Embed Size (px)
Citation preview
University of St Andrews
School of Mathematics and Statistics
20th Annual Scottish Fluid Mechanics Meeting
Sponsored by
Dantec Dynamics
andthe Edinburgh Mathematical Society
Mathematical Institute
North Haugh
St Andrews, KY16 9SS
Lecture Theatre A
25 May 2007
A very warm welcome to all of you to this 20th meeting of Scottish (and a little beyond!)fluid mechanics. Up here in the Far North we should be used to fluids, or so one might think,given normal climate conditions. Nonetheless, a meeting like this is a great opportunity tomeet each other and learn about the diverse research in this field being carried out (mainly) inScotland.
In the scientific programme listed overleaf, we have reduced the number of speakers relativeto past years and increased their allotted time. We hope this makes for a relaxed meeting. Dofeel free to participate!
We are immensely grateful for the generous support given by DANTEC Dynamics Limitedand by the Edinburgh Mathematical Society (EMS), which has helped keep registration feesto a minimum. We encourage you to visit the display put up by DANTEC and discuss anypotential opportunities their presence may afford. Also, the current President of the EMS, DrColin Campbell, is right here in room 209!
Special thanks are very much due to Ms Valerie Sturrock, Ms Tricia Watson, and Mr PeteLindsay who have worked hard to organise all aspects of the meeting. We hope you enjoy lunchat the Gateway (see map below or follow one of the organisers if this proves incomprehensible),and that you make the most of the breaks (in the Physics building common area next door —accessed by a bridge from the Mathematics Institute) to catch up with friends and colleagues.Finally, if you have time after the meeting, we can investigate other aspects of fluids in town....
We hope you enjoy your day, and if there is any way we can help to make it better, pleasesee one of us.
Magda Carr David Dritschel Ali Mohebalhojeh Jean Reinaud Richard Scott Chuong Tran
20th Annual Scottish Fluid Mechanics Meeting:
PROGRAMME
ORAL PRESENTATIONS
09:45 Welcome to Participants
Session 1 in Lecture Theatre A Chaiman: D.G. Dritschel.
10:00 R.K. Scott, St AndrewsPotential vorticity mixing and jet formation in
planetary atmospheres.
10:20A.J.S. Cuthbertson and P.A. Davies,
Dundee
Deposition from particle-laden, round, turbulent,
bouyant jets.
10:40M. Carr, St Andrews, P.A. Davies, Dundee,
D. Fructus, J. Grue and A. Jensen, Oslo
Stability Characteristics of Large Amplitude
Internal Solitary Waves.
11:00 T. Sherwin, SAMSIrregular near bed velocity profiles in a fast
flowing deep water cascade in the North Atlantic.
11:20 Coffee Break
Session 2 in Lecture Theatre A Chairman: S. Wilson.
11:40 T. Mullin, Manchester Transition to Turbulence in a Pipe.
12:00 A. Creech, Heriot-WattA simple CFD model of power-extracting axial
flow turbines.
12:20 A. Robertson and I.J. Taylor, StrathclydeInvestigating the Effect of Rivulets on Stay Cables
using a Discrete Vortex Method.
12:40S. Coleman, Auckland and V. Nikora,
Aberdeen
SWAT.nz: New Zealand Programme of Sand
Waves and Turbulence Research.
13:00 Lunch Break at the Gateway
Session 3 in Lecture Theatre A Chairman: T. Mullin.
2:20E. Arlemark, K. Dadzie and J. Reese,
Strathclyde
Modeling rarefaction effects of gas in
microchannel flows.
2:40S. Wilson, D. Dunn, B. Duffy, Strathclyde
and K. Sefiane, S. David, Edinburgh
Droplet Evaporation: Mathematical Modelling and
Experiment.
3:00R. Simitev, Glasgow and F.H. Busse,
BayreuthInertial convection in rotating fluid spheres.
3:20 S. K. Tarikere, AberdeenDesign & Development of Water Abrasive Jet
Cutting using CFD.
3:40 Coffee Break
Session 4 in Lecture Theatre A Chairman: P.A. Davies.
4:00 B. Duffy, Strathclyde Thin-Film Flow on a Rotating Cylinder.
4:20 R. MacDonald and M. Newton, EdinburghApplications of Particle Image Velocimetry in
musical acoustics research.
4:40M. Smith, J.J. Kobine and F.A. Davidson,
Dundee
Free and forced motion in an asymmetric
liquid-column oscillator.
POSTERS PRESENTATIONS
- J. Sullivan, S. Wilson and B. Duffy,
StrathclydeAir-Blown Rivulet Flow.
- R.R. Bambrey, J.N. Reinaud and D.G.
Dritschel, St Andrews
Energy transfers during the interaction between two
co-rotating quasi-geostrophic vortices.
- D.G. Dritschel, R.K. Scott and C.V. Tran
St AndrewsThe inviscid limit of two-dimensional turbulence.
- C.V. Tran, St Andrews An upper bound for passive scalar diffusion in shear flows.
- C. Greated, Edinburgh SOUND and COAST public engagement exhibitions.
- M. Inall, SAMSObservations and models of exchange and turbulent mixing in
fjords.
Back to the main page.
