1er Symposium canadien en analyse numérique et calcul scientifique
Jeudi, 20 juin
Salle Des Plaines B
10h00
A Numerical Model for the Study of Turbulent Free Surface Flows
Dufour, Steven and Roland Rivard
Département de mathématiques et de génie industriel, École Polytechnique de Montréal

A numerical model for the simulation of turbulent free surface flows is proposed. This model is based on large-eddy simulations (LES) and on a marker-variable based Eulerian interface capturing strategy. The LES model is introduced in the context of free surface flows problems, using the Wall-adapting local eddy-viscosity (WALE) model for the turbulent viscosity. The filtered free surface capturing model is also introduced. The proposed methodology is validated based on two well-documented free surface flow problems where turbulence is observed: the Rayleigh-Taylor instability and the two-phase transient mixing layer problem.

10h30
Dynamics of a Periodically Heated Fluid Layer
Floryan, J. M.
Department of Mechanical and Materials Engineering, The University of Western Ontario

Natural convection in an infinite horizontal layer subject to a periodic heating along the lower wall has been investigated. The heating maintains the same mean temperatures at both walls while producing sinusoidal temperature variations along one horizontal direction with its spatial distribution characterized by the wave number $\alpha$ and the amplitude expressed in terms of a Rayleigh number $Ra_p$. The primary response of the system has the form of a stationary convection which consists of rolls with axis orthogonal to the heating wave vector and structure determined by the particular values of $Ra_p$ and $\alpha$. It is shown that for sufficiently large $\alpha$ convection is limited to a thin layer adjacent to the lower wall with a uniform conduction zone emerging above it; the temperature in this zone becomes independent of the heating pattern and varies in the vertical direction only.

Linear stability of the above system has been considered and conditions leading to the emergence of a secondary convection have been identified. The secondary convection gives rise either to the longitudinal rolls, to the transverse rolls, or to the oblique rolls at the onset, depending on $\alpha$. The longitudinal rolls are parallel to the primary rolls and the transverse rolls are orthogonal to the primary rolls, and both of them result in striped patterns. The oblique rolls lead to the formation of convection cells with aspect ratio dictated by their inclination angle and formation of rectangular patterns.

Two mechanisms of instability have been identified. In the case of $\alpha=0(1)$, the parametric resonance dominates and leads to the pattern of instability that is locked-in with the pattern of the heating according to the relation $\delta_{cr} = \alpha/2$, where $\delta_{cr}$ denotes the component of the critical disturbance wave vector parallel to the heating wave vector. The second mechanism, Rayleigh-Bénard (RB) mechanism, dominates for large $\alpha$ where the instability is driven by the uniform mean vertical temperature gradient created by the primary convection with the critical disturbance wave vector $\delta_{cr} \rightarrow 1.56$ for $\alpha \rightarrow \infty$ and the fluid response becoming similar to that found in the case of a uniformly heated wall. Competition between these mechanisms gives rise to non-commensurable states in the case of longitudinal rolls and appearance of soliton lattices, to the formation of very distorted transverse rolls, and to the appearance of the wave vector component in the direction perpendicular to the forcing direction. A rapid stabilization is observed when the heating wave number is reduced below $\alpha \approx 2.2$ and no instability is found when $\alpha < 1.6$ in the range of $Ra_p$ considered. It is shown that $\alpha$ plays the role of an effective pattern control parameter and its judicious selection provides means for creation of a wide range of flow responses.

The relevant mean flow solution has been determined using discretization based on Fourier expansions in the streamwise direction and spectral collocation method for the transverse direction. The resulting system of nonlinear algebraic equations has been solved using linearization based on the use of convective terms from previous iteration. The linear stability problem has lead to an eigenvalue problem for system of ordinary differential equations. These equations have been discretized using spectral collocation method. The eigenvalues have been determined either by computing spectra of very large matrices, or by tracing eigenvalues using inverse iteration method, or by tracing eigenvalues using Newton-Raphson procedure.

