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 high-frequency 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 well-suited 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 field-driven 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 flow-induced 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 electro-hydrodynamic 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 field-driven 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 (NAG3-2698), NSF (#0307099), and NSF-Sandia 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 far-field wake behind a towed (or self-propelled) 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 Gledzer-Ohkitani-Yamada (GOY) type can provide an excellent testbed for new ideas and methods for two- and three-dimensional 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 two-dimensional wake consisting of two counter-rotating 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 three-dimensional 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 (2-S), whereas in others, there are four (2-P). However, in the case of a 2-S wake, there is a spatial evolution resulting in a 2-P 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  97331-5503, USA
A New Exact Solution of a Nonlinear Model of Coastal Upwelling

A two-dimensional, frictionless, nonlinear model of coastal upwelling is discussed. Included in the model is cross-shore advection of density and momentum, continuous density stratification, and a geostrophic along-shore flow (often called the semigeostrophic approximation). This model has been solved previously at steady-state and as an initial-value problem. However, the previous solution to the initial-value problem is inconsistent with the steady-state solution. A new solution to the initial-value problem is presented that tends to the existing steady-state 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 closed-form 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 Interface-Vorticity and Interface Dynamics

Laser-Induced 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 two-layer 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 Ri-1/2 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 two-dimensional 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 spectral-finite difference procedure for numerically integrating the full Navier-Stokes 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 Laminar-Turbulent Transition in Flows over Rough Surfaces

It is known that distributed surface roughness has strong effect on the laminar-turbulent 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 2-D s-transformed 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 non-breaking waves. The effectiveness of the TLD is discussed through predictions of coupling frequencies of the tank-structural system for different tank sizes and mass ratios between fluid and structure. Good agreement is achieved between numerical model and first-order potential theory outside the resonance region. When the free-surface 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 visco-plastic 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 slow-moving or near-stationary 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 two-shock regular and three-shock 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 two-shock 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
Large-amplitude 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 near-resonant response of the barotropic tide in the Gulf of Maine-Bay 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, large-amplitude, 5-min. period internal wave trains occur, mainly in association with maximum ebb tide.

DAVID M. HOLLAND, New York University, 251 Mercer St., MC-0711
Internal hydraulic jumps and mixing in two-layer flows

Internal hydraulic jumps in two-layer 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

Hydrodynamics of the interaction between a fluid-filled circular cylindrical shell and an external hydrodynamic shock wave is considered. A semi-analytical solution of the corresponding linear diffraction-radiation 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 "head-on" 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 Mass-Spectrometer 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 ion-source 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 ion-source and interface regions of atmospheric pressure ionization (API) mass spectrometer systems. In the case of the neutral gas, the Navier-Stokes equations are solved using a commercial code. For the ions, a new five-moment continuum-based model and parallel multi-block numerical solution procedure has been developed. The latter accounts for the coupling between the ion and neutral flows and incorporates the effects of ion-neutral 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 two-dimensional for Re > 180. To investigate this question we have calculated two- and three-dimensional flow through periodic rotated square tube bundles with tight spacing P/D=1.5. The calculations were done using a high resolution parallelized pseudo-spectral 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 three-dimensional streamwise vorticity at Re=200, while at Re=1000 the flow becomes three-dimensional 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, semi-rigid polymer molecules have an elongated rod-like 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 (time-dependent) 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 specially-designed 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 [(B0)\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 rod-like semi-flexible molecules in general.

FOTINI LABROPULU, Luther College, University of Regina
MHD stagnation point flow of a viscoelastic fluid with heat transfer

The steady two-dimensional 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, two-dimensional 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 pre-industrial times to the present-day. 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 Navier-Stokes Equations

Coupled methods require a significant amount of computer time and storage in order to obtain a solution, particularly for multi-dimensional 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 fractional-time step which allows satisfying the boundary conditions without introducing artificial boundary conditions for the pressure, namely a generalization of the Douglas-Rachford 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 2-D 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 three-dimensional 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 so-called 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 micro-flows 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 non-circular 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 cross-sectional 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 solitary-like 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 computer-aided design of aerodynamic shapes, a representation technique is required to interpret a real-valued 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 quasi-random 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 m2/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 fluid-structure 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 fluid-structure 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 wind-driven 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 forward-cascading 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 well-described 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 Doppler-shifted 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 top-hat shaped N2 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 top-hat, 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 finite-depth 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 deep-water 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 quasi-steady 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 cross-equatorial 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 sub-inertial 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 ocean-I. Stationary flow patterns on a sphere. Deep-Sea Res. 6(1960), 140-154.

[2]
P. L. Richardson, On the crossover between the Gulf Stream and the Western Boundary Undercurrent. Deep-Sea Res. 24(1977), 139-159.

[3]
D. Nof, The translation of isolated cold eddies on a sloping bottom. Deep-Sea Res. 30(1983), 171-182.

[4]
W. R. Holland, The role of mesoscale eddies in the general circulation of the ocean-Numerical experiments using a wind-driven quasi-geostrophic model. J. Phys. Oceanogr. 8(1978), 363-392.

[5]
G. E. Swaters, On the baroclinic instability of cold-core coupled density fronts on sloping continental shelf. J. Fluid Mech. 224(1991), 361-382.

LAURETTE S. TUCKERMAN, LIMSI-CNRS, Orsay, France
Computational Study of Turbulent-Laminar Bands in Couette Flow

Recent experiments by Prigent and Dauchot have shown that the remarkable spiral turbulence state of Taylor-Couette flow also occurs in plane Couette flow. In both cases, a pattern of alternating turbulent and laminar bands appears at a well-defined 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 turbulent-laminar 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, antigen-antibody 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 integro-differential 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, two-dimensional flow of a viscous incompressible fluid around an infinitely long straight cylinder. For low Reynolds number, the well-known 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 cross-sectional shape of the cylinder. The drag coefficient for a symmetric cylinder of specific cross-sectional shape, and which is asymptotically correct to within all logarithmic terms, is given in terms of a single shape-dependent parameter determined from the solution to a canonical Stokes problem. The resulting hybrid asymptotic-numerical 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 two-dimensional 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. Johns, 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 non-Newtonian 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 shear-thinning models: power-law, Carreau and Cross models, and for a range of yield stress models: Herschel-Bulkley, Bingham, generalised Casson and Robertson-Stiff. Dispersion effects are generally reduced by shear-thinning behaviour and by having a large yield stress.

Joint work with Ian A. Frigaard (UBC).