Postal address: |
Mr Duncan Sutherland School of Mathematics and Statistics F07 University of Sydney NSW 2006 Australia |
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Office: | Room 807 Carslaw Building |
Email: | duncans@maths.usyd.edu.au |
FAX: | +61 2 9351 4534 |
The problem of a dipole rebounding from a rigid wall in a viscous fluid has been very well studied using a variety of numerical techniques. Recently, Romain Nguyen van yen, Kai Schneider and Marie Farge (Phys. Rev. Lett. 106, 184502 (2011)) used a volume penalisation method to investigate the energy dissipation over the rebound as the viscosity approaches zero. The penalisation method approximates a no-slip boundary condition and intrinsically introduces some non-zero slip length at the boundary, which also vanishes as the viscosity approaches zero. The results of Nguyen van yen et. al. surprisingly indicate that energy dissipating structures persist in the vanishing viscosity limit. We consider the problem of a dipole incident on a rigid wall with a Navier slip boundary condition, which reduces to the standard no-slip boundary condition in the case of zero slip length. We find no energy dissipating structures for any fixed slip length, but we recover the results of Nguyen van yen et. al. in the case where the slip length is proportional to the viscosity. Hence it appears that the observation of Nguyen van yen et. al. is an artifact of their numerical method. We then proceed to study the vorticity generation at the rigid wall in more detail. In any bounded domain the walls act as a source of enstrophy which constantly injects small scale vortices into the flow. To do this we study the number and location of the critical points, either minima, maxima, or saddles of the streamfunction and vorticity. We will discuss the techniques for identifying and classifying the fixed points, as well as some of the difficulties of interpretation that the boundaries present. Results showing the motion of critical points in time and the variation in the number of critical points over the simulation for bounded geometries will also be presented.
In an unbounded domain, the well-known forward energy cascade forces energy to large spatial scales. Large numbers of small scale vortices will therefore merge into domain-filling structures. In bounded domains, the walls act as a source of enstrophy which constantly injects small scale vortices that disturb the formation of organised, domain-filling circulation cells. Fayeza Al Sulti and Koji Ohkitani, ``Vortex merger and topological changes in two-dimensional turbulence'', \emph{Physical Review E}, $86.1\,016309,\, 2012$ studied vortex merger in unbounded domains by counting the number of elliptic and hyperbolic critical points of vorticity and streamfunction. To identify and classify the critical points Al Sulti and Ohkitani identified points wher $\nabla f=0$ and classified them using the eigenvalues of the Hessian matrix, but identifying the zeros is difficult numerically. We develop a more reliable and efficient method using a simple nearest neighbour comparison method for finding minima $(x_i,y_i)$ of $\vert\nabla f\vert^2$ and then compute the Poincar{\'e}-Hopf index of $(x_i,y_i)$ of the vector field $\bm (f_x,f_y)$ to classify it as a elliptic, hyperbolic, or regular point of the scalar field. Results showing the motion of critical points in time and the variation in the number of critical points over the simulation for bounded geometries will be presented.
We consider the problem of a viscous dipole colliding with one wall of a periodic channel. The collision results in a layer of vorticity generated near the boundary. This layer rolls up into two monopoles, one either side of the original dipole, which then detach and rebound from the wall along with the original dipole. To treat this we use a Fourier spectral method in the periodic direction, a compact finite difference scheme in the streamwise direction, and an influence matrix method to enforce the no-slip boundary conditions exactly. Romain Nguyen van yen, Kai Schneider and Marie Farge (Phys. Rev. Lett. 106, 184502 (2011)) have recently studied a similar problem, using a penalisation technique, which treats the boundary conditions approximately. They find energy dissipating structures near the wall, which persist as the viscosity tends to zero. We are currently testing their conclusions across a range of Reynolds numbers.
A difficulty with the streamfunction-vorticity formulation for viscous flows is that there are no boundary conditions specified for the vorticity. This can be overcome using a linear correction technique known as the influence matrix method. We consider the influence matrix method for a channel geometry, periodic in the streamwise direction, with no-slip boundary conditions in the spanwise direction. A linear multistep method is used for the time integration. For the spatial integration a standard Fourier method is used in the periodic direction. The spanwise direction is treated using either a Chebyshev spectral approach or a fourth order compact finite difference scheme. For given spatial resolution, the compact difference scheme is faster and more stable while the Chebyshev approach is more accurate. Both methods become increasingly more costly as the Reynolds number is increased. The aim is to incorporate the ideas described here into a hybrid viscous contour advection algorithm.
I will present a review of the use of Chebyshev methods for the solution of the stream-function vorticity formulation of flow in a channel, periodic in the streamwise direction and with Dirichlet boundary conditions on the channel wall. The vorticity equation is handled with standard pseudo-spectral techniques, while Poisson's equation for the streamfunction is solved by transforming in both x and y and deriving a recursion relation for the Chebyshev coefficients in y. The recursion relation can be expressed as a quasipentadiagonal matrix equation, which may be solved in O(N) operations. Example flows used to test convergence and accuracy of the scheme will be discussed. Preliminary results for the study of geophysical turbulence and the formation of jets on the beta-plane will be discussed.
