Large-wavelength instabilities in free-surface Hartmann flow at low magnetic Prandtl numbers

We study the linear stability of the flow of a viscous electrically conducting capillary fluid on a planar fixed plate in the presence of gravity and a uniform magnetic field. We first confirm that the Squire transformation for MHD is compatible with the stress and insulating boundary conditions at the free surface, but argue that unless the flow is driven at fixed Galilei and capillary numbers, the critical mode is not necessarily two-dimensional. We then investigate numerically how a flow-normal magnetic field, and the associated Hartmann steady state, affect the soft and hard instability modes of free surface flow, working in the low magnetic Prandtl number regime of laboratory fluids. Because it is a critical layer instability, the hard mode is found to exhibit similar behaviour to the even unstable mode in channel Hartmann flow, in terms of both the weak influence of Pm on its neutral stability curve, and the dependence of its critical Reynolds number Re_c on the Hartmann number Ha. In contrast, the structure of the soft mode's growth rate contours in the (Re, alpha) plane, where alpha is the wavenumber, differs markedly between problems with small, but nonzero, Pm, and their counterparts in the inductionless limit. As derived from large wavelength approximations, and confirmed numerically, the soft mode's critical Reynolds number grows exponentially with Ha in inductionless problems. However, when Pm is nonzero the Lorentz force originating from the steady state current leads to a modification of Re_c(Ha) to either a sublinearly increasing, or decreasing function of Ha, respectively for problems with insulating and conducting walls. In the former, we also observe pairs of Alfven waves, the upstream propagating wave undergoing an instability at large Alfven numbers.Comment: 58 pages, 16 figure

Interplay among unstable modes in films over permeable walls

The stability of open-channel flows (or film flows) has been extensively investigated for the case of impermeable smooth walls. In contrast, despite its relevance in many geophysical and industrial flows, the case that considers a permeable rather than an impermeable wall is almost unexplored. In the present work, a linear stability analysis of a film falling over a permeable and inclined wall is developed and discussed. The focus is on the mutual interaction between three modes of instability, namely, the well-known free-surface and hydrodynamic (i.e. shear) modes, which are commonly observed in open-channel flows over impermeable walls, plus a new one associated with the flow within the permeable wall (i.e. the porous mode). The flow in this porous region is modelled by the volume-averaged Navier-Stokes equations and, at the wall interface, the surface and subsurface flow are coupled through a stress-jump condition, which allows one to obtain a continuous velocity profile throughout the whole flow domain. The generalized eigenvalue problem is then solved via a novel spectral Galerkin method, and the whole spectrum of eigenvalues is presented and physically interpreted. The results show that, in order to perform an analysis with a full coupling between surface and subsurface flow, the convective terms in the volume-averaged equations have to be retained. In previous studies, this aspect has never been considered. For each kind of instability, the critical Reynolds number (${\mathit{Re}}_{c}$) is reported for a wide range of bed slopes ($\theta$) and permeabilities ($\sigma$). The results show that the free-surface mode follows the behaviour that was theoretically predicted by Benjamin and Yih for impermeable walls and is independent of wall permeability. In contrast, the shear mode shows a high dependence on $\sigma$: at $\sigma = 0$ the behaviour of ${\mathit{Re}}_{c} (\theta )$ recovers the well-known non-monotonic behaviour of the impermeable-wall case, with a minimum at \theta \sim 0. 05\textdegree . However, with an increase in wall permeability, ${\mathit{Re}}_{c}$ gradually decreases and eventually recovers a monotonic decreasing behaviour. At high values of $\sigma$, the porous mode of instability also occurs. A physical interpretation of the results is presented on the basis of the interplay between the free-surface-induced perturbation of pressure, the increment of straining due to shear with the increase in slope, and the shear stress condition at the free surface. Finally, the paper investigates the extent to which Squire's theorem is applicable to the problem presented herei

Stability of two-dimensional forced Navier-Stokes flow on a bounded circular domain