Potential vorticity mixing and jet formation in planetary
atmospheres
R. K. Scott
University of St Andrews
The zonal jets on the giant gas planets provide a striking example ofthe interplay between wave-like and turbulent motions residing on a back-ground planetary potential vorticity gradient. The classical picture is ofturbulent energy cascading to larger scales, where it eventually accumulatesin zonal modes as frequencies of eddy motions project increasingly onto fre-quencies of Rossby waves. A length scale for jet separation results fromsimple phenomenological arguments. This talk will review some of the basictheory with an emphasis on the underlying potential vorticity dynamics andexplore certain aspects of jet formation arising from the effects of sphericalgeometry, including: (i) an alternative scaling for jet separation based on theconstraint of global angular momentum conservation; (ii) the confinementof equatorial jets to low-latitudes arising from the latitudinal dependenceof the Rossby deformation radius; (iii) cyclone-anticyclone asymmetry inbarotropic and shallow water dynamics and the implications for equatorialand polar regions. Examples will be shown from high resolution forced-dissipative numerical simulations of a single layer model, extending earlierwork on freely decaying geostrophic turbulence. The choice of large scaledissipation turns out to play a crucial role in jet formation and the relevanceof single layer models to the planetary atmospheres will be discussed in ageneral context.
1
DEPOSITION FROM PARTICLE-LADEN, ROUND, TURBULENT, BUOYANT JETS
A.J.S. Cuthbertson1 and P.A. Davies1
1Department of Civil Engineering, University of Dundee, Dundee DD1 4HN.Email: [email protected]; [email protected]
A series of scaled laboratory experiments was conducted to investigate the influence of a negatively-buoyant particulate load on the mean behaviour of round, turbulent buoyant jets discharged horizontally into confined, stationary and co-flowing ambient fluid bodies. The principal focus was on determining the form and structure of sedimentation patterns forming on the container bed as a result of particulate fall-out from the margins of the developing jet. Quantitative measurements of buoyant jet trajectories and longitudinal particle deposition distributions were made using the 2-d particle tracking facilities provided by the Digimage flow visualisation software package. Additionally, a light intensity visualisation technique was developed to determine quantitatively the spatial distribution of particle deposition.
The experiments results demonstrated that, for the particle concentrations tested (up to ~0.1% by volume), the mean trajectories of the buoyant jets in both stationary and co-flowing ambients are unaffected by the presence of the particle load. This result is valid over a wide range of source Froude numbers Fr0 typical of prototype-scale wastewater outfall discharges. For discharges into still ambient conditions, good agreement was also demonstrated between the measured trajectories and theoretically-derived, power-law relationships for clear, round buoyant jet discharges into an unconfined ambient environment. For particle-laden discharges into moving ambients, the measured buoyant jet trajectories were found to be well represented by a modified form of this power-law. Synoptic flow fields showed that the deposition trajectories of particulates within the ambient fluid are governed by the relative strengths of (i) the radial inflow generated as a result of fluid entrainment at the margins of the rising buoyant jet, (ii) the ambient cross-flow velocity, and (iii) the settling velocity of the particles. For discharges into still ambient conditions, the effects of entrainment-induced radial inflow and particulate settling were shown to combine to produce deposition trajectories back towards the buoyant jet source, consistent with observations from previous studies of particle-laden turbulent plumes and two-dimensional, turbulent buoyant jets. Discharges into co-flowing ambient fluids revealed the relative magnitudes of the co-flowing velocity and particle settling velocity to be dominant in determining the resulting particle deposition trajectories, which resulted in the downstream transportation of particles away from the buoyant jet source.
In both cases, the resulting particle sedimentation patterns on the channel bed exhibited two distinct deposition regions, (i) a narrow band associated with near-source fall-out from the buoyant jet margins and (ii) a diffuse zone associated with fall-out from the radially-expanding surface gravity current. Laterally-averaged probability density functions of particulate deposition revealed the longitudinal distributions to be either single peak or bimodal in nature, dependent on the relative proportion of particulate fall-out from these two mechanisms. Parameterisations and scaling arguments were successfully applied to deposition length scales associated with these sedimentation patterns. This provided a parametric framework within which the quantitative characteristics of particulate deposition behaviour could be adequately represented.
Stability Characteristics of Large Amplitude Internal Solitary Waves
M. Carr (University of St Andrews), P. A. Davies (University of Dundee), D. Fructus, J. Grue & A. Jensen (University of Oslo).
Presenting Author:Dr Magda CarrSchool of Mathematics and StatisticsUniversity of St AndrewsKY16 9SSPhone: +44 (1334) 463715Email: [email protected]
Laboratory study has been carried out to investigate the instability of an internal solitary wave of depression in a shallow stratified fluid system. The experimental campaign has been supported by theoretical computations and has focussed on two- and three-layered stratifications. In the two layer regime the lower layer is homogeneous and the upper layer is linearly stratified. In the three layer regime the lower layer is homogeneous, the middle layer is linearly stratified and the upper layer is either homogeneous or stratified. It is found that the onset and intensity of breaking are affected dramatically by changes in the background stratification. Manifestations of a combination of shear and convective instability are seen on the leading face of the wave in the two layer regime. It is shown that there is interplay between the two instability types and convective instability induces shear by enhancing isopycnal compression. In the three layer regime shear instability alone is seen on the pycnocline beginning at the trough of the wave and extending throughout the tail. Comparison between the experimental data and a fully nonlinear numerical solution allows for the precise evaluation of the Richardson number, Ri, at every point in the fluid. It is found that the criterion for shear instability is determined by not only the smallest values of Ri induced by the flow but also by the ratio of the horizontal extent of the small Ri region to the wavelength.
