An Analysis of Higher Order Boundary Conditions for the Wave Equation.

Thanks to the use of the Cagniard-De Hoop method, we derive an analytic solution in the time domain for the half-space problem associated with the wave equation with Engquist-Majda higher order boundary conditions. This permits us to derive new convergence results when the order of the boundary condition tends to infinity, as well as error estimates. The theory is illustrated by numerical results

Studies In Small Scale Thermal Convection

The effect of non-Fourier heat transfer and partial-slip boundary conditions in Rayleigh-Bénard are analyzed theoretically. Non-Fourier fluids possess a relaxation time that reflects the delay in the response of the heat flux to a change in the temperature gradient while the partial slip boundary condition assumes that the fluid velocity and temperature are not equal to that of the wall. Both non-Fourier and partial-slip effects become important when small-scale heat transfer applications are investigated such as convection around micro- and nano-devices as suggested by the extended heat transport equations from kinetic theory. Other applications are also known to exhibit one or both of these effects such as low-temperature liquids, nanofluids, granular flows, rarefied gases and polymer flows. Small scale effects are measured by the Knudsen number. From this, non-Fourier effects can be estimated, measured non-dimensionally by the Cattaneo number and modelled using the frame indifferent Cattaneo-Vernotte equation which is derived from Oldroyd’s upper-convected derivative. Linear stability of non-Fourier fluids shows that the neutral stability curve possesses a stationary Fourier branch and an oscillatory branch intersecting at some wave number, where for small relaxation time, no change in stability is expected from that of a Fourier fluid. As the relaxation time increases to a critical value, both stationary and oscillatory convection become equally probable. Past this value, oscillatory instability is expected to occur at a smaller Rayleigh number and larger wave number than for a Fourier fluid. Non-linear analysis of weakly non-Fourier fluids shows that near the onset of convection, the convective roll intensity is stronger than for a Fourier fluid. The bifurcation to convection changes from subcritical to supercritical as the Cattaneo number increases and the instability of the convection state for a non-Fourier fluid is shown to occur via a Hopf bifurcation at lower Rayleigh number and higher Nusselt number than for a Fourier fluid. When hydrodynamic slip and temperature jump boundary conditions are considered, a significant reduction in the minimum critical Rayleigh number and corresponding wave number are found. Depending on the sign used for second-order coefficients, critical conditions can be greater than or less than that for first-order boundary conditions

Comparing macroscopic continuum models for rarefied gas dynamics : a new test method

We propose a new test method for investigating which macroscopic continuum models, among the many existing models, give the best description of rarefied gas flows over a range of Knudsen numbers. The merits of our method are: no boundary conditions for the continuum models are needed, no coupled governing equations are solved, while the Knudsen layer is still considered. This distinguishes our proposed test method from other existing techniques (such as stability analysis in time and space, computations of sound speed and dispersion, and the shock wave structure problem). Our method relies on accurate, essentially noise-free, solutions of the basic microscopic kinetic equation, e.g. the Boltzmann equation or a kinetic model equation; in this paper, the BGK model and the ES-BGK model equations are considered. Our method is applied to test whether one-dimensional stationary Couette flow is accurately described by the following macroscopic transport models: the Navier-Stokes-Fourier equations, Burnett equations, Grad's 13 moment equations, and the regularized 13 moment equations (two types: the original, and that based on an order of magnitude approach). The gas molecular model is Maxwellian. For Knudsen numbers in the transition-continuum regime (Kn less-than-or-equals, slant 0.1), we find that the two types of regularized 13 moment equations give similar results to each other, which are better than Grad's original 13 moment equations, which, in turn, give better results than the Burnett equations. The Navier-Stokes-Fourier equations give the worst results. This is as expected, considering the presumed accuracy of these models. For cases of higher Knudsen numbers, i.e. Kn > 0.1, all macroscopic continuum equations tested fail to describe the flows accurately. We also show that the above conclusions from our tests are general, and independent of the kinetic model used

