efficiency is not clear yet, but significant differences are observed in both the fluid velocities and bubble sizes. In the experiments in the top part of the table the velocities vary between 5 and 18 m/s, while U0∼

1m/s in the bottom part. Although this might be relevant, several researchers have shown that drag reduction results are indepent of the velocity when plotted as a function of α. The non-dimensional bubble size d+ = du_τ/ν differs by about two orders of magnitude between the data in the top and bottom part of table 1.2. In the typical large-scale experiments summarized in the top part of the table, bubbles are typically generated by the injection of air through porous or perforated plates or even through elongated slits. In these situations the bubble size is determined by the local shear and the majority of bubbles have a diameter (d) in the range

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Table 1.2: Overview of some of the maximum drag reduction results reported in literature. Data that could not be derived from the reported literature is estimated and put between brackets. The abbreviations stand for towing tank (TT), experiment (Exp), direct numerical simulation (DNS), turbulent boundary layer (TBL), channel flow (Ch) and Taylor-Couette flow (TC).

Authors type U0 d+ DR α DR/α ta

[m/s] [-] [%] [%] [-] [mm]

Watanabe et al. (1998) TT: TBL 7 (> 102₎ ₅₀ ₍₅₀₎ ₍₁₎ _2.9

Madavan et al. (1985a) Exp: TBL 4.6 (_{∼ 10}2) 60 35 1.7 (9) Madavan et al. (1985a) Exp: TBL 16.8 (> 102₎ ₅₀ ₃₅ _1.4 ₍₂₎

Sanders et al. (2006) Exp: TBL 12 4_×102 76 32 2.4 (7)

Sanders et al. (2006) Exp: TBL 18 2_×102 60 25 2.4 (6)

Shen et al. (2006) Exp: TBL 11.1 2_×102 28 20 1.4 (2.1)

Shen et al. (2006) Exp: TBL 11.1 18 4 1.6 2.5 (0.3)

Kodama et al. (2000) Exp: Ch 5 (_{∼ 10}2) 25 8 3.1 1.2

Kodama et al. (2000) Exp: Ch 10 (_{∼ 10}2) 18 10 1.8 1.5

Kitagawa et al. (2005) Exp: Ch 5 (_{∼ 10}2) 7 2 3.5 0.3

van den Berg et al. (2007) Exp: TC 4 > 102 ₂₆ ₈ _3.3 _(2.5)

van Gils et al. (2011) Exp: TC 6.3 (> 102₎ ₇ ₄ _1.8 _(1.5)

van Gils et al. (2011) Exp: TC 25.1 (> 102₎ ₄₀ ₄ ₁₀ _(1.5)

Ferrante & Elghobashi (2004) DNS: TBL 0.83 2.4 20 2 10 0.2 Ferrante & Elghobashi (2005) DNS: TBL 0.83 1.5 22 1 22 0.1 Ferrante & Elghobashi (2005) DNS: TBL 1.97 3.3 19 1 19 0.1 Ortiz-Villafuerte & Hassan (2006) Exp: TBL 0.08 < 1 42 5.1 8.2 0.4

Jacob et al. (2010) Exp: TBL 0.75 5 25 0.1 250 (0.04)

Hara et al. (2011) Exp: Ch 1.1 3 30 0.03 1000 0.001

0.2 < d < 1 mm. This typically implies that those bubbles are much larger than the smallest length scales of the fluid, which are of the order of the viscous length scale. The bubbles in the bottom half of the table are generated by electrolysis and have a size of about 30< d < 150 µm. In combination with the lower velocities, and therefore larger fluid scales, this results in bubble sizes that are of the order of the viscous length scale. Whether this is essential for the higher drag reduction efficiency is not yet clear, and is one of the key questions of the work described in this thesis.