This research is concerned with the stability of a two-dimensional, electromagnetically forced, zonal flow on a circular domain. Flows like these are found in nature (e.g. shear flow in the atmosphere, Jovian disk) and experiment (e.g. plasma flow in a Fusion reactor) and a requirement for experiments is often that these types of flows remain stable and axi-symmetric. A numerical method is developed based on a spectral expansion into an infinite system of ordinary differential equations for velocity functions resulting from a Stokes eigenvalue problem. The system is truncated to gain a finite-dimensional system which is useful for computations of both equilibrium flows and strongly disturbed flows. Numerical results are compared to both finite difference method results and analytical results for the equilibrium basic flow. Both linear and nonlinear stability are explored for the Navier-Stokes equations on the circular domain and for the system of ordinary differential equations. Differences in stability and the evolution of perturbations are explained on the basis of discrepancies between infinite-dimensional partial differential equations like the Navier-Stokes equations and a finite-dimensional system of ordinary differential equations resulting from a Galerkin truncation. On the basis of both stability analyses a control system is developed which stabilizes the system of ordinary differential equations to stay in a desired equilibrium. It is argued that this control system is also usable for the control of the Navier-Stokes equations. This research is concerned with the stability of a two-dimensional, electromagnetically forced, zonal flow on a circular domain. Flows like these are found in nature (e.g. shear flow in the atmosphere, Jovian disk) and experiment (e.g. plasma flow in a Fusion reactor) and a requirement for experiments is often that these types of flows remain stable and axi-symmetric. A numerical method is developed based on a spectral expansion into an infinite system of ordinary differential equations for velocity functions resulting from a Stokes eigenvalue problem. The system is truncated to gain a finite-dimensional system which is useful for computations of both equilibrium flows and strongly disturbed flows. Numerical results are compared to both finite difference method results and analytical results for the equilibrium basic flow. Both linear and nonlinear stability are explored for the Navier-Stokes equations on the circular domain and for the system of ordinary differential equations. Differences in stability and the evolution of perturbations are explained on the basis of discrepancies between infinite-dimensional partial differential equations like the Navier-Stokes equations and a finite-dimensional system of ordinary differential equations resulting from a Galerkin truncation. On the basis of both stability analyses a control system is developed which stabilizes the system of ordinary differential equations to stay in a desired equilibrium. It is argued that this control system is also usable for the control of the Navier-Stokes equations

Electro-Magnetic Ship Propulsion Stability under Gusts

The purpose of this study is to analytically investigate the effect of Stuart number as well as magnetic and electrical angular frequency on the velocity distribution in a magneto-hydro-dynamic pump. Results show that as Stuart number approaches zero the velocity profile becomes similar to that of fully developed flow in a pipe. Furthermore, for high Stuart number there is a frequency limit for stability of fluid flow in certain direction of flow. This stability frequency is depending on geometric parameters of channel. Furthermore stability frequency of electro-magnetic field is independent of gusts frequency and fluid thermo-physical properties

Internal waves in fluid flows. Possible coexistence with turbulence

Waves in fluid flows represents the underlying theme of this research work. Wave interactions in fluid flows are part of multidisciplinary physics. It is known that many ideas and phenomena recur in such apparently diverse fields, as solar physics, meteorology, oceanography, aeronautical and hydraulic engineering, optics, and population dynamics. In extreme synthesis, waves in fluids include, on the one hand, surface and internal waves, their evolution, interaction and associated wave-driven mean flows; on the other hand, phenomena related to nonlinear hydrodynamic stability and, in particular, those leading to the onset of turbulence. Close similarities and key differences exist between these two classes of phenomena. In the hope to get hints on aspects of a potential overall vision, this study considers two different systems located at the opposite limits of the range of existing physical fluid flow situations: first, sheared parallel continuum flows - perfect incompressibility and charge neutrality - second, the solar wind - extreme rarefaction and electrical conductivity. Therefore, the activity carried out during the doctoral period consists of two parts. The first is focused on the propagation properties of small internal waves in parallel flows. This work was partly carried out in the framework of a MISTI-Seeds MITOR project proposed by Prof. D. Tordella (PoliTo) and Prof. G. Staffilani (MIT) on the long term interaction in fluid flows. The second part regards the analysis of solar-wind fluctuations from in situ measurements by the Voyagers spacecrafts at the edge of the heliosphere. This work was supported by a second MISTI-Seeds MITOR project, proposed by D. Tordella (PoliTo), J. D. Richardson (MIT, Kavli Institute), with the collaboration of M. Opher (BU)

