


16th Canadian Symposium on Fluid Dynamics / 16ième Symposium Canadien sur la dynamique des fluides (Org: Richard Karsten, Acadia University and/et Serpil Kocabiyik, Memorial University)
 ANDREAS ACRIVOS, The Levich Institute, The City College of New York
Front propagation in suspensions driven by AC dielectrophoresis

When exposed to a spatially nonuniform highfrequency electric field,
a particle immersed in a liquid experiences a time average
dielectrophoretic force which is proportional to the particle volume,
the relative particle polarization, and the square of the field
strength. Compared to other available methods, AC dielectrophoresis is
particularly wellsuited for the manipulation of tiny particles in
microfluidics. However, the exposure of a suspension to a field
generates not only the dielectrophoretic force acting on each
particle, but also the dipolar interparticle interactions due to their
polarization. Furthermore, the fielddriven motion of the particles is
accompanied by their hydrodynamic interactions. In our recently
published papers (Appl. Phys. Lett. 83(2003), 4866; Phys. Rev. E 69(2004), 021402), we reported observations of new
field and flowinduced collective phenomena in the behavior of
suspensions, resulting in the formation and propagation of a distinct
front between the regions enriched with and depleted of particles. We
also proposed an electrohydrodynamic model for these phenomena, which
encompasses equations for the electric field, the suspension flow, and
the particle motion coupled together. The theoretical predictions were
found to be quantitatively consistent with the experiments even though
the model contains no fitting parameters.
The purpose of the present study is to examine analytically and
numerically the structure of the model equations referred to above in
the special case when, via a similarity transformation, these
equations reduce to a single o.d.e. for the particle concentration. We
establish the existence of shock solutions to this o.d.e. and
determine the location of the concentration front as well as the
dependence of the front velocity on the bulk particle concentration of
the suspension, the particle polarizability, and the field
strength. In particular, we demonstrate that the appearance of the
front can be caused either by the electrically induced local phase
separation of a suspension or by the rapid local growth of the
suspension viscosity due to the fielddriven particle accumulation in
a certain area. We show further that these similarity solutions
capture the principal features of the concentration profiles even
under conditions when the similarity transformation is not strictly
applicable.
The work was supported, in part, by grants from NASA (NAG32698), NSF
(#0307099), and NSFSandia National Laboratory (#0331001).
 YAKOV AFANASYEV, Memorial University of Newfoundland, Department of Physics
and Physical Oceanography, St. John's, NL, Canada
Vortex dipoles and wakes in a viscous fluid: asymptotic
theory, numerical simulations and laboratory experiments

Vortex dipoles are often observed in the Earth's oceans and atmosphere
as well as in a variety of other flows including engineering
applications. They are generated by a localized force acting in a
viscous fluid. The dipoles are closely related to wakes, which are
generated by a translating force. The translating localized force (or
force doublet) simulates the farfield wake behind a towed (or
selfpropelled) body. Wakes often become unstable, forming von Karman
vortex streets. Asymptotic solutions for the class of flows induced by
a localized (point) force or a force doublet both in 2D and 3D space
are discussed and compared with recent laboratory experiments and
direct numerical simulations.
 JOHN C. BOWMAN, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Spectral Reduction of the GOY Shell Model of Turbulence

Shell models of the GledzerOhkitaniYamada (GOY) type can provide an
excellent testbed for new ideas and methods for two and
threedimensional turbulence. In this talk, we use the GOY model to
study the method of spectral reduction (Bowman, Shadwick, and
Morrison, Phys. Rev. Lett. 83(1999), 5491). This decimation scheme
exploits the smoothness of moments of the underlying probability
distribution function to replace neighbouring shells by a reduced
number of representative shells with enhanced couplings. We show how
spectral reduction may be used to derive subgrid models.
 JAMES H. BUCHHOLZ, Princeton University, Princeton, NJ, USA
Wake Structures and Thrust Produced by Low Aspect Ratio
Flapping Panels

Under certain forcing conditions, flapping foils of high aspect
ratio (s/c; s=span, c=chord) have been shown to
shed a nominally twodimensional wake consisting of two
counterrotating vortices shed per flapping cycle. For foils of low
aspect ratio, streamwise vorticity shed from the sides of the panel
connects with spanwise vorticity shed by the trailing edge to form a
highly threedimensional wake structure. Digital particle image
velocimetry (DPIV) and flow visualization are used to investigate
vortical structures within the wakes of rigid and flexible flapping
flat panels with s/c < 1. In some cases, two vortices are shed per
flapping cycle (2S), whereas in others, there are four (2P).
However, in the case of a 2S wake, there is a spatial evolution
resulting in a 2P type of wake structure further downstream.
Detailed investigation of the flow reveals the wake to consist of
trains of vortex rings. The mean thrust produced by these panels is
estimated using DPIV data. The influence of Strouhal number, Reynolds
number, and various length ratios on thrust and wake structure will be
discussed.
 PAUL CHOBOTER, Oregon State University, College of Oceanic and Atmospheric
Sciences, 104 COAS Admin. Bldg., Corvallis, Oregon 973315503,
USA
A New Exact Solution of a Nonlinear Model of Coastal Upwelling