Finally, it is interesting to note that it has also been reported that additional drag reduction can be obtained when bubbles are so large that they become deformable. Bubbles will deform when inertia forces on the surface are larger than the surface tension force, which tends to maintain a spherical shape of the bubble. This force ratio is expressed by the Weber number,

We= ρV

2_d

σ , (1.8)

with surface tensionσand a velocity scale, V , that is representative for the velocity difference over the bubble surface. Lu et al. (2005) simulated turbulent channel flow with 16 large bubbles

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and found drag reduction when the bubbles were deformable (We= 0.4) and drag increase for nearly rigid bubbles (We= 0.2). van den Berg et al. (2005) added bubbles to the flow between two concentric cylinders of which the inner one rotated (Taylor-Couette flow; TC) and observed only drag reduction at their highest rotation rates. Inspired by the results of Lu et al., they argued that at these higher velocities, the velocity fluctuations become sufficiently large to cause bubble deformation i.e. the Weber number becomes larger than unity. As a verification, a comparable concentration of rigid buoyant particles was added to the flow and no significant drag reduction was observed. Recent experiments in a much larger Taylor-Couette facility also show a much larger bubble drag reduction at velocities that correspond to Weber numbers larger than unity (van Gils et al., 2011). Although the analysis is elegant and quite convincing, it should be realised that there are differences between Taylor-Couette flow, channel flow and boundary layer flow. The most obvious one is the direction of gravity, which is parallel to the wall in TC flow (along the cylinder axis) and normal to the wall in the other flow geometries. In TC flow with inner cylinder rotation a centrifugal force pushes bubbles towards the inner cylinder, which can be interpreted as a velocity dependent, effective gravitational acceleration. Secondly, the different flow geometries lead to different large scale flow structures. At large Reynolds numbers, a large range of spatial and temporal scales might be present, so that the small scale structures become independent of the large scale structures and are therefore universal for all flow geometries. When the bubble-fluid interaction occurs at these small scales, then the drag reduction mechanism might be identical for all flow geometries, but this can not be expected when the bubbles interact with large scale fluid structures.

As a final remark about the possibly higher drag reduction efficiency of deformable bubbles, it is noted that the drag reduction efficiencies as reported in table 1.2 are not significantly higher for the Taylor-Couette experiments with deformable bubbles than for the channel and boundary layer experiments in the top of the table. In addition, bubble deformation is observed in several of the experiments summarized in table 1.2. The work by Kitagawa et al. (2005) for example focuses on the deformation and orientation of bubbles channel flow. It can even be argued that when the flow geometry under study has sufficiently length, the bubble size might approach an equilibrium. Bubbles may coalesce and grow until surface tension is no longer sufficient to prevent large deformation of the bubbles by the local shear, and the bubbles are split in two or more smaller bubbles (Hinze, 1955). As a result, the maximum bubble size that occurs in equilibrium flow is by definition slightly deformable. Therefore, at this time the effect of bubble deformability on drag reduction is not fully understood.

The three different bubble-size regimes defined in this section are summarized in figure 1.2. In this figure the bubble size is not compared with the viscous length scale (δ_ν), but with the Kolmogorov length scale (η), which is based on the fluid dissipation (see section 3.5.1). Since dissipation occurs at the smallest fluid scales, the Kolmogorov length scale represents the scale of the smallest fluid structures in the flow. Therefore, the Kolmogorov length scale is con-ceptually a more appropriate choice than the viscous length scale, but since it is difficult to determine in an experiment, the latter one is used in table 1.2. Note that although ηchanges with the wall-normal distance, it is roughly of the same order asδ_ν;δ_ν<η< 10δν.

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d_≤η& We₁ d>η& We< 1 d>η& We> 1

ηDR 1 ηDR∼ 1 ηDR> 1

Figure 1.2: Schematic representation of possible bubble size regimes. Bubbles that have a size of about the Kol-mogorov length scaleηare undeformable (We_{1) and seem to have a high drag reduction efficiency}ηDR. In the

intermediate bubble size regime bubbles are larger than the Kolmogorov length scale, but are still undeformable. Observed drag reduction ratios are about as large as the volume fraction of air in the boundary layer, giving

ηDR∼ 1. Finally, when bubbles are so large that they are deformable (We > 1) they also seem to be more effective

for achieving drag reduction.

1.2.4 Analogies in the effect of bubble drag reduction, ultrasonic forcing

and uniform blowing

It is interesting to briefly describe two alternative methods to generate local drag reduction in turbulent boundary layers, and to compare them to drag reduction by bubble injection.

