Linear instability of asymmetric Poiseuille flows

We compute solutions for the Orr-Sommerfeld equations for the case of an asymmetric Poiseuille-like parallel flow. The calculations show that very small asymmetry has little effect on the prediction for linear instability of Poiseuille-like flow but that moderate asymmetry, such as found in channel flow near an elongated wall vortex, has a large effect and that instability can occur at much lower (less than 100) Reynolds numbers. We give some characterisation of the instability

Spectral method for the unsteady incompressible Navier-Stokes equations in gauge formulation

A spectral method which uses a gauge method, as opposed to a projection method, to decouple the computation of velocity and pressure in the unsteady incompressible Navier-Stokes equations, is presented. Gauge methods decompose velocity into the sum of an auxilary field and the gradient of a gauge variable, which may, in principle, be assigned arbitrary boundary conditions, thus overcoming the issue of artificial pressure boundary conditions in projection methods. A lid-driven cavity flow is used as a test problem. A subtraction method is used to reduce the pollution effect of singularities at the top corners of the cavity. A Chebyshev spectral collocation method is used to discretize spatially. An exponential time differencing method is used to discretize temporally. Matrix diagonalization procedures are used to compute solutions directly and efficiently. Numerical results for the flow at Reynolds number Re = 1000 are presented, and compared to benchmark results. It is shown that the method, called the spectral gauge method, is straightforward to implement, and yields accurate solutions if Neumann boundary conditions are imposed on the gauge variable, but suffers from reduced convergence rates if Dirichlet boundary conditions are imposed on the gauge variable

A hydro-elastic model of hydrocephalus

We combine elements of poroelasticity and of fluid mechanics to construct a mathematical model of the human brain and ventricular system. The model is used to study hydrocephalus, a pathological condition in which the normal flow of the cerebrospinal fluid is disturbed, causing the brain to become deformed. Our model extends recent work in this area by including flow through the aqueduct, by incorporating boundary conditions which we believe more accurately represent the anatomy of the brain and by including time dependence. This enables us to construct a quantitative model of the onset, development and treatment of this condition. We formulate and solve the governing equations and boundary conditions for this model and give results which are relevant to clinical observations

On the fractal characteristics of a stabilised Newton method

In this report, we present a complete theory for the fractal that is obtained when applying Newton's Method to find the roots of a complex cubic. We show that a modified Newton's Method improves convergence and does not yield a fractal, but basins of attraction with smooth borders. Extensions to higher-order polynomials and the numerical relevance of this fractal analysis are discussed

A new perspective on the stability of unsteady stream-function vorticity calculations

The stability of a numerical solution of the Navier-Stokes equations is usually approached by considering the stability of an advection-diffusion equation for either a velocity component, or in the case of two-dimensional flow, the vorticity. Stability restrictions for discretised advection-diffusion equations are a very serious constraint, particularly when a mesh is refined, so an accurate understanding of the stability of a numerical procedure is often of equal or greater importance than concerns with accuracy. The stream-function vorticity formulation provides two equations, one an advection-diffusion equation for vorticity and the other a Poisson equation between the vorticity and the stream-function. These two equations are usually not coupled in stability considerations, commonly only the stability of time marching of the advection diffusion equation is taken into account. In this work, we derive a global time-iteration matrix for the full system and show that this iteration matrix is far more complicated than that for just the advection-diffusion equation. We show how for a model system, the complete equations have much tighter stability constraints than would be predicted from the advection-diffusion equation alone

Finite Difference Approximation of a Convection Diffusion Equation Near a Boundary

Techniques for solving evolutionary convection diffusion equations are almost universally based on analysis of infinite domain situations. Almost all practical problems involve physical domains with boundaries. For a number of numerical schemes with Dirichlet boundary conditions, the numerical algorithm can be used without alteration near a boundary. The development of higher order methods such as Quickest or second order upwinding (including many schemes with flux limiters) can introduce difficulty near an inflow boundary, since for points adjacent to the boundary there are insufficient upstream points for the high order scheme to be applied without alteration. Usually reliance is placed on ad-hoc solutions for individual problems. Recently Morton & Sobey (1993) showed how analytic evolutionary solutions could be used to derive arbitrary accuracy finite difference and finite element schemes for constant coefficient convection diffusion. In this paper we continue that work by considering analytic solutions for evolution on the half line $x \leq 0$. We use an exact evolutionary operator to derive finite difference approximation schemes which maintain accuracy near a boundary. These schemes are applied and compared for a simple test problem which has an exact solution. As might be expected, it is not just accuracy but also stability, which dominates solution of time evolution problems.\ud \ud The work reported here forms part of the research programme of the Oxford-Reading Institute for Computational Fluid Dynamics

Fractal Characteristics of Newton's Method on Polynomials

In this report, we present a simple geometric generation principle for the fractal that is obtained when applying Newton's method to find the roots of a general complex polynomial with real coefficients. For the case of symmetric polynomials $z^{\nu}-1$ , the generation mechanism is derived from first principles. We discuss the case of a general cubic and are able to give a description of the arising fractal structure depending on the coefficients of the cubic. Special cases are analysed and their characteristics, including scale factors and an approximate fractal dimension, are derived. The theoretical results are confirmed via computational experiments. An application of the theory in turbulence modelling is presented

The brightness and spatial distributions of terrestrial radio sources

Faint undetected sources of radio-frequency interference (RFI) might become visible in long radio observations when they are consistently present over time. Thereby, they might obstruct the detection of the weak astronomical signals of interest. This iss

Effect of boundary vorticity discretisation on explicit stream-function vorticity calculations

The numerical solution of the time dependent Navier-Stokes equations in terms of the vorticity and a stream function is a well tested process to describe two-dimensional incompressible flows, both for fluid mixing applications and for studies in theoretical fluid mechanics. In this paper, we consider the interaction between the unsteady advection-diffusion equation for the vorticity, the Poisson equation linking vorticity and stream function and the approximation of the boundary vorticity, examining from a practical viewpoint, global error and iteration stability. Our results show that most schemes have very similar global stability constraints although there may be small stability gains from the choice of method to determine boundary vorticityCentro de Matemática da Universidade de Coimbr