Stationary and travelling crossflow vortices in three-dimensional boundary layers: nonlinear interactions within a common critical layer

Three-dimensional boundary layers, such as that over a swept wing, are known to exhibit a crossflow instability, which manifests itself in the form of stationary vortices and travelling-wave vortices. Despite their smaller linear growth rates, the former tend to be the dominant cause of transition to turbulence when free-stream turbulence is relatively low. Travelling-wave vortices, which have higher growth rates, dominate at elevated levels of free-stream turbulence. Recent experiments found that the development of stationary vortices may be affected by free-stream disturbance levels representative of flight conditions, suggesting that travelling-wave vortices may play a role. It was also observed that travelling-wave vortices may be affected by stationary modes. Prompted by these observations, we carry out a theoretical study of nonlinear mutual interactions between stationary and travelling-wave vortices. In order to fix the idea, the base flow is taken to be a Falkner-Skan-Cooke boundary layer. The eigenvalue problem, consisting of the Rayleigh equation and homogeneous boundary conditions, is solved. It was found that there exists a pair of stationary and travelling vortices that share a common critical level, where the base flow velocity projected to the directions of the wave vectors is equal to the phase speeds of the vortices. This pair is particularly significant because effective nonlinear interactions take place in the critical layer, a thin region surrounding the common critical level. The mutual and self nonlinear interactions are analysed by employing the nonlinear non-equilibrium critical-layer approach. The governing equations and solutions for the disturbances are expanded asymptotically both in the main boundary layer and inside the critical layer. The analysis of the outer expansion determines the eigenmodes at leading-order, and at the next order leads to solvability conditions involving the velocity jumps across the critical layer. The analysis of the inner expansions, in particular of the inter-modal interactions, provides jumps, which are combined with the solvability conditions to obtain the integro-differential amplitude equations. For the Falkner-Skan-Cooke baseflow under consideration, the stationary and travelling vortices interact at the quadratic level to force the sum mode. The latter interacts with the stationary mode to regenerate the travelling vortices. The resulting nonlinear effects cause the travelling vortices to amplify rapidly in the form of a super-exponential growth. It is also noted that alternative forms of interactions, involving the difference mode, may be possible in general, and for other base flows, travelling vortices may cause stationary vortices to amplify super-exponentially instead. The analysis is then extended to include the nonlinear self-interactions of the stationary and travelling vortices, and the resulting integro-differential amplitude equations contain derivatives of the amplitudes within the integrals. The amplitude equations are solved numerically. We also investigated the interactions of travelling vortices with the distortion induced by distributed surface roughness, which is modelled by a wavy wall with its height being constant or spatially modulated. Of importance is the distortion sharing the same critical level of the travelling vortices. The form of the interaction is similar to that found for the stationary and travelling eigenmodes, however the presence of a viscous wall layer and an associated `blowing velocity' alters the interaction. A more realistic case, roughness consisting of a continuum of wavenumbers, which has previously not been considered, is also investigated. The contribution of the interaction is found to be a superposition of the Fourier modes. These amplitude equations are also solved numerically, and the solutions indicate that roughness elements of moderate height can have significant effects on the instability.Open Acces