Park & Sung (2005) generated cavitation bubbles of less then 80 µm diameter in a turbulent boundary layer (U0= 0.15 m/s) by ultrasonic forcing. During the contraction of the ultrasound

transducers the local pressure in the fluid reduces, which causes gas bubbles to expand. During the subsequent expansion of the ultrasound transducers and pressure increase, these bubbles implode into a large number of much smaller bubbles. In this way, directly underneath the transducers a local gas concentration of up to 25% is created. These bubbles dissolve again, within 15 mm downstream of the transducers. As a result of this forcing, up to 60% drag reduction is created within 15 mm downstream of the transducer. This drag reduction is very local, however, and completely diminishes within 250 mm. The flow modifications as a result of this forcing were measured by PIV. A strong increase in wall-normal velocity is observed together with a decrease in streamwise velocity near the wall, which is consistent with the observed drag reduction. In addition, both the streamwise and wall-normal velocity fluctuations decrease near the wall (z+< 50) and increase away from the wall. Finally, it is observed that the location of maximum Reynolds stress u0w0 moves away from the wall, and it is suggested that this is caused by streamwise vortices that are lifted up from the wall.

The properties of a turbulent boundary layer can also be manipulated by blowing or suction through a permeable wall, i.e. air into an air-flow or water into a water-flow. These techniques can for example be used on wings to increase lift or prevent flow separation (Schlichting, 1979). Continuous blowing through a permeable wall leads to sustained reduction in skin friction drag (Kametani & Fukagata, 2011), while local blowing through a spanwise slot leads to local drag

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reduction (Park & Choi, 1999) only. In the numerical simulations by Park & Choi (1999) the integral momentum thickness (θ) decreases shortly after injection, but shows a net increase about 60δ_ν further downstream. As a consequence, it can be concluded that the total drag in the calculation domain has increased. In addition, the lifting of streamwise vortices and an increase in turbulent fluctuations downstream of the slot is observed, just as in the experiments with ultrasonic forcing by Park & Sung (2005).

When bubbles are injected into a turbulent boundary layer by a porous material or through a slit, they also displace fluid away from the wall. Thereby, they can be expected to increase the wall-normal velocity and displace vortical structures away from the wall. Unfortunately, very limited experimental information is available on flow modifications in turbulent boundary layers with bubbles. Nevertheless, there are reports of an increase in wall-normal velocity (Ortiz-Villafuerte & Hassan, 2006) and a (small) increase in turbulent fluctuations (Jacob et al., 2010). In the numerical simulations of Ferrante & Elghobashi (2004, 2005), on the other hand, both a reduction in streamwise velocity, increase in wall-normal velocity and a displacement of vortical structures away from the wall is observed. Therefore, ultrasonic forcing, blowing, and bubble injection might generate a comparable local forcing of the fluid, which results in a local reduction in skin friction. This does not exclude the possibility that there are additional mechanisms by which bubbles can reduce the skin friction drag, but in the region directly downstream of gas injection the local forcing of the fluid by gas injection may be the dominant mechanism.

1.2.5 Possible drag reduction mechanisms

In order to investigate possible bubble drag reduction mechanisms, it is instructive to recall that the shear stress in a two-dimensional turbulent boundary layer is given by

τ=ρ ν∂U_∂(z) z − u0w0 , (1.9)

with a wall-normal direction z. The shear stress consists of a viscous part and a part that results from turbulent fluctuations, which is known as the Reynolds stress. Velocity fluctuations vanish near the wall, so a viscous subregion is defined (z < 5δ_ν) where the shear stress is dominated by the viscous contribution. Outside this region the shear stress is dominated by the Reynolds stress. It is clear that the shear stress in a two-phase flow boundary layer changes as a result of the change in mean or effective density and viscosity of the fluid mixture, even in the absence of flow modification (Legner, 1984). Madavan et al. (1985b) implemented a spatially varying mixture density and viscosity in a numerical simulation of a two-dimensional developing boundary layer. The simulations show that the reduction in mixture density as a result of the volumetric concentration of bubbles reduces the drag. The presence of bubbles leads to a viscosity increase, which reduces the drag reduction efficiency when bubbles are present in the viscous region, while it increases drag reduction efficiency when the bubbles are located in the buffer layer (5< z/δ_ν < 30). This last effect, i.e. drag reduction from local

On the effect of turbulence on bubbles: Experiments and numerical simulations of bubbles in wall-bounded flows