A twodimensional, frictionless, nonlinear model of coastal upwelling
is discussed. Included in the model is crossshore advection of
density and momentum, continuous density stratification, and a
geostrophic alongshore flow (often called the semigeostrophic
approximation). This model has been solved previously at steadystate
and as an initialvalue problem. However, the previous solution to
the initialvalue problem is inconsistent with the steadystate
solution. A new solution to the initialvalue problem is presented
that tends to the existing steadystate solution. The key to the
method of solution is a coordinate tranformation that renders the
nonlinear coupled set of PDEs linear, and thus provides a systematic
way to find the exact closedform solution to the fully nonlinear
system. The transformation itself is not new; our contribution lies
in the application of novel boundary conditions in transformed space.
The dynamical insights provided by the new solution are discussed.
 ALINE COTEL, University of Michigan
Thermal Impingement at a Stratified InterfaceVorticity and
Interface Dynamics

LaserInduced Fluorescence and Particle Image Velocimetry measurements
are performed for a thermal impinging on a stratified interface. The
thermal fluid is released from a cylinder at the bottom of a tank
containing a stable twolayer stratified environment. The Reynolds
number varies from 3000 to 8000 and the Richardson number from 1 to
16. The Richardson and Reynolds numbers are based on the thermal
quantities before impingement and on the initial density difference
across the interface. Maximum penetration height, rebound distance,
lateral spreading velocity and transport are evaluated. The
penetration and rebound heights follow a Ri1/2 power law. A
baroclinic eddy generated at the interface generally combines with one
of the thermal vortices to form a vortex pair. The pair remains near
the interface and propagates horizontally away from the impingement
zone. From these measurements, a significant level of lateral
transport is observed providing a more accurate representation of
transport in stratified flows.
 SERGE D'ALESSIO, University of Waterloo, Waterloo, Ontario N2L 3G1
Sheared Flow Past A Cylinder

The twodimensional problem concerning the unsteady uniform shear flow
of a viscous incompressible fluid past a cylinder will be
presented. The flow is calculated using two methods. The first takes
the form of a double series solution which is valid for small times
following the start of the motion and for large Reynolds numbers. The
second method involves a spectralfinite difference procedure for
numerically integrating the full NavierStokes equations expressed in
terms of a stream function and vorticity. The results will demonstrate
that for small times and moderately large Reynolds numbers the two
solutions are in very good agreement. A wide range of Reynolds numbers
has been considered and comparisons with previous studies will also be
discussed.
 ROBERT W. DERKSEN, University of Manitoba, Winnipeg, Manitoba R3T 5V6
The optimization of a biplane configuration

This talk will examine the optimization of a biplane configuration to
obtain the minimum drag to lift ratio configuration. The optimization
used a genetic algorithm, Differential Evolution, to select the
optimum combination of fourteen parameters such as wing span and chord
lengths, sweep and stagger angles, and airfoil sections. The
biplane's performance was calculated using the well established
biplane theory due to Max Munk. This is an interesting problem in
aerodynamic design and optimization as it has parameters that are both
continuous, such as wing span, and parameters that are discreet and
obtained from a limited data base. The optimization exercise clearly
demonstrated the power and value of Differential Evolution as a tool
to solve optimization problems.
 J. MACIEJ FLORYAN, University of Western Ontario
On the LaminarTurbulent Transition in Flows over Rough Surfaces

It is known that distributed surface roughness has strong effect on
the laminarturbulent transition in shear layers. This is one of
classical problems in fluid mechanics that is still not
understood. Resolution of this problem is of considerable practical
importance, particularly in the design of laminar airfoils in
aeronautical applications. Recent progress in this area will be
discussed.
 JANNETTE B. FRANDSEN, Louisiana State University, Dept. Civil & Environmental
Engineering, 3418C CEBA, Baton Rouge, LA 70803, USA
Frequency predictions in water tank structural systems

A fully nonlinear 2D stransformed finite difference solver has been
developed based on inviscid flow equations in rectangular tanks. The
fluid equations are coupled to an elastic support structure. Sloshing
motions are simulated during structural vibration cycles at and
outside resonance. The wave tank acts as a Tuned Liquid Damper
(TLD). Results of liquid sloshing induced by horizontal base
excitations are presented for small to steep nonbreaking waves. The
effectiveness of the TLD is discussed through predictions of coupling
frequencies of the tankstructural system for different tank sizes and
mass ratios between fluid and structure. Good agreement is achieved
between numerical model and firstorder potential theory outside the
resonance region. When the freesurface amplitudes become large in the
coupled system, the numerical peaks are larger and the troughs become
lower as time evolves compared to the linear solution. Nonlinearities
were found to reduce the system displacement significantly, e.g.,
system resonance shifted to beating response, but no shift in system
frequencies was observed.
 IAN A. FRIGAARD, University of British Columbia
On static bubbles in a viscoplastic fluid

We consider the propagation of a gas bubble in a cylindrical
column filled with a viscoplastic fluid. Because of the yield
stress of the fluid, it is possible that a bubble will remain
trapped in the fluid indefinitely. We investigate this case of
slowmoving or nearstationary bubbles. Using variational principles
we develop two stopping conditions for axisymmetric bubbles,
i.e. for a given bubble we can calculate a critical Bingham
number above which the bubble will not move. The first condition
is dependent on the bubble length as well as on the general shape
of the bubble. The second stopping condition is essentially a
comparison principle. We illustrate the stopping conditions by
application to specific bubble shapes. The analytical
results are quite general and not particularly sharp. To get
sharper bounds requires a numerical solution in which the
unyielded regions are resolved. We present our preliminary
numerical results.
Joint work with N. Dubash (Imperial College) and J. Y. Zhang (UBC).
 JAMES GOTTLIEB, Institute for Aerospace Studies, University of Toronto
Seeking Solutions to Unsolved Problems Involving
Shock Reflections in Gases and Condensed Matter