11h00
Special Solutions in Smectic Electroconvection
Pugh, M.C. (1) and S.W. Morris (2)
(1) Department of Mathematics, University of Toronto, (2) Department of Physics, University of Toronto

We discuss electroconvection in a free submicron-thick liquid crystal film in an annular geometry. The film is flat in the xy plane; seen from above it looks like a DVD. (Seen from above, it has two boundaries: concentric circles.) A voltage is applied across the film, from the inner boundary to the outer boundary; this voltage provides a convective forcing. Because of the annular geometry, the dynamics are periodic in the azimuthal direction and the only boundaries are those at which the convective forcing is applied. The liquid crystal is in smectic A phase, forming a nearly-perfect two-dimensional fluid because the film does not change thickness, even while flowing. Also, the inner electrode can be rotated and so the experiment can be used to study the interplay between a stabilizing force applied via the boundary (Couette shear) and convection. We present preliminary numerical simulations of special solutions such as convection cells, oscillatory convection cells, undulating convection cells, and localized vortex solutions

11h30
Fouling of viscoplastic fluids in uneven channels
Roustaei, Ali and Ian A. Frigaard
University of British Columbia

Due to the nature of viscoplastic fluids, fouling is a very common phenomenon in the flow of yield stress fluids in non-smooth/complex geometries. By fouling we mean that the fluid becomes stationary in layers attached to the wall of the duct. Although yield stress is necessary to have fouling in the flow, it is not sufficient and we need to have effect of geometry as well. For example the flow of a yield stress fluid along a uniform plane channel or circular pipe exhibits maximal shear stress at the wall with unyielded fluid is found only in the center of the duct. So neither effect alone is able to cause fouling. From another perspective, fouling is often associated with physicochemical changes in the fluid in the flowing fluid, e.g. dried deposits of milk solids [1,2], precipitation of asphaltene deposits [3,4]. However we only consider the combination of rheological and geometric causes of fouling. We take the simplest non-trivial case, of a Bingham fluid in Stokes flow along a channel with a sinusoidal wavy-wall, as studied in [5,6] for long-thin wavy-walled channels. The geometry is described by a wavelength and amplitude of the wall oscillation and by the channel width and the fluid is characterized by Bingham number (B). We carry out an extensive computational study of these flows over wide range of the three dimensionless parameters of the flow. Results suggest that fouling layers develop if the dimensionless amplitude of the wall perturbation exceeds some critical value. Fouling can occur over a range of channel aspect ratios, and progressively at larger Bingham number. At moderate B, when a sufficiently significant fouling has appeared the fluid appears to self-select the flowing region, i.e. the shape of the fouling layer. This can be partly understood as the selection of a new length-scale for the flow in the widest part of the channel. Following this line of reasoning we are able to establish the relevant similarity scalings to collapse our results, but understanding is not complete. We have also developed empirical expressions that give both necessary and sufficient conditions for fouling to occur.

#### References

[1] P.A. Cole, K. Asteriadou, P.T. Robbins, E.G. Owen, G.A. Montague, P.J. Fryer, Comparison of cleaning of toothpaste from surfaces and pilot scale pipework, Food and Bioproducts Processing 88 (2010) 392–400.

[2] G.K. Christian, P.J. Fryer, The effect of pulsing cleaning chemicals on the cleaning of whey protein deposits. Trans. IChemE C, 84 (2006) 320–328.

[3] R.A. Almehaideb, Asphaltene precipitation and deposition in the near wellbore region: a modeling approach. J. Petr. Sci. Eng. 42 (2004) 157-170.

[4] D. Eskin, J. Ratulowski, K. Akbarzadeh, S. Pan, Modelling asphaltene deposition in turbulent pipeline flows. Can. J. Chem. Eng. 89 (2011) 421-441.

[5] I. Frigaard, D. Ryan, Flow of a visco-plastic fluid in a channel of slowly varying width, J. non-Newt. Fluid Mech. 123 (2004) 67–83.

[6] A. Putz, I. Frigaard, D.M. Martinez, On the lubrication paradox and the use of regularisation methods for lubrication flows, J. non-Newt. Fluid Mech. 163 (2009) 62–77.

12h00
Absorbing boundary conditions for quantum relativistic equations
Lorin, Emmanuel
School of Mathematics and Statistics, Carleton University

Absorbing boundary conditions are derived for the Klein-Gordon and Dirac equations modeling quantum particles subject to a classical electromagnetic field in high frequency regime. The proposed approach is based on the approximation of pseudo-differential operators using techniques developed by Antoine, Barucq and Besse. Numerical schemes are derived and analyzed to illustrate the accuracy of the derived boundary conditions.

This is a joint work with X. Antoine (Univ. Lorraine) and J. Sater (Carleton Univ.).