Irregular near bed velocity profiles in a fast flowing deep water cascade in the North Atlantic
Toby SherwinScottish Association for Marine Science
The overflow of deep water across the Wyville Thomson Ridge at the southern end of
the Faroe-Shetland Channel carries cold Norwegian Sea Deep Water down a narrow
gully and into the Rockall Trough. It is intermittent with maximum transports of
order 2×106 m3 s-1 (making it at times a significant ocean current). The 3-dimensional
nature of the flow is emphasised. Earth’s rotation results in the body of the flow
being geostrophic, with isopycnal surfaces lifted to the right looking downstream.
During high flow the velocity profile is greatly sheared in the bottom 50 - 60 m, and
maximum speeds (> 1.5 m s-1) are observed up to 100 m above the seabed. Ekman
dynamics (the effect of bottom friction dominating Earth’s rotation near the sea bed)
result in a strong leftward flow (looking downstream) at the bottom which is
compensated by a return flow in the upper layers. Internal tidal currents become
significantly amplified during such events (compared with other times) but appear to
have a marginal impact on bottom stress. All these dynamics can be explained by
conventional theory. However, and confusingly, downstream velocities can be
anomalously large close to the seabed on occasion.
Transition to turbulence in a pipe.
T. MullinManchester Centre for Nonlinear Dynamics,
Department of Physics and Astronomy,The University of Manchester,
Manchester M13 9PL, UK
The puzzle of why fluid motion along a pipe is observed to become tur-bulent as the flow rate is increased remains the outstanding challenge ofhydrodynamic stability theory, despite more than a century of research. Theissue is both of deep scientific and engineering interest since most pipe flowsare turbulent in practice even at modest flow rates. All theoretical workindicates that the flow is linearly stable i.e. infinitesimal disturbances decayas they propagate along the pipe and the flow will remain laminar. Finiteamplitude perturbations are responsible for triggering turbulence and thesebecome more important as the non-dimensionalized flow rate, the Reynoldsnumber Re, increases. We will show that scaling of the amplitude of theperturbations with Re gives insights into origins of the turbulent motion.
A simple CFD model of power-extracting axial flow turbinesAngus CW CreechHeriot Watt University
In terms of renewable energy, Scotland has both wind and tidal power in abundance. A common method of power extraction in either case is via axial flow turbines. While the flow dynamics of the individual turbine are relatively well understood, the interference effect and performance of turbine farms remain comparative unknowns.
This short presentation will discuss the implementation of an hradaptive finite element model for axialflow marine turbines, and how this can be deployed as an effective tool for simulating turbine farms, at relatively little computational expense.
Investigating the effect of rivulets on stay cablesusing a Discrete Vortex Method
Andy Robertson∗ & Ian J. Taylor
Department of Mechanical EngineeringUniversity of Strathclyde
Glasgow, G1 1XJ
Precipitation and wind are both environmental loadings incorporated within anybridge design code. The simultaneous occurrence of these can result in the genera-tion of rainwater rivulets upon the cables of cable-stayed bridges. Within a certainrange of this parameter space the interaction of these rivulets with the structural prop-erties of the longer stays can induce a large amplitude, velocity restricted vibration.A phenomenom commonly referred to as Rain Wind Induced Vibration (RWIV). Thegoverning mechanism for this aeroelastic response is not asyet fully understood andis subsequently not incorporated within any design literature. However with ever ex-panding maximum spans and corresponding cable lengths, thenecessity to mitigateagainst RWIV continues to increase. RWIV has thus become an area of active inves-tigation amongst the wind engineering community.
Previous research has concentrated on full scale and wind tunnel experimentation,with little computational analysis undertaken. The work presented herein representsone phase of the ongoing development and validation of such anumerical method.The modelling approach of which, proposes to quasi-steadily link the unsteady aero-dynamic field determined by a discrete vortex method, (DIVEX), with the currentsurface geometry ascertained from a thin film solver based onlubrication theory, tocapture rivulet motion.
Specifically the work to be presented, investigates the addition and forced oscil-lation of fixed shape artificial rivulets to a plain cylinder.In total three geometriccases, all governed by a single angular parameter are examined for a range of exter-nal loading conditions and frequencies. The results provide an indication as to howboth the time-averaged and local aerodynamic forces are affected by the inclusion ofthese rivulets and the vibrations they potentially induce.Primarily these purport to apotential galloping instability caused by the asymmetric profile in the instance of astatic rivulet. With an increasingly complex response ensuing with the addition of aprescribed oscillation.