Regular and Mach wave reflection of shock waves from surfaces goes
back 125 years to the time of Mach, and the first solutions for
twoshock regular and threeshock Mach reflections dates back 55 years
to von Neumann. In spite of intense and increasing research activity
from von Neumann's time to the present, the physical understanding of
some shock reflections from surfaces in gases and condensed matter are
incomplete. For example, the experimentally measured transition from
regular to Mach reflection of a planar shock from a wedge occurs at
wedge angles some seven degrees lower than predicted by twoshock
theory, and this was deemed the von Neumann paradox by Birkoff in
1950. This and other regular and Mach reflection paradoxes are
described, available analytical and numerical solutions are
introduced, and interesting discoveries arising from the search to
resolve the paradoxes are discussed. A new principle of shock
reversibility is reported for the first time. This principle predicts
the sound speed behind shocks for which the shock and flow velocities
are measured in condensed matter, without using an equation of state.
 ALEXANDER E. HAY, Dalhousie University, Department of Oceanography, Halifax, NS
B3H 4J1
Largeamplitude internal tides and internal waves in the
lower Bay of Fundy

The highest tides in the world ocean occur in the Bay of Fundy, the
result of a nearresonant response of the barotropic tide in the Gulf
of MaineBay of Fundy system at the M2 tidal frequency. The
barotropic tide and associated residual circulation in the Bay of
Fundy system have received extensive treatment in the literature. In
contrast, the effects of stratification on the dynamics within the
Bay, including internal waves and the internal tide, have been little
studied. Results are presented from two Acoustic Doppler Current
Profiler (ADCP) deployments in the lower Bay of Fundy. The results
demonstrate that the velocity amplitudes associated with the internal
tide can be large: 30 to 50% of the barotropic tidal current
amplitude. In addition, largeamplitude, 5min. period internal wave
trains occur, mainly in association with maximum ebb tide.
 DAVID M. HOLLAND, New York University, 251 Mercer St., MC0711
Internal hydraulic jumps and mixing in twolayer flows

Internal hydraulic jumps in twolayer flows are studied, with
particular emphasis on their role in entrainment and mixing. For
highly entraining internal jumps, a new closure is proposed for the
jump conditions. The closure is based on two main assumptions:
(i) most of the energy dissipated at the jump goes into
turbulence, and
(ii) the amount of turbulent energy that a stably stratified
flow may contain without immediately mixing further is bounded by a
measure of the stratification.
As a consequence of this closure, surprising bounds emerge, for
example on the amount of entrainment that may take place at the
location of the jump. These bounds are probably almost achieved by
highly entraining internal jumps, such as those likely to develop in
dense oceanic overflows. The values obtained here are in good
agreement with the existing observations of the spatial development
of oceanic downslope currents, which play a crucial role in the
formation of abyssal and intermediate waters in the global ocean.
 SERGUEI IAKOVLEV, Department of Engineering Mathematics, Dalhousie University,
Halifax, Nova Scotia B3J 2X4
Shock loading on a fluidfilled cylindrical shell

Hydrodynamics of the interaction between a fluidfilled circular
cylindrical shell and an external hydrodynamic shock wave is
considered. A semianalytical solution of the corresponding linear
diffractionradiation problem is obtained, and the internal pressure
field is simulated. A variety of hydrodynamic phenomena in the
internal fluid is observed. In particular, primary and secondary
reflection and focusing of the internal shock wave are studied,
allowing for a better understanding of the influence that the internal
fluid has on the shell. Hydroelastic aspects of the interaction are
addressed as well. Two types of radiated hydrodynamic waves are
observed, the first one being induced by the incident shock wave, and
the second one is due to the "headon" elastic waves propagating in
the shell. The simulated pressure patterns appear to be in a good
agreement with available experimental observations.
 MANISH JUGROOT, University of Toronto Institute for Aerospace Studies,
4925 Dufferin Street, Toronto, Ontario M3H 5T6, Canada
Modelling of Neutral and Ion Transport in MassSpectrometer
Systems

This research was joint work with Clinton P. T. Groth.
The transport of ions through rapidly expanding and/or jet flows of
neutral gases is important to the operation of mass spectrometers,
such as liquid chromatography (LC) / mass spectrometry (MS) systems
used extensively in the trace analysis of biological fluids for
metabolites and natural biopolymers and in drug design. The
performance of the mass spectrometers is highly dependent on both the
gas and ion transport from the source region to the mass detectors,
and gaining an improved understanding of ionsource and interface
region flows and related transport phenomena is an active area of
research.
This presentation will review our experiences in the application of
modern numerical methods to the modelling of the neutral gas and ion
transport in the ionsource and interface regions of atmospheric
pressure ionization (API) mass spectrometer systems. In the case of
the neutral gas, the NavierStokes equations are solved using a
commercial code. For the ions, a new fivemoment continuumbased
model and parallel multiblock numerical solution procedure has been
developed. The latter accounts for the coupling between the ion and
neutral flows and incorporates the effects of ionneutral collision
processes and externally applied electric fields. The modelling has
been applied to several key MS flow regions including the dissolvation
chamber, skimmer interface region, and quadrupole region. The
capability of controlling the charged particle motions through a
combination of directed neutral flow and applied electric field is
demonstrated. Moveover, the numerical modelling is proving to be
extremely helpful in the design and optimization of the next
generation of MS systems.
This research was supported by MDS SCIEX, the Natural Sciences and
Engineering Research Council of Canada, and the Canadian Foundation
for Innovation.
 NICHOLAS KEVLAHAN, McMaster University, Hamilton, Ontario
Suppression of 3D flow instabilities in tightly packed tube bundles