Bifurcation contrasts between plane Poiseuille flow and plane Magnetohydrodynamic flow

The stability characteristics of an incompressible viscous pressure-driven flow of an electrically conducting fluid between two parallel boundaries in the presence of a transverse magnetic field are compared and contrasted with those of Plane Poiseuille flow (PPF). Assuming that the outer regions adjacent to the fluid layer are perfectly electrically insulating, the appropriate boundary conditions are applied. The eigenvalue problems are then solved numerically to obtain the critical Reynolds number Rec and the critical wave number ac in the limit of small Hartmann number (M) range to produce the curves of marginal stability. The non-linear two-dimensional travelling waves that bifurcate by way of a Hopf bifurcation from the neutral curves are approximated by a truncated Fourier series in the streamwise direction. Two and three dimensional secondary disturbances are applied to both the constant pressure and constant flux equilibrium solutions using Floquet theory as this is believed to be the generic mechanism of instability in shear flows. The change in shape of the undisturbed velocity profile caused by the magnetic field is found to be the dominant factor. Consequently the critical Reynolds number is found to increase rapidly with increasing M so the transverse magnetic field has a powerful stabilising effect on this type of flow

Exact coherent structures in the transitional regime of shear and centrifugal flows

Tesi en modalitat de compendi de pubicacionsTurbulence is one of the major concerns for most technological problems involving fluid motion. Specially in aeronautics, a turbulent boundary layer results in structural stresses, vibrations and higher aircraft drag, leading to a significant increase in fuel consumption. Therefore, trying to comprehend the origin of turbulence by studying its most common transition routes is a crucial first step towards its effective control. Transition to turbulence of an homogeneous flow is frequently mediated by transient visits to highly nonlinear laminar coherent structures that usually are at the threshold between laminarity and turbulence. From a dynamical systems point of view, these structures are invariant sets in the infinite-dimensional Navier-Stokes phase space that here we aim to identify in different canonical flows. With the use of direct numerical simulations together with Newton-Krylov solvers and Arnoldi iteration method for the linear stability analysis, invariant sets such as equilibria, relative equilibria or periodic orbits are accurately computed and tracked along the parameter space to understand the transition mechanisms. From a mathematical perspective, dynamical systems and bifurcation theory provide the suitable framework to understand the hydrodynamic instabilities and transition to turbulence from a deterministic point of view. In addition, the use of spectral methods for the spatial discretisation is particularly convenient due to the exponential convergence of the numerical solutions. In the first work, the onset of transition in two-dimensional Plane Poiseuille flow is analysed. A new family of Tollmien-Schlichting waves, breaking the usual half-shift and reflect symmetry of the classical ones, has been identified and tracked across parameter space. In addition, the role of another classical travelling wave family that did not participate in the localisation mechanisms has been clarified. We continue by analysing the nonlinear mode competition in purely hydrodynamic and also magnetised Taylor-Couette flow. Finite-amplitude mixed-mode solution branches, arising from both purely hydrodynamic and magneto-rotational instabilities, are identified. These nonlinear mode interactions are efficiently computed in suitable skewed computational domains instead of the classical orthogonal ones, allowing for a significant reduction of the required computational resources. Finally, the generalisation of extensional flows between biorthogonally stretching-shrinking parallel plates is analysed. Under the assumption of the self-similar ansatz, three-dimensional steady equilibria of the Navier-Stokes equations are identified and systematically tracked in parameter space, to cover all possible configurations of the acceleration rates and thus unfold all occurring bifurcations. After the explorations, up to seven different families of steady solutions have been identified, some