1
SWAT.nz: New Zealand Programme of Sand Waves and Turbulence Research Stephen E. Coleman1 and Vladimir I. Nikora2 1 Department of Civil and Environmental Engineering, The University of Auckland, Private Bag 92019, Auckland, New Zealand, ph 64-9-373 7599, fax 64-9-373 7462, email [email protected] 2 Department of Engineering, University of Aberdeen, Aberdeen, AB24 3UE, United Kingdom, ph 44-1224-272787, fax 0044-1224 272 497, email [email protected] Abstract: When water flows over a particulate bed causing particles to move, regular patterns of waves develop in the bed. Waves in sediment beds beneath tranquil water flows are typically described as ripples or dunes, ripples being small fine-sediment bed waves that do not influence the water surface, and dunes being bed waves that occupy a significant portion of the flow depth, causing the water surface to be disturbed. Ripples and dunes are common features of river beds, sea floors, geological strata, and many hydro-transport engineering systems. These wave trains are intriguing to the eye, and significant to the environment in controlling scour, hyporheic flows, and movement of the granular particles and any attached contaminants (including organic molecules, inorganics, and disease-carrying micro-organisms). Flow over a planar sediment bed gives the origin of both ripples and dunes as small sand-wavelets, different mechanisms acting to grow the respective bed forms for the flow. Both ripples and dunes grow at rates reducing with time to attain magnitudes in equilibrium with the applied flow. These magnitudes are typically described as a function of sediment properties for ripples, and flow properties (principally) for dunes. Despite these observations, the respective effects of grain movements, flow-bed instabilities or bed-wave-flow interactions (as reflected by bulk-flow properties, turbulent flow structures, boundary-layer development, or bed-roughness magnitudes) on bed-wave generation and development processes remain unclear. Methods promoted to predict the types and sizes of bed waves occurring in equilibrium with a flow suffer significantly from the lack of understanding of the physical mechanisms controlling bed-wave development to equilibrium. Numerical simulations of erodible-bed development similarly suffer from a lack of appropriate turbulence models for bed waves developing to equilibrium. A three-year New-Zealand-government-funded project was begun in 2003 to address the lack of understanding of subaqueous sand waves. The project combined two research teams, from The University of Auckland and from the National Institute of Water and Atmospheric Research (NIWA). The goal of the project was to clarify the most intriguing problem of submerged particulate waves: what causes these waves to form and grow. The aim of this presentation is to provide an overview of the design and progress of the investigative effort, along with preliminary results and investigative directions.
Modelling rarefaction effects in micro-channel flows
Erik J. Arlemark, S. Kokou Dadzie and Jason M. Reese
Department of Mechanical Engineering,
University of Strathclyde, Glasgow, G1 1XJ, UK
Abstract
Studying fluid flow at the nano and micro scale has a number of engineering applica-tions such as in the fabrication of pumps without moving parts, micro-turbines andlow-drag aircraft. However, the theoretical prediction of flow in nano and micro de-vices remains a critical issue. It is now known that, at the nano and micro scale, flowsmay not be predicted by the set of Navier-Stokes equations with no slip boundaryconditions which constitute the classical continuum fluid mechanics model [1]. Theseproblems are due to rarefaction and surface effects which become important in microand nano scales in contrast to macro scale flows [2]. Another important effect for microand nano gas flows is the fluid compressibility due to changes in density or pressure[1, 2]. Some of these effects can be accounted for in the framework of the Navier-Stokesmodel by using of slip boundary conditions, but not all .
We use two different models to improve gas flow predictions in nano and microchannels. On the one hand, we develop a way to account for surface effects by calcu-lating the molecular mean free path varying with distance from the surface. On theother hand, we use new continuum fluid mechanics equations, built in the frameworkof kinetic theory, that account for sensitive local density variations [3, 4]. This hydro-dynamic model differs from the usual Navier-Stokes model by some additional termsdue to density gradients. We show comparisons with other theoretical predictions andexperimental data.
References
[1] G. Karniadakis, A.Beskok, N. Aluru, Microflows and Nanoflows, Springer, (2005).
[2] M. Gad-el-Hak, The Fluid Mechanics of Microdevices - The Freeman ScholarLecture. Journal of Fluids Engineering, Vol. 121, pp. 5-33, (1999).
[3] S. Kokou Dadzie, Jason M. Reese and Colin R. Gallis, A volume-based descriptionof gas flows with localised mass-density variations, (unpublished), (2007).
[4] S. Kokou Dadzie, Jason M. Reese and Colin R. Gallis, A constitutive model forvolume-based descriptions of gas flows, (unpublished), (2007).
DROPLET EVAPORATION: MATHEMATICAL MODELLING AND EXPERIMENT
Gavin Dunna, Stephen Wilsona, Brian Duffya, Samuel Davidb and Khellil Sefianeb
aDepartment of Mathematics, University of Strathclyde, Livingstone Tower,
26 Richmond Street, Glasgow G1 1XH, United KingdombSchool of Engineering & Electronics, University of Edinburgh, Sanderson Building,
Kings Buildings, Mayfield Road, Edinburgh EH9 3JL, United Kingdom
A mathematical model for the evaporation of an axisymmetric sessile droplet of liquid whose contactline is pinned by surface roughness (or other) effects is developed and analysed. We compare thepredictions of our mathematical model with recently obtained experimental results.
The droplet rests on a substrate and evaporates under controlled conditions. In particular, the dropletis neither heated nor cooled externally, and the atmosphere surrounding the droplet is at room tem-perature, so that evaporation is limited by the diffusion of the vapour away from the droplet surface.In the experiments each droplet is much smaller than the capillary length, and so we assume that itsfree-surface profile has the shape of a spherical cap.
PSfrag replacements
dV
dt(nL/s)
R (mm)
Acetone
Methanol
Water
Al Exp.
PTFE Exp.
Al Theory
PTFE Theory
Deegan Theory
Al +Buoyancy
PTFE +Buoyancy
0.8 1.0 1.2 1.4 1.6 1.8
1
2
3
10
20
30
40
50
60
Current models capture many of the key physical processes but do not include all the physical effectsthat are often significant in practice. For instance, previous work neglects the thermal conductivity ofthe substrate, but experiments reveal that this can play an important role in the evaporative process.For example, for a droplet of acetone on a poorly conducting substrate, such as PTFE, the temperatureon the droplet surface is approximately 8K less than the temperature of the surrounding atmosphere;this considerably reduces the concentration of water vapour at the surface of the droplet and so theevaporative mass flux is significantly reduced compared to that on a better conductor. Other physicaleffects that have been investigated experimentally include reducing the pressure in the surroundingatmosphere and replacing the surrounding air with various other gases. In this talk we report onextending the existing mathematical models to include all of these physical effects.