In this paper we study the generation of streamwise vorticity in
tightly packed tube bundles. In particular, we would like to
understand which conditions (if any) enable the flow to remain
twodimensional for Re > 180. To investigate this question we have
calculated two and threedimensional flow through periodic rotated
square tube bundles with tight spacing P/D=1.5. The calculations
were done using a high resolution parallelized pseudospectral code
with penalization. We considered two types of bundles: fixed cylinder
and moving cylinder bundles. The natural frequency of the moving
cylinders is tuned to match the Strouhal frequency in order to
maximize the moving cylinder effect. We found that cylinder motion
completely suppresses threedimensional streamwise vorticity at
Re=200, while at Re=1000 the flow becomes threedimensional in both
the moving and fixed cylinder cases. However, the spanwise
correlation length is significantly larger when the cylinder is
moving.
 MARIA L. KILFOIL, McGill University, Montreal
Two ways to reorient liquid crystal polymers

Linear, semirigid polymer molecules have an elongated rodlike shape,
and in the right conditions may align spontaneously along a common
direction, exhibiting characteristics of nematic liquid crystals. This
class of molecules is called liquid crystalline polymers. When exposed
to body forces associated with magnetic or electric fields, or with
coupling to the flow field, the (timedependent) director orientation
can be described by the balance of the viscous torques.
This talk will present measurements of the director dynamics for a
lyotropic liquid crystal polymer under competing magnetic and
extensional or shearing flow fields, interpreted on the basis of the
Leslie and Ericksen model. We use speciallydesigned nuclear magnetic
resonance (NMR) method to perform localized spectroscopy, sensitive to
molecular orientation, to monitor in real time the director
orientation under pure planar extension; and compare the results to
predictions arising from the torque balance equation solved for this
geometry. In particular we explore the intriguing case of extensional
flow around a stagnation point in the plane perpendicular to a static
magnetic field [(B_{0})\vec], where the theory predicts a sudden flip in
director orientation at a critical extension rate
E¢_{c}. Flows in which there is a stagnation
point are important because as the molecular trajectories approach the
stagnation point, the polymer chains may be stretched far from
equilibrium. These studies reveal some key information about the
dynamics of this model liquid crystal polymer, and of rodlike
semiflexible molecules in general.
 FOTINI LABROPULU, Luther College, University of Regina
MHD stagnation point flow of a viscoelastic fluid with heat transfer

The steady twodimensional flow of an incompressible Walter's B fluid
impinging on an infinite flat plate in the presence of a magnetic
field is considered. We examined the stagnation point flow and heat
transfer of the Walter's B fluid on a linearly stretching sheet when
the velocity of the sheet and the free stream velocity are not
equal. The problem may be regarded as a combination of two problems,
twodimensional stagnation point flow and flow over a stretching sheet
in an ambient fluid. Numerical and analytical solutions of the
governing equations, including the energy equation, have been
obtained. Particular cases of the present solutions are compared with
those available in literature.
 ULRIKE LOHMANN, Dalhousie University
Aerosols and Climate: The Cloud Connection

The anthropogenic component of sulfate and carbonaceous aerosols has
substantially increased the global mean aerosol amount from
preindustrial times to the presentday. Some aerosols exert an
indirect effect by acting as centers for cloud droplets and thereby
affecting the reflectivity, precipitation formation, and lifetime of
warm clouds. For a constant amount of cloud water, a higher cloud
droplet number causes an increase in cloud reflectivity. Reductions
in precipitation efficiency due to more but smaller cloud droplets
slow down the precipitation formation and increase cloud lifetime. The
cooling from both indirect effects can partly offset the greenhouse
gas warming. It is, however, still very poorly constrained from
climate model simulations and thus is an important source of
uncertainty in projections of future climate change.
In this talk, I will give an overview over these various indirect
aerosol effects including the ability of anthropogenic aerosols to
enhance or reduce precipitation.
 ROSSITZA MARINOVA, Enabled Simulation & Optimization Software, #304, 10240 124
Street, Edmonton, AB T5N 3W6
Efficient Coordinate Operator Splitting for Incompressible
NavierStokes Equations

Coupled methods require a significant amount of computer time and
storage in order to obtain a solution, particularly for
multidimensional problems. On the other hand, they are more
efficient in general. A splitting procedure can reduce in order of
magnitude the number of operations per iteration comparing with
application of direct solvers. We employ operator splitting leaving
the system coupled at each fractionaltime step which allows
satisfying the boundary conditions without introducing artificial
boundary conditions for the pressure, namely a generalization of the
DouglasRachford scheme. Since the equations to be solved are
conservation laws, the numerical scheme should also preserve these
laws. We choose approximations of the differential operators for which
the numerical scheme preserves the integral properties of the
respective differential problem. It is not a trivial task to
construct finite difference schemes especially in the case of
operator splitting. The splitting algorithm is verified through
various benchmark problems.
 PIER MARZOCCA, Clarkson University
Linear/Nonlinear Indicial Methods for 2D Lifting Surfaces
Aeroelasticity