Traditional medical ultrasonography has used the backscattering of acoustic signals from tissue to investigate an object of interest. Microbubble contrast agents, which are typically administered intravenously into a patient's circulatory system, were first approved for clinical studies in 1991. Modern contrast agents are typically a gas filled microbubble, which may be preformed with a shell of, for example, lipid, polymer or albumin. Contrast agents enhance backscatter, due to the high compressibility of the gas, relative to body tissue. The oscillatory response of the microbubble to an applied ultrasound field is complicated and frequency dependent. The bubble has a size dependent linear resonance frequency. Typically the contrast agent size is 1-7 um with resonant frequencies in the 2-15 MHz range, which is used for clinical applications. As the amplitude of the driving field is increased, the bubble response is nonlinear, which leads to sub and higher harmonic generation. New techniques have been developed for use in specialised applications to utilise this phenomenon. I will be discussing the some of the models that have been developed to describe the response of a single bubble to a driving pressure. This requires consideration of the surrounding fluid, internal gas and the shell layer of the microbubble. The discussion will focus on a second order ODE, called the Rayleigh-Plesset (RP) equation. The RP equation may be extended to include effects due to fluid viscosity and surface tension. Similarly, it is possible to include damping of the bubble oscillation due to radiation at the bubble wall. For contrast agents with encapsulating shells the equation of motion, due to Church, of the bubble wall is, again, an extension of the RP equation. Some simulation results will be presented, and limitations of the models will be discussed.
The speaker has found a unique way to cram almost everything he finds interesting (fluid mechanics, cycling, hair and free food) into one conveniently sized SUMS talk. In 1989 Laurent Fignon lost the Tour de France to Greg LeMond, by 8 seconds. Anecdotal evidence suggests it was calculated that if Fignon had a haircut, he would have lowered his aerodynamic drag enough that he would have won. This talk will introduce basic fluid dynamics, and discuss how computational fluid dynamics is used to simulate the aerodynamics of cyclists and bicycles. The talk is aimed at being fairly accessible to a general audience.
A sequential hermaphrodite fish, such as Wrasse or Clownfish, is born as one sex and later under goes transition to the opposite sex. This is kinky. This talk was originally presented as assessment for the applied maths honours course 'Populations and Disease'. The assignment was to find three papers with a common theme in the area of mathematical biology and present a short talk about them. A continuously structured population model, due to Calsina and Ripoll, will be discussed. The model assumes a random age transition between the sexes. The fecundity (birth) and mortality functions are arbitrary and only restricted by obvious biological considerations. A stage structured population model including the effects of fishing, by Armsworth, will also be discussed. Time permitting, the talk will include discussion of a third paper about optimal breeding stragies for sequential hermaphrodites. The speaker is also available for private parties.
Duncan Sutherland is perpetually irritated by annoying pedestrians getting in his way. To understand the mathematics behind modelling pedestrian dynamics, he consulted google and found the following paper: Burstedde, C. et al. 'Simulation of pedestrian dynamics using a two-dimensional cellular automaton', Physica A, (2001) 507-525, which he decided to discuss in a SUMS talk. This talk will introduce the basic concepts of cellular automata, and will discuss the model of Burstedde et al. A pedestrian is modelled as a particle which makes a transition between cells with some probability p_ij. The most interesting feature of Burstedde's model is the concept of a floor field. The floor field modifies the probabilty p_ij. A static floor field is used to model topological features such as walls and doors, and also features which are attractive or repulsive to pedestrians, such as emergency exits or fires. A dynamic floor field changes with the pedestrian flow and is used to model long range interactions.
In this talk I will discuss solitons, a solitary wave which propagates with unchanged wave profile, even after interaction with other solitons. A modon is a localised vortex dipole, which display similar interaction behaviour, to solitons. The shape of the modon, however, does not completely remain unchanged by the interaction. I will present movies of numerical solutions showing the behaviour of soliton-soliton and modon-modon interactions and a simple overview of the methods used to obtain them.
Modern contrast agents are a gas filled microbubble, which may be preformed with a shell of, for example, lipid, polymer or albumin. The oscillatory response of a microbubble to an applied ultrasound field is complicated and frequency dependent. In this project models are derived for the radius of the bubble as a function of time in response to a time dependent driving pressure. The most fundamental of the nonlinear models is a second order, ordinary differential equation, called the Rayleigh-Plesset (RP) equation. This applies to a bubble in a simplified liquid, assumed infinite in domain, inviscid, incompressible (uniform density) and with zero surface tension. For small changes in radius the RP equation reduces to a linear oscillator equation. The RP equation may be modified by allowing non-zero surface tension and viscosity, resulting in an equation referred to as the RPNNP equation. This is the first step to constructing a shelled bubble model. In the RP equation the fluid is assumed incompressible, and this implies an infinite sound speed. This limits the accuracy of the RP equation to moderate amplitude oscillations. The RP model was extended by Herring and Trilling, by allowing for a constant sound speed, assuming the fluid disturbance was a diverging spherical wave. Several models exist for the encapsulating shell. This thesis reviews the models derived by Church. The encapsulating shell is modelled as a Newtonian fluid, or as a viscoelastic material with a constant shear modulus and viscosity. The equation of motion obtained is an extension of the RP equation. The models including radiation damping and shell layers, all have a persistent form, which is the same as the fundamental Rayleigh-Plesset equation.