This work is supported by the EPSRC via research grants GR/S59451 and GR/S59444.
Inertial convection in rotating fluid spheres
R.D. Simitev, Department of Mathematics, University of Glasgow, G12 8QW, UK
F.H. Busse, Institute of Physics, University of Bayreuth, D-95440, Germany
Since stellar interiors as well as metallic planetary cores are characterized by
rather small Prandtl numbers much attention has been focused on the problem
of the onset of convection in rotating fluid spheres in the case of small P . Here
we present some examples of the variety of patterns encountered in a rotat-
ing sphere of a low Prandtl number fluid. The preferred mode near onset is
equatorially-attached and can travel in the prograde as well in the retrograde
directions, depending on the parameters of the problem. Moreover, the azi-
muthal wavenumber m of convection does not increase monotonically with the
Coriolis parameter as is usually found for the columnar mode at values of P of
the order of unity or higher. In addition to the simple ‘single-cell’ modes mul-
ticellular modes can be observed. The equatorially-attached modes were first
found by Zhang and Busse. They represent inertial oscillation which becomes
excited when viscous dissipation is balanced by the energy provided by thermal
buoyancy. The fact that both energies can be regarded as small perturbations
has led Zhang to solve the problem of the onset of convection by an asymptotic
analysis. The buoyancy term and viscous dissipation are introduced in the equa-
tion of motion as small perturbations of inviscid inertial waves and the balance
of the two terms is then used for the determination of the critical value of the
Rayleigh number. Here we extend this approach to case of a spherical boundary
of low thermal conductivity on the one hand and to an alternative method of
analysis on the other hand which will allow us to obtain explicit expressions for
the dependence of the Rayleigh number on the azimuthal wavenumber.
Design & Development of Water Abrasive Jet Cutting using CFDShashi Tarikere1, 2, Shuisheng He1, Douglas Morrison1, Kenny Anderson2
1 Department of Engineering, University of Aberdeen2 Circle Technical Services Ltd, Kintore, Aberdeenshire, Scotland
The Abrasive Water Jet Cutting System is proven to be an eco friendly solution for decommissioning of wellheads in the offshore industry. This project concentrates on analysing the Abrasive Water Jet Cutting process using Computational Fluid Dynamics (CFD). In simple terms, the Abrasive Water Jet (AWJ) Cutting involves accelerating a mixture of water and abrasive particles through a nozzle (orifice) at a certain pressure which then performs the cutting action.
In this study we have identified key parameters of the AWJ system and categorised them into two groups, one being System Parameters which included Pressure, Abrasive Particle Diameter, Abrasive Particle Flow Rate, Stand Off Distance and other being Geometry of the Nozzle which included Nozzle Diameter, Nozzle Length, Nozzle Inlet Angle. A CFD study was carried out to examine the effect of each parameter identified in a view to arrive at an optimised system conditions and optimum nozzle design.
A 2D axisymmetric computational domain has been used with appropriate boundary conditions to simulate the real multistring cutting scenario. Simulations have been performed using Fluent, a CFD Solver which is coupled with userdefinedfunctions (UDF). Turbulence modelling is done using standard kε turbulence model. Particle transport modelling is performed using the Discrete Phase Model (DPM) and a Particle Erosion model was used to model the wear and rate of material removal.
A systematic parametric study has been carried out to investigate the effects of varying the design parameters of the nozzle (diameter, length, taper angle) and system parameters (particle diameter, particle flow rate, pressure). The results have been studied in terms of the particle velocity, slip velocity (velocity difference between the fluid & particle), particle kinetic energy, wear rate in the nozzle, and material removal rate on the target wall. A design improvement should give high particle velocity, low slip velocity, less wear in the nozzle and high material removal rate. Based on the investigation, new nozzle designs have been arrived at and
prototype nozzles are being made. Experiments will be carried out and results will be compared with CFD simulations.
This is a KTP (Knowledge Transfer Partnership) Project with University of Aberdeen as the academic partner and Circle Technical Services Ltd, Kintore, Aberdeenshire as the industrial partner.
A thin rivulet on a horizontal rotating
cylinder
By B. R. DUFFY AND S. K. WILSON
Department of Mathematics, University of Strathclyde,
Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH
e-mail: [email protected] & [email protected]
In recent years there has been an explosion of interest in the paradigm free-surface
problem of flow of a film of viscous fluid on the inside or outside of a cylinder rotating
about its (horizontal) axis. Such flows are of relevance to, for example, coating
processes in industry, but also they have come to be regarded as paradigm free-
surface problems in the mechanics of viscous fluids, throwing up a variety of interesting
phenomena such as shock formation and complicated three-dimensional instabilities.
We use the lubrication approximation to investigate an analogous three-dimensional
problem involving a thin rivulet of fluid (of finite axial extent) on a rotating cylinder.
In particular, we determine the maximum weight of fluid that can be supported on the
outside of the cylinder (for a prescribed rotation rate), and we consider the possibility
of re-circulation of fluid.
Ω
Cylinder,
radius R
Rivulet
Horizontal
axisg
A thin rivulet on the outside of a rotating cylinder.