The principal aim of this study is the development of a linear and
nonlinear indicial based models for the calculation of unsteady
aerodynamic loads, flutter instability and aeroelastic response.
Issues related to the formulation and generation of linear/nonlinear
aerodynamic indicial functions through a combined CFD and analytical
procedure are addressed and in this context a unified functional
representation of the unsteady aerodynamic airfoil theory in
subsonic/supersonic compressible and transonic flight speed regimes is
presented. This study shows that the linear indicial theory gives
excellent results as long as the aerodynamic nonlinearities (shock
waves, separation, and high angle of attack) included in the
aeroelastic systems are weak. Nonlinear indicial model better predict
the aerodynamic loads and consequently the aeroelastic response in
presence of significant aerodynamic nonlinearities. Comparison and
validation of the aerodynamic model, flutter and aeroelastic response
are presented and pertinent conclusions are outlined.
 CATHERINE MAVRIPLIS, George Washington University, Washington, DC, USA
An Adaptive Spectral Element Method for Complex Fluid Flow

The spectral element method offers high accuracy and geometrical
flexibility for improved resolution capabilities of complex fluid
flow. Adaptivity further enhances the power of the method by
allocating resources when and where they are needed in a spatially and
temporally developing complex flow. The adaptive method uses the
mortar formulation to retain spectral accuracy with nonconforming
grids and a posteriori error estimators to guide the refinement and
coarsening. Calculations of 2D premixed flames wrinkled by a synthetic
turbulent velocity field and 3D advancing heat sources will be
presented and discussed.
 JULIO MILITZER, Dalhousie University, P. O. Box 1000, Halifax, NS B3J 2X4
Numerical Simulations of Vortex Induced Vibrations in Long
Cylinders with the Numerical Wind Tunnel

Long cylindrical risers ( > 2000m) are required for deep water
exploration/production of petroleum or natural gas. The flow of
seawater around these cylinders is subject to vortex shedding. This is
an unsteady oscillatory phenomenon, which causes the pressure
distribution around the cylinders to fluctuate. If the vortex shedding
frequency is equal to one of the natural frequency modes, this will
cause the cylinders to vibrate with what is known as Vortex Induced
Vibrations (VIV). To simulate these flows we have developed a
Computational Fluid Dynamics (CFD) tool called the Numerical Wind
Tunnel (NWT). The program incorporates: automatic generation of an
anisotropic Cartesian mesh, Immersed Boundary Method for boundary
condition specification, automatic grid refinement/coarsening, Large
Eddy Simulation (LES) turbulence model and code parallelization. One of
the features of the NWT is that it is capable of dealing with flows
with moving boundaries, without having to recalculate the mesh.
Results for two and threedimensional simulations will be presented.
 PATRICK MONTGOMERY, University of Northern British Columbia, 3333 University Way,
Prince George, BC
Deceleration of purely rotating curling rocks through wet
friction

The standard method for describing the motion of a solid body on an
ice sheet is with a sliding coefficient of friction: this is called
`dry friction'. If a thin liquid layer exists between sliding body
and the ice, the socalled `wet friction' is typically modelled by
finding an appropriate power law. In this presentation, a model for
wet friction of a curling rock undergoing only rotational motion is
derived from first principles. The subsequent multiphase flow problem
is simplified to the point where exact solutions may be obtained. The
interface between the water and ice phases is considered to be a
Stefan problem, and the resulting fluid layer is responsible for the
viscous drag forces that slow and eventually stop the rotating solid
body. As a special case, a similar problem obtained by neglecting
curvature is examined where the problem may be expressed in Cartesian
coordinates rather than radial coordinates. Thus instead of a
rotating curling rock, the physical problem is represented by a
finite skate blade moving in a straight line over an ice surface with
a fluidized interface.
 YURI MUZYCHKA, Memorial University of Newfoundland
Asymptotic Methods and Scaling Principles in Laminar Internal Flows

Internal flows appear in a host of fundamental engineering
applications ranging from small scale microflows in MEMS devices to
large scale process flows in compact heat exchangers. The focus of the
present topic is on the development of robust models using several key
concepts: asymptotic analysis, scaling principles, and characteristic
length scales. Two fundamental problems from fluid dynamics for which
a number of exact and approximate solutions exist, are examined. These
are: laminar developing flows in finite ducts and laminar commencement
flows in infinite ducts. Both circular and noncircular ducts are
considered. Both problems contain convenient asymptotic behavior,
which allows for the construction of simple predictive models. In the
case of laminar entrance flow, short and long duct solutions are
examined for a variety of duct shapes and a simple model is developed
for the dimensionless mean wall shear stress. In the case of laminar
commencement flows, short time and long time solutions are examined
for a variety of duct shapes and simple models are developed for
predicting the time varying area average velocity and perimeter
average shear stress. Scaling principles are used to show the order of
magnitude asymptotic characteristics. Both problems contain the
classic fully developed flow problem, which is shown to be easily
modeled by means of using a more appropriate characteristic length
scale, the square root of crosssectional area, in the definition of
dimensionless mean wall shear. By means of an asymptotic correlation
method, these asymptotes are combined to yield robust models which are
valid for any value of the dimensionless duct length or dimensionless
time. Finally, application of these principles is also demonstrated
for adiabatic two phase flow, by combining the asymptotic single phase
characteristics in circular tubes.
 RICHARD PELTIER, University of Toronto, Department of Physics, 60 St. George
Street, Toronto, Ontario M5S 1A7, Canada
On the resonant generation of large amplitude internal
solitary and solitarylike waves