of them related in pairs with symmetries. When increasing wall acceleration rates, the solution branches interact by means of fold and codimension-2 cusp bifurcations, increasing the complexity of the topology of equilibria. Besides the specific interest attached to each one of the three problems we have addressed, these have further served as a proof-of-concept for the applicability and suitability of the methods and tools developed in the course of this thesis, which may assist in tackling a vast range of problems across a huge variety of physics and engineering disciplines.La turbulència és una de les principals preocupacions per a la majoria de problemes tecnològics relacionats amb el moviment de fluids. Especialment en el cas de l'aeronàutica, una capa límit turbulenta produeix tensions estructurals, vibracions i una major força d'arrossegament de l'aeronau que resulten en un increment significatiu del consum de combustible. Per tant, intentar comprendre l'origen de la turbulència, tot estudiant-ne les rutes de transició més habituals, és un primer pas indispensable cap al seu control efectiu. La transició a la turbulència d'un flux homogeni sovint es caracteritza per visites transitòries a estructures coherents, laminars i altament no-lineals, que acostumen a trobar-se al llindar entre la laminaritat i la turbulència. Des del punt de vista dels sistemes dinàmics, aquestes estructures són conjunts invariants en l'espai de fase infinit-dimensional de les equacions de Navier-Stokes, que aquí es pretén identificar en diferents fluxos canònics. Mitjançant la integració temporal de les equacions, resolutors de Newton-Krylov i el mètode iteratiu d'Arnoldi per a l'anàlisi d'estabilitat lineal, els diferents conjunts invariants siguin equilibris, equilibris relatius o òrbites periòdiques són acuradament calculats i continuats al llarg de l'espai de paràmetres per tal d'entendre els mecanismes involucrats en la transició. Des d'una perspectiva matemàtica, els sistemes dinàmics i la teoria de bifurcacions proporcionen el marc adequat per a comprendre les inestabilitats hidrodinàmiques i la transició a la turbulència des d'un punt de vista determinista. A més, l'ús de mètodes espectrals per a la discretització espaial resulta particularment convenient degut a la convergència exponencial de les solucions numèriques. En el primer treball, s'analitza l'inici de la transició del flux bidimensional de Poiseuille pla. En aquest cas, una nova família d'ones de Tollmien-Schlichting, que trenca la clàssica simetria de translació i reflexió, ha estat identificada i continuada al llarg de l'espai de paràmetres. A més, s'ha aclarit el rol d'una vella família d'ones viatgeres que en estudis previs no participava dels mecanismes de localització. A continuació, s'analitza la competició entre modes no lineals en el flux purament hidrodinàmic i també hidromagnètic de Taylor-Couette. Branques de solucions d'amplitud finita, en forma de modes mixtes, han estat identificades sorgint d'inestabilitats purament hidrodinàmiques i magnètiques. Aquestes interaccions de modes no lineals són eficientment calculades mitjançant dominis computacionals inclinats, enlloc dels clàssics ortogonals, permetent una reducció significativa dels recursos computacionals necessaris. Finalment, s'analitza la generalització dels fluxos extensibles entre plaques paral·leles que s'estiren i s'encongeixen biortogonalment. Sota la hipòtesi d'autosimilitud, s'identifiquen fluxos estacionaris tridimensionals de les equacions de Navier-Stokes i s'estenen al llarg de l'espai de paràmetres, tot estudiant totes les possibles configuracions d'acceleració de les plaques i trobant totes les bifurcacions existents. En finalitzar les exploracions s'han identificat un total de set famílies de solucions, algunes d'elles relacionades per simetries. La complexitat de la topologia d'aquests equilibris creix notablement en incrementar l'acceleració de les plaques, quan les diferents branques de solucions interaccionen per mitjà de bifurcacions de node-sella i punts de codimensió-2 en forma de bifurcacions de cúspide. Al marge de l'interès específic de cada un dels tres problemes estudiats, aquests també han servit com a demostració conceptual de l'aplicabilitat i idoneïtat dels mètodes i eines desenvolupats en el transcurs d'aquesta tesi, que poden ajudar a abordar un ampli ventall de problemes en una gran varietat de disciplines de la física i l'enginyeria.Postprint (published version