Free and forced motion in an asymmetric liquid-column oscillator M. J. Smith1, J. J. Kobine1 & F. A. Davidson2 1Division of Civil Engineering, University of Dundee, Dundee DD1 4HN, UK 2Division of Mathematics, University of Dundee, Dundee DD1 4HN, UK The results of a combined theoretical, numerical and experimental study of liquid oscillations in an asymmetric U-tube are presented. The configuration under investigation is analogous to that of the tuned liquid-column damper (TLCD) used to suppress oscillatory motion in large semi-supported structures. The liquid motion is described by a second-order ordinary differential equation that is nonlinear when the widths of the two vertical columns are unequal. It is shown that this asymmetry can be used as a tuning parameter to determine the natural frequency of free oscillations in the system, in addition to the known tuning effect of the connecting chamber height. The effects of viscous damping and periodic forcing are considered, leading to a description of likely initial and long-term resonance behaviour in a practical asymmetric device. Keywords: Structural damping, tuning, perturbation analysis, nonlinear resonance
AcknowledgementsJ. M. Sullivan gratefully acknowledges the sup-port of the Engineering and Physical SciencesResearch Council via a studentship and the ad-ditional support of Solution Canvas which en-abled her to attend the 2007 SET presentations.
ReferencesS. K. Wilson and B. R. Duffy, “Unidirectional flow of a thin rivulet on a vertical substrate subject toa prescribed uniform shear stress at its free surface,” Phys. Fluids 17, 108105-1–108105-4 (2005).S. K. Wilson and B. R. Duffy, “When is it energetically favourable for a rivulet of perfectly wettingfluid to split?” Phys. Fluids 17, 078104-1–07801-3 (2005).
• Flow subject to transverse shear•More sophisticated model for the
external airflow•Genuinely coupled model
Ongoing Work
• n rivulets: each height hmn, flux Qn and energy En
• (n + 1) subrivulets: each height hm(n+1), flux Qn and energy E(n+1)
• (n + 1) rivulet state is most favourable when nEn − (n + 1)E(n+1) > 0
•More complicated expression for energy difference of two states (omitted for brevity)
Summary
3
2
1
−2 −1 10
4
PSfrag replacements
n = 2 4 53 6 10 50 100 1000
τ
hm = −9τ
5
hm = −6τ
5
hm
τc =
(
2
3
)1
3
n →∞
τ = τc =
(
2
3
)1
3
infinitely infinitelymany rivulets many rivulets
unfavourableto split
hmn = khm, hm =9τ (1− nk2)
5(nk3 − 1)
n Rivulets → (n + 1) Rivulets
∆E =π
7680
[
252h3m(1− nk5) + 875τh2
m(1− nk4) + 800τ 2hm(1− nk3) + 960(1− nk2)
]
h2m
•Original rivulet: height hm, flux Q and energy E
• Subrivulet: height hmn, flux Qn and energy En
• Splitting is favourable to occur when ∆E = E − nEn > 0
Total energy and flux of a rivulet with α and W scaled out for convenience,
E =π
7680
[
252h3m + 875τh2
m + 800τ2hm + 960]
h2m, Q =
π
24[5hm + 9τ ] h2
m
Rivulet → n Rivulets
We determine the conditions under which it is ‘energetically favourable’ for a rivulet tosplit into subrivulets, and when n rivulets is the most energetically favourable state.
• Total Energy=Kinetic Energy+Surface Energy
KE =1
2
∫ +a
−a
∫ h
0u2 dz dy, SE =
1
W
[∫ +a
−a
(
1 +(
h′)2
)1
2
dy − 2a
]
Simplified Equations
Energetics
• τ < 0: flow pattern may change as a rivulet moves round a large cylinder
PSfrag replacements
IV
IV
VII
III
III
free surfacecylinder
g
−τ (> 0)
α = αc1
α = αc2
PSfrag replacements
IV
V
V
V
V
II
III
free surface
cylinder
g
−τ (> 0)
α = αc1
α = αc2
τ < 0τ > 0
• τ > 0: single solution for each Q for π/2 < α ≤ π, all type I• τ < 0: behaviour is more interesting, types II-V
0.5 0.6 0.7 0.8 0.9 1
2.5
2
1.5
1
0.5
0
PSfrag replacements
Q = 1
Q = 2
Q = 3
Q = 4
Q = 5
I
α/π
hm
hm = −τ
sin α
hm = −6τ
5 sin α
hm = −2τ
sin α
hmc
αc/π
Q = 0
Q = −1
Q = Qcrit
Q = 5
Q = 5
Q = 1
Q = −5
Q = −2
Q = −1
Q = −2
Q = −1/2
I
IIIIIIVV 0.5 0.6 0.7 0.8 0.9 1
7
6
5
4
2
3
1
0
PSfrag replacementsQ = 1
Q = 2
Q = 3
Q = 4
Q = 5
I
α/π
hm
hm = −τ
sin α
hm = −6τ
5 sin α
hm = −2τ
sin α
hmc
αc/π
Q = 0
Q = −1
Q = Qcrit
Q = 5
Q = 5
Q = 1
Q = −5
Q = −2
Q = −1
Q = −2
Q = −1/2
Q = −1/2
I
II
III
IV
V
V
Velocity and pressure,
u =sin α
2(2h− z) z + τz, p = (h− z) cos α− h′′
• 0 ≤ α ≤ π/2: no solutions• π/2 < α ≤ π: simple solution corresponding to pendent rivulet
Semiwidth a, free surface profile h and volume flux Q are given by
a =π
m, h =
hm
2(1 + cos my) , Q =
π
24m(5 sin αhm + 9τ ) h2
m
where m =√
| cos α| and hm = h(0) is the maximum height of the rivulet.