This work by Dr. Marek Stastna and I has been motivated by
observations of large amplitude internal solitary waves (ISW's) in the
coastal ocean which have become commonplace, yet detailed
understanding of their origins has remained somewhat obscur. The most
promising candidate mechanism for their generation is that involving a
resonant process in which the initial disturbance at the level on
which the ISW's are able to propagate horizontally is induced by
topographic forcing from below. By employing an analytical
representation of the vertical density stratification that serves as a
generic model of the coastal ocean, we find, using both precise
numerical and analytical methods, that even with small amplitude
compact topography it is possible to generate a wide variety of
extremely stable, highly nonlinear response structures. Aside from the
well known upstream propagating internal solitary waves, we find
upstream propagating dissipationless bores, extremely large amplitude
disturbances trapped over the topography, and solitary waves that
are swept downstream without changing form. We demonstrate that the
conjugate flow concept provides a useful general means of
understanding and predicting the fluid response. This work is a
significant generalization to previous work that was devoted to the
development of understanding of the flows observed in Knight Inlet on
the coast of British Columbia using acoustic imaging methods
(M. Stastna and W. R. Peltier, Upstream propagating solitary
waves and breaking internal waves in flow over the sill in Knight
Inlet. Proc. Roy. Soc. Ser. A, in press, 2004).
 FRANCIS POULIN, UCSD
An Oscillating Jet in the Cape Cod Bay

During the spring months, the Cape Cod Bay is a roaming ground for the
North Atlantic right whale, perhaps the most endangered whale species
in the world. The whales are observed to travel along the topographic
steps that run parallel to the shore, eating plankton patches that
form in the coastal water.
In this region, off the coast of Provincetown, there is an oscillatory
current with the same period as that of the ambient tides. The
location of the current and its periodicity suggest that the
topography and tides play fundamental roles in generating the jet.
This current, depending on its velocity profile, may become unstable
and generate vortices. It is likely that the local surface
convergences and divergences in the tidal flows and vortices are
related to the aggregation of the copepods (Calanus Finmarchicus),
which are the right whale's primary food source. Understanding the
dynamics of this jet is essential to predicting the spatial and
temporal patterns of the codepods, which will in turn help us
understand the likely locations and feeding history of the whales.
In this talk we discuss results of the first phase of this study, that
of the oscillatory jet in the Cape Cod Bay. This jet is rather
complicated since it involves complex topography and coastlines,
bottom and lateral friction, stratification and numerous other
effects. Rather than study this system in fine detail, we investigate
an idealized model that captures the essential features. In the
context of this model, we first compute possible profiles for the
oscillating jet. We then solve the linear stability problem to
determine how the growth rates depend on the various parameters.
Finally, and most importantly, we study the nonlinear problem to
observe the time evolution of the instability process along with its
equilibration. This provides some insight into how the instabilities
are related to fluid transport across the shelf.
 TIM ROGALSKY, Canadian Mennonite University, 500 Shaftesbury Blvd.,
Winnipeg, MB R3P 2N2
Influence of Airfoil Representation on Aerodynamic Design

In the computeraided design of aerodynamic shapes, a representation
technique is required to interpret a realvalued vector as a geometric
shape. A global search method can then be used to search for the
vector that optimizes the design objective. Designers are beginning to
realize that the geometric representation method used can profoundly
influence the end results. Not all methods are capable of representing
the optimal shape, which has an obvious effect on global
convergence. The dimension of the vector directly affects the size of
the solution space, thereby influencing the convergence
rate. Furthermore, several new results indicate that the types of
parameters used can influence both globality and rate of
convergence. In particular, the incorporation of aerodynamic
quantities (such as leading edge radius, or trailing edge angle) can
significantly improve the convergence characteristics. This talk will
survey the representation methods available for airfoils, present a
new representation method, and examine the influence on convergence
for a particular aerodynamic design method.
 BARRY R. RUDDICK, Department of Oceanography, Dalhousie University, Halifax,
Nova Scotia
Differential mixing of heat and salt by breaking internal waves

A laboratory experiment has demonstrated differential mixing of heat
and salt due to breaking internal waves. Following McEwan's classic
experiment, a paddle was pivoted at resonant frequency to excite the
gravest internal wave mode. Energy was transferred to higher modes
via a cascade of resonant triad interactions, leading to quasirandom
overturn/mixing events. Fluxes of heat and salt were deduced from
profiles measured before and after mixing periods, after correction
for thermal heat exchange through the tank walls. The ratio of salt
to heat eddy diffusivities was found to be approximately 0.6 when the
waves were strongly pumped (thermal eddy diffusivity approximately 2×10^{6} m^{2}/s), decreasing to less than 0.2 at weaker pumping
rates. Corrections for molecular diffusion, estimates of viscous
dissipation of turbulent kinetic energy, and the possibility of heat
flux via sidewall Stokes boundary layer effects, will be discussed.
 SIV SIVALOGANATHAN, University of Waterloo, Department of Applied Mathematics
The use of quasilinear viscoelasticity theory in calculating
the ventricular boundary motion in hydrocephalus

Hydrocephalus is a condition arising when an excessive accumulation of
cerebrospinal fluid (CSF) in the brain causes enlargement of the
ventricular cavities. Modern treatments of shunt implantation are
effective but have an unacceptably high rate of failure. One of the
most common factors causing shunt failure is the misplacement of the
proximal catheter tip. This can be remedied if the ventricular
boundary configuration (due to decompression) can be predicted. In
this talk, we report on a theoretical method of calculating the
ventricular boundary speed using quasilinear viscoelasticity theory.
 JOHN STOCKIE, Simon Fraser University, Department of Mathematics, Burnaby,
BC V5A 1S6
Parametric Resonance in Immersed Elastic Boundaries