Exact coherent structures in the transitional regime of shear and centrifugal flows

Tesi en modalitat de compendi de pubicacionsTurbulence is one of the major concerns for most technological problems involving fluid motion. Specially in aeronautics, a turbulent boundary layer results in structural stresses, vibrations and higher aircraft drag, leading to a significant increase in fuel consumption. Therefore, trying to comprehend the origin of turbulence by studying its most common transition routes is a crucial first step towards its effective control. Transition to turbulence of an homogeneous flow is frequently mediated by transient visits to highly nonlinear laminar coherent structures that usually are at the threshold between laminarity and turbulence. From a dynamical systems point of view, these structures are invariant sets in the infinite-dimensional Navier-Stokes phase space that here we aim to identify in different canonical flows. With the use of direct numerical simulations together with Newton-Krylov solvers and Arnoldi iteration method for the linear stability analysis, invariant sets such as equilibria, relative equilibria or periodic orbits are accurately computed and tracked along the parameter space to understand the transition mechanisms. From a mathematical perspective, dynamical systems and bifurcation theory provide the suitable framework to understand the hydrodynamic instabilities and transition to turbulence from a deterministic point of view. In addition, the use of spectral methods for the spatial discretisation is particularly convenient due to the exponential convergence of the numerical solutions. In the first work, the onset of transition in two-dimensional Plane Poiseuille flow is analysed. A new family of Tollmien-Schlichting waves, breaking the usual half-shift and reflect symmetry of the classical ones, has been identified and tracked across parameter space. In addition, the role of another classical travelling wave family that did not participate in the localisation mechanisms has been clarified. We continue by analysing the nonlinear mode competition in purely hydrodynamic and also magnetised Taylor-Couette flow. Finite-amplitude mixed-mode solution branches, arising from both purely hydrodynamic and magneto-rotational instabilities, are identified. These nonlinear mode interactions are efficiently computed in suitable skewed computational domains instead of the classical orthogonal ones, allowing for a significant reduction of the required computational resources. Finally, the generalisation of extensional flows between biorthogonally stretching-shrinking parallel plates is analysed. Under the assumption of the self-similar ansatz, three-dimensional steady equilibria of the Navier-Stokes equations are identified and systematically tracked in parameter space, to cover all possible configurations of the acceleration rates and thus unfold all occurring bifurcations. After the explorations, up to seven different families of steady solutions have been identified, some of them related in pairs with symmetries. When increasing wall acceleration rates, the solution branches interact by means of fold and codimension-2 cusp bifurcations, increasing the complexity of the topology of equilibria. Besides the specific interest attached to each one of the three problems we have addressed, these have further served as a proof-of-concept for the applicability and suitability of the methods and tools developed in the course of this thesis, which may assist in tackling a vast range of problems across a huge variety of physics and engineering disciplines.La turbulència és una de les principals preocupacions per a la majoria de problemes tecnològics relacionats amb el moviment de fluids. Especialment en el cas de l'aeronàutica, una capa límit turbulenta produeix tensions estructurals, vibracions i una major força d'arrossegament de l'aeronau que resulten en un increment significatiu del consum de combustible. Per tant, intentar comprendre l'origen de la turbulència, tot estudiant-ne les rutes de transició més habituals, és un primer pas indispensable cap al seu control efectiu. La transició a la turbulència d'un flux homogeni sovint es caracteritza per visites transitòries a estructures coherents, laminars i altament no-lineals, que acostumen a trobar-se al llindar entre la laminaritat i la turbulència. Des del punt de vista dels sistemes dinàmics, aquestes estructures són conjunts invariants en l'espai de fase infinit-dimensional de les equacions de Navier-Stokes, que