Flow Patterns•All possible flow patterns can be categorised into types I to V•Downward flow is coloured, upward flow is white
VI II III IVPSfrag replacements
+a−a
Rivulet Solutions for Varying α
Rivulet Solution
Governing Equations
0 = sin α + uzz, 0 = −py, 0 = −pz − cos α
Boundary Conditions
p = −h′′, h′ = 0 on z = h
h(±a) = 0, h′(±a) = 0u(y, 0) = 0
PSfrag replacements
x
y
zg
τ
free surface z = h(y)
α
substrate z = 0
+a
+a
−a
−a
Q
Problem Set-UpConsider the steady unidirectional flow of athin rivulet with semiwidth a and prescribedvolume flux Q of a perfectly wetting fluidsubject to a prescribed uniform longitudinalshear stress τ at its free surface on a planarsubstrate inclined at an angle α to the hori-zontal.
• velocity u = u(y, z)i, pressure p = p(x, y, z)
• constant density ρ, viscosity µ, and surfacetension γ
• no slip at the substrate• stress balance at the free surface• zero contact angle at the contact lines
Thin film flows occur in a variety of physical situations in industry, biology and the natural environment; for example they occur inside heat exchangers, as thetear film in the human eye, and as mud and lava flows. We investigate flow of a thin fluid film subject to a shear stress at its free surface due to an externalairflow, a situation that can be seen in many contexts including rain moving on a car windscreen and in the air-knife coating process in manufacturing.
Background & Governing Equations
J. M. Sullivan, B. R. Duffy & S. K. WilsonDepartment of Mathematics, University of Strathclyde
26 Richmond Street, Glasgow G1 1XH, UK
Air-Blown Rivulet Flow
Energy transfers during the interaction between two co-rotating quasi-geostrophicvortices.
Ross R Bambrey, Jean N Reinaud and David D Dritschel
.Mathematical InstituteUniversity of St Andrews
North Haugh, St Andrews, KY16 9SS, UK
Oceanic and atmospheric meso-scale flows are dynamically dominated by the slowevolution of, and the interaction between vortices – swirling masses of fluid which can beidentified as coherent volumes of potential vorticity (PV). Such flows are strongly influ-enced by both the planetary rotation and the stable density stratification of the fluid.
The simplest dynamical system which takes into account both these two dominanteffects is the three-dimensional quasi-geostrophic (QG) model. Until recently, little wasknown about how two QG-vortices would interact in the general case. Even for equal-PV vortices, the interaction depends on 5 parameters: the vortices height-to-width aspectratios, their volume ratio, their vertical offset, and their horizontal separation.
Two major questions arise
(i) Under which conditions a pair of equal-PV vortices is likely to strongly interact ?
(ii) What is the outcome of the strong interaction ?
The answer to the first question can be provided by the investigation of the stabilityof a pair of equal-PV vortices in mutual equilibrium. The margin of stability indicates theonset of a strong interaction. We determine the margin of stability by using a simplifiedapproach where vortices are modelled as ellispoidal volumes of uniform PV, at a min-imum numerical cost. Any other approach would be impractical for a large parameterspace.
The answer to the second question involves the investigation of the non-linear evolu-tion of a pair of vortices settled just beyond their margin of stability, using the Contour-Advective Semi-Lagrangian Algorithm (CASL). We show that the self-energy of the vor-tices is, in average, transfered to larger scales (in physical space) confirming the trendobserved in spectral space for QG turbulence. This is a consequence of the partial mergerbetween vortices. However, there is a large number of persistent small-scale vorticesgenerated during the non-linear interactions, but they carry a limited amount of energy.More interestingly, intermediate-size vortices tend to be destroyed during the interaction.Another important result is that vortex merger is not as efficient for the inverse energycascade as one may think. Complete vortex merger is rare, and vortices usually onlymerge partially.
The inviscid limit of two-dimensional turbulence
David Dritschel, Richard Scott and Chuong Tran
University of St Andrews
Recently, Tran and Dritschel (J. Fluid Mech. 559, 107–116, 2006) demonstrated that acentral hypothesis made by Batchelor (Phys. Fluids 12, 233–239, 1969) in his theory of two-dimensional turbulence is invalid. Batchelor assumed that, analogous to the energy dissipationin three-dimensional turbulence, the enstrophy dissipation in two-dimensional turbulence wouldapproach a finite, positive value in the limit of vanishing viscosity. In fact, the enstrophydissipation vanishes. The implications of this have been explored in a paper by Dritschel,Tran and Scott (J. Fluid Mech., under review), who showed that many aspects of Batchelor’stheory — in particular the 1/k enstrophy spectrum for large wavenumber k — are preservedby replacing the enstrophy dissipation in his theory by the enstrophy itself.
Further results have recently been obtained for the large-scale behaviour of two-dimensionalturbulence, which has attracted much less attention. Tran and Dritschel (Phys. Fluids 18,121703, 2006) demonstrate that the large-scale energy spectrum cannot be shallower than k3
for k sufficiently small, and obtain bounds on the amplitude of the spectrum proportional tothe energy (per unit area) and time squared.
Very recently, we have used a point vortex model to explore the emergence of large-scaleorder, as indicated in the statistical theory of Onsager (Nuovo Cimento Suppl. 6, 279–287,1949). Our results suggest that there is a third, intermediate range of scales, characterisedby an energy spectrum proportional to k. This occurs despite the absence of any net angularmomentum.