We examine the stability of fluid flows containing immersed elastic
boundaries, where the fluidstructure interaction is driven by
periodic variations in the elastic properties of the solid material.
These systems can give rise to "parametric resonance", and using
Floquet theory we derive an eigenvalue problem which can be solved
numerically to determine values of the forcing frequency and fluid
viscosity for which resonance occurs. Numerical simulations of the
fluidstructure interaction problem are performed in 2D using the
"immersed boundary method", which verifies the existence of the
parametric resonances suggested by the theory. We then discuss
applications of the results to biological systems such as cardiac
muscle fibre and the cochlea in the inner ear.
 DAVID STRAUB, McGill University, Atmospheric and Oceanic Sciences, 805
Sherbrooke W., Montreal, Quebec H3A 2K6
Mechanical energy input and dissipation in winddriven ocean
circulation

Traditional models of ocean circulation have that mechanical energy is
input by the winds to the ocean general circulation at large scales
and is dissipated primarily in the bottom boundary layer. This view is
challenged on two fronts. First, it is argued that a weak dependence
of wind stress on the surface ocean velocity can lead to a much more
pronounced reduction in the wind power input. Specifically, energy
input by the winds at large scales is taken out at mesoscales. Basic
scaling arguments and numerical simulations are presented to support
this claim. A second point is that a significant energy sink could
result from energy transfers out of the "balanced" modes of flow and
into forwardcascading unbalanced modes (such as the gravity wave
field). These transfers can occur either over rough topography or in
regions where local measures of the Rossby number are O(1). We
concentrate on the latter, and argue that such transfers are both
generic and welldescribed by hydrostatic dynamics.
 BRUCE SUTHERLAND, University of Alberta, Edmonton, AB
Reflection, Transmission and Tunnelling of Internal Waves

The path followed by internal waves propagating through a fluid with
varying stratification and background horizontal flow is often
assessed by way of ray theory. In particular, this theory predicts
that waves reflect from a level where the Dopplershifted frequency of
the waves equal the background buoyancy frequency. Thus, without more
careful consideration of the limitations of ray theory, one might
conclude that internal waves reflect from regions that are locally
mixed (for example, due to wave breaking or double diffusive
convection). In reality, if the mixed region is sufficiently thin,
incident internal waves can partially transmit across it.
The linear equations of motion are solved to predict the transmission
coefficient of internal waves across mixed regions in two
circumstances. For waves with fixed horizontal wavenumber incident
upon a tophat shaped N^{2} profile (with corresponding continuous
density profile), the maximum transmission occurs for nonhydrostatic
waves with frequency w = N/Ö2. For waves incident upon a
mixed region with discontinuous density jumps on either flank of the
tophat, as might occur due to localised mixing of a continuously
stratified fluid, resonant coupling occurs between interfacial and
vertically propagating internal waves that permits perfect
transmission even for finitedepth gaps.
 GORDON E. SWATERS, Applied Mathematics Institute, Department of Mathematical &
Statistical Sciences, and Institute for Geophysical Research,
University of Alberta, Edmonton, Canada
Meridional flow of grounded source driven abyssal currents in
a basin with topography

Stommel and Arons [1] showed that the Sverdrup vorticity balance
predicts the equatorward flow of a source driven abyssal water
mass. Thus, in the immediate vicinity of the region of deepwater
production in high latitudes, there is an intrinsic tendency for
preferential equatorward abyssal flow. Away from the source region,
much of the abyssal circulation is strongly characterized by the
isopycnal field being grounded against sloping topography (e.g., the
deep western boundary undercurrent in the North Atlantic, Richardson
[2]) and the flow being in geostrophic balance. Indeed, as shown by
Nof [3], a fully grounded (i.e., compactly supported) abyssal water
mass, in the fully nonlinear but reduced gravity dynamical limit,
moves nondispersively and steadily in the along slope direction (in a
right (left) handed sense in the northern (southern) hemisphere),
regardless of the height or vorticity field within the abyssal water
mass.
These two results provide a compelling scenario for the initiation and
maintenance of grounded abyssal flow. That is, in high latitude source
regions where deep water is produced (often over sloping topography),
the Sverdrup vorticity balance initiates equatorward flow. One
produced, this abyssal flow can become grounded and geostrophically
adjusted, maintaining a Nof balance which permits sustained basin
scale meridional quasisteady and coherent propagation regardless of
the spatial structure of the water mass. Of course, this picture
leaves out many important dynamical processes such as diabatic
effects, baroclinicity, instability and mixing. In addition, such a
scenario cannot explain crossequatorial abyssal currents where the
underlying assumptions of geostrophically balanced grounded flow must
necessarily break down.
The principal purpose of the present contribution is to briefly
describe some results from a model for the subinertial evolution and
meridional flow of source driven grounded abyssal currents over
sloping topography and their baroclinic interaction with the overlying
water column.
Succinctly summarized, the model is an amalgamation, with the
inclusion of variable topography and mass conserving up and
downwelling, of the two layer QG model used by Holland [4] to
investigate the baroclinic evolution of the wind driven circulation
and the QG/PG abyssal current model of Swaters [5] used to investigate
the baroclinic instability of grounded geostrophic flow.
References
 [1]

H. Stommel and A. B. Arons,
On the abyssal circulation of the world oceanI. Stationary
flow patterns on a sphere.
DeepSea Res. 6(1960), 140154.
 [2]

P. L. Richardson,
On the crossover between the Gulf Stream and the Western Boundary
Undercurrent.
DeepSea Res. 24(1977), 139159.
 [3]

D. Nof,
The translation of isolated cold eddies on a sloping bottom.
DeepSea Res. 30(1983), 171182.
 [4]

W. R. Holland,
The role of mesoscale eddies in the general circulation of the
oceanNumerical experiments using a winddriven quasigeostrophic
model.
J. Phys. Oceanogr. 8(1978), 363392.
 [5]

G. E. Swaters,
On the baroclinic instability of coldcore coupled density fronts
on sloping continental shelf.
J. Fluid Mech. 224(1991), 361382.
 LAURETTE S. TUCKERMAN, LIMSICNRS, Orsay, France
Computational Study of TurbulentLaminar Bands in Couette Flow

Recent experiments by Prigent and Dauchot have shown that the
remarkable spiral turbulence state of TaylorCouette flow also occurs
in plane Couette flow. In both cases, a pattern of alternating
turbulent and laminar bands appears at a welldefined Reynolds
number. The pattern is tilted with respect to the streamwise (or
azimuthal) direction and its wavelength is much larger than the gap;
the angle and wavelength depend systematically on Reynolds number. We
have numerically simulated these turbulentlaminar patterns for plane
Couette flow. In our computational approach, we replace the very large
lateral dimensions of the experiment by a narrow and periodically
repeating rectangle which is tilted with respect to the streamwise
direction. In this way we determine which angles and lengths support
turbulent bands.
 HENRY VAN ROESSEL, University of Alberta, Edmonton, Alberta T6G 2G1
Some Results on Coagulation Equations

An important phenomenon in a wide variety of processes in physics,
chemistry, biology, medicine and engineering is the formation of large
clusters by the union of many separate small elements. Examples
include, but are not limited to, polymerization processes in polymer
science, coagulation processes in aerosol and colloidal physics,
percolation and nucleation in phase transitions and critical
phenomena, antigenantibody aggregation in immunology, rouleaux
formation by red blood cell adhesion in hematology and crystallization
in materials science.
One of the earliest attempts to understand coagulation, and the first
to derive a mathematical model, was Smoluchowski. He
made the assumption that collisions are binary and fluctuations in
density are small in order that collisions occur at random. The
coagulation equations of Smoluchowski comprise an infinite set of
coupled ordinary differential equations. In addition to this set of
coupled ODEs, which models the coagulation of discrete sized
particles, a generalized continuous model that consists of an
integrodifferential equation, which describes the coagulation of
particles of arbitrary size, is also commonly studied.
In applications one might wish to exercise some control over the
coagulation process. For instance, it may be desirable to increase or
restrict the limiting number of particles of a particular size. One
might attempt to achieve this is by the introduction of a source of
particles of some prescribed size to enhance the coagulation process
to arrive at some desired limiting state. The long time
behaviour
of solutions as well as the effects of source terms on solutions will be discussed.
 MICHAEL J. WARD, University of British Columbia, Vancouver, BC V6T 1Z2
Summing Logarithmic Expansions in a Low Reynolds Number Flow

We consider the classic problem of slow, steady, twodimensional flow
of a viscous incompressible fluid around an infinitely long straight
cylinder. For low Reynolds number, the wellknown asymptotic
expansions of the drag coefficient and of the flow field start with
infinite logarithmic series. We show that these entire infinite series
are contained in the solution to a certain related problem that does
not involve the crosssectional shape of the cylinder. The drag
coefficient for a symmetric cylinder of specific crosssectional
shape, and which is asymptotically correct to within all logarithmic
terms, is given in terms of a single shapedependent parameter
determined from the solution to a canonical Stokes problem. The
resulting hybrid asymptoticnumerical method is illustrated, and
compared to various previous theories, for cylindrical bodies that are
symmetric or asymmetric with respect to the free stream. The
asymptotic structure of this problem is found to be very similar to
that for nonlinear elliptic problems in a twodimensional container
with small holes.
 TIMOTHY WEI, Rutgers University, Mech. & Aero. Eng'g. / 98 Brett Rd.
Fundamental Fluid Dynamics and Olympic Swimming

The world of competitive swimming is dynamic. Swimmers today are
bigger, stronger and faster than they ever have been. The training
regimen of an elite athlete includes not only endless practice of his
or her skills, but also a carefully planned diet, strength and
endurance training, and hours of mental preparation. Within this
framework, researchers from Rutgers and George Washington Universities
have teamed with USA Swimming to develop advanced, fluid dynamics
based training and analysis tools for current and future Olympic
swimmers. The focus of this presentation will be on the objectives,
methodologies and early outcomes of measurement and computations of
flow around swimmers. Movies of flow measurements around swimmers,
including Beth Botsford, the 1996 Olympic Gold Medalist in the 100 m
backstroke, will be presented.
 F. MARY WILLIAMS, Institute for Ocean Technology, St. Johns, Newfoundland
Flow past a cylinder: practical applications in ocean
engineering

A classical problem in fluid dynamics has serious and expensive
implications in ocean engineering. As offshore drilling platforms move
into deeper water, the length of mooring lines and risers subjected to
subsea currents extends to thousands of metres. The talk presents the
context and scale of the practical problem, and then provides a review
of recent experimental and numerical investigations at the Institute
for Ocean Technology.
 JIANYING ZHANG, University of British Columbia
Taylor dispersion for generalised nonNewtonian fluids

We consider laminar Taylor dispersion for a generalised Newtonian
fluid flowing in a pipe. Using a multiple timescales method we
derive an expression for the Taylor dispersion coefficient, which
can be evaluated by simple quadrature. We present results for a
variety of shearthinning models: powerlaw, Carreau and Cross
models, and for a range of yield stress models: HerschelBulkley,
Bingham, generalised Casson and RobertsonStiff. Dispersion effects
are generally reduced by shearthinning behaviour and by having a
large yield stress.
Joint work with Ian A. Frigaard (UBC).