Modelling Fluid Structure Interaction problems using Boundary Element Method

This dissertation investigates the application of Boundary Element Methods (BEM) to Fluid Structure Interaction (FSI) problems under three main different perspectives. This work is divided in three main parts: i) the derivation of BEM for the Laplace equation and its application to analyze ship-wave interaction problems, ii) the imple- mentation of efficient and parallel BEM solvers addressing the newest challenges of High Performance Computing, iii) the developing of a BEM for the Stokes system and its application to study micro-swimmers.First we develop a BEM for the Laplace equation and we apply it to predict ship-wave interactions making use of an innovative coupling with Finite Element Method stabilization techniques. As well known, the wave pattern around a body depends on the Froude number associated to the flow. Thus, we throughly investigate the robustness and accuracy of the developed methodology assessing the solution dependence on such parameter. To improve the performance and tackle problems with higher number of unknowns, the BEM developed for the Laplace equation is parallelized using OpenSOURCE tech- nique in a hybrid distributed-shared memory environment. We perform several tests to demonstrate both the accuracy and the performance of the parallel BEM developed. In addition, we explore two different possibilities to reduce the overall computational cost from O(N2) to O(N). Firstly we couple the library with a Fast Multiple Method that allows us to reach for higher order of complexity and efficiency. Then we perform a preliminary study on the implementation of a parallel Non Uniform Fast Fourier Transform to be coupled with the newly developed algorithm Sparse Cardinal Sine De- composition (SCSD).Finally we consider the application of the BEM framework to a different kind of FSI problem represented by the Stokes flow of a liquid medium surrounding swimming micro-organisms. We maintain the parallel structure derived for the Laplace equation even in the Stokes setting. Our implementation is able to simulate both prokaryotic and eukaryotic organisms, matching literature and experimental benchmarks. We finally present a deep analysis of the importance of hydrodynamic interactions between the different parts of micro-swimmers in the prevision of optimal swimming conditions, focusing our attention on the study of flagellated \u201crobotic\u201d composite swimmers

Wave radiation in simple geophysical models

Wave radiation is an important process in many geophysical flows. In particular, it is by wave radiation that flows may adjust to a state for which the dynamics is slow. Such a state is described as “balanced”, meaning there is an approximate balance between the Coriolis force and horizontal pressure gradients, and between buoyancy and vertical pressure gradients. In this thesis, wave radiation processes relevant to these enormously complex flows are studied through the use of some highly simplified models, and a parallel aim is to develop accurate numerical techniques for doing so. This thesis is divided into three main parts. 1. We consider accurate numerical boundary conditions for various equations which support wave radiation to infinity. Particular attention is given to discretely non-reflecting boundary conditions, which are derived directly from a discretised scheme. Such a boundary condition is studied in the case of the 1-d Klein-Gordon equation. The limitations concerning the practical implementation of this scheme are explored and some possible improvements are suggested. A stability analysis is developed which yields a simple stability criterion that is useful when tuning the boundary condition. The practical use of higher-order boundary conditions for the 2-d shallow water equations is also explored; the accuracy of such a method is assessed when combined with a particular interior scheme, and an analysis based on matrix pseudospectra reveals something of the stability of such a method. 2. Large-scale atmospheric and oceanic flows are examples of systems with a wide timescale separation, determined by a small parameter. In addition they both undergo constant random forcing. The five component Lorenz-Krishnamurthy system is a system with a timescale separation controlled by a small parameter, and we employ it as a model of the forced ocean by further adding a random forcing of the slow variables, and introduce wave radiation to infinity by the addition of a dispersive PDE. The dynamics are reduced by deriving balance relations, and numerical experiments are used to assess the effects of energy radiation by fast waves. 3. We study quasimodes, which demonstrate the existence of associated Landau poles of a system. In this thesis, we consider a simple model of wave radiation that exhibits quasimodes, that allows us to derive some explicit analytical results, as opposed to physically realistic geophysical fluid systems for which such results are often unavailable, necessitating recourse to numerical techniques. The growth rates obtained for this system, which is an extension of one considered by Lamb, are confirmed using numerical experiments

Four simplified gradient elasticity models for the simulation of dispersive wave propagation

Gradient elasticity theories can be used to simulate dispersive wave propagation as it occurs in heterogeneous materials. Compared to the second-order partial differential equations of classical elasticity, in its most general format gradient elasticity also contains fourth-order spatial, temporal as well as mixed spatial temporal derivatives. The inclusion of the various higher-order terms has been motivated through arguments of causality and asymptotic accuracy, but for numerical implementations it is also important that standard discretization tools can be used for the interpolation in space and the integration in time. In this paper, we will formulate four different simplifications of the general gradient elasticity theory. We will study the dispersive properties of the models, their causality according to Einstein and their behavior in simple initial/boundary value problems