The presentation will put together these pieces of the puzzle, to help better understandthe role of vortex dynamics and vortex cluster formation in shaping turbulence over a vastrange of spatial scales. Theory will be supported by ultra-high Reynolds number simulations oftwo-dimensional turbulence, using both the Navier–Stokes equations and the Euler equations.
An upper bound for passive scalar diffusion in shear flows
Chuong V. TranSchool of Mathematics and Statistics, University of St Andrews
St Andrews KY16 9SS, United Kingdom
Abstract
This study is concerned with the diffusion of a passive scalar Θ(r, t) advected by generaln-dimensional shear flows u = u(y, z, · · · , t)x having finite mean-square velocity gradients.The unidirectionality of the incompressible flows conserves the stream-wise scalar gradient,∂xΘ, allowing only the cross-stream components to be amplified by shearing effects. Thisamplification is relatively weak because an important contributing factor, ∂xΘ, is conserved,effectively rendering a slow diffusion process. It is found that the decay of the scalar variance〈Θ2〉 satisfies d〈Θ2〉/dt ≥ −Cκ1/3, where C > 0 is a constant, depending on the fluidvelocity gradients and initial distribution of Θ, and κ is the molecular diffusivity. This resultgeneralizes to axisymmetric flows on the plane and on the sphere having finite mean-squareangular velocity gradients.
1
Public Engagement: the COAST and SOUND exhibitions
Marianne Greated, Glasgow School of Art and The University of [email protected]
Clive Greated, The University of [email protected]
The University of Edinburgh has produced two art exhibitions entitled COAST and SOUND. The paintings and soundscapes for the exhibitions are by Glasgow artist Marianne Greated and are inspired by different aspects of environmental pollution in Scotland and the research that is going on to address these issues. There are also posters explaining the scientific background, video interviews with scientists and a working model wave tank. As well as engaging the general public in different aspects of environmental fluid mechanics and acoustics, the exhibitions seek to break down barriers that are sometimes set up to delineate art from science.
The issues addressed in COAST primarily relate to pollution and erosion around the coastline, both of which are of concern in Scotland. They arise out of work at Edinburgh University on water wave dynamics, and the environmental effects of nuclear radiation. The SOUND exhibition relates to the everincreasing sound levels in the environment which are one of the inevitable consequences of technological advance. These can have a profound effect on the lives of people in the community and also wildlife. Recent issues in Scotland relate for example to the sound generated to wind turbines, a topic that has been researched at Edinburgh.
COAST and SOUND are sponsored by EPSRC, The British Council, The Royal Commission for the 1851 Exhibition The Scottish Executive and the Danish Cultural Institute. They have been shown throughout the UK, in Greece and in Denmark and have been seen by well over 100,000 visitors. Currently COAST is showing at Denmark’s Aquarium in Copenhagen. The next showing of SOUND will be at the Waterfront in Belfast and next year it shows in India and Belarus where a television film is to be made about the project.
SILL DYNAMICS AND ENERGY TRANSFORMATION IN A JET FJORD
MARK INALL, FINLO COTTIER, COLIN GRIFFITHS
Scottish Association for Marine Science, Dunstaffnage Marine Laboratory, Oban. PA37 1QA, Scotland.
Email: [email protected]
TOM RIPPETH
School of Ocean Science, University of Wales Bangor, Menai Bridge, Anglesey LL59 5EY, Wales.
A detailed set of observations are presented of the tidal forcing and basin response of Loch Etive, a jet-type fjordic system on the west coast of Scotland. The characteristics of the tidal jet observed during a spring tide are discussed in detail, and with reference to laboratory studies of Baines and Hoinka (1985, J. Atmos. Sci. 42, 1614-30). Key results are: 1) Directly measured dissipative energy losses due to skin friction and an internal hydraulic transition are relatively small, each accounting for approximately 10% of the mean flow energy loss, and 2) Production and dissipation of turbulent kinetic energy in flow over the topographic feature are in approximate balance throughout the transition from sub- to super-critical mean flow. The study is two-dimensional and a closed energy budget through direct measurement remains elusive; it is concluded that horizontal aspects of barotropic form drag such as eddy-shedding are responsible for as much as 35% of the mean flow energy loss. Although the system is categorised as a jet basin (i.e. the mode-1 densimetric Froude number exceeds 1), a mode-1 baroclinic wave response is observed throughout the spring/neap cycle. Of the dissipated tidal energy approximately 30% is lost to a baroclinic tide. The ratio between loss to bottom friction, barotropic form drag, and baroclinic wave drag is estimated to be 1:4:1 (1:4:3.3) at springs (neaps). Despite this, and even during a spring tide, a 20 m amplitude baroclinic mode-1 wave is observed to propagate along the full length of the basin at a speed of 0.2 m/s, somewhat slower than the predicted linear mode-1 phase speed. It is demonstrated that the large amplitude baroclinic response is generated some distance from the sill and on the flanks of the topography where the flow remains sub-critical. It is suggested that a large baroclinic response may therefore be a feature of all jet basins, contrary to previous assumptions and expectations. A hydrographic section supports the implication of the dissipation of the baroclinic wave towards the loch-head. The stratification of the upper layers is observed to decrease rapidly landward of the 40 m isobath, a possible signature of enhanced diapycnal mixing in the shallower reaches towards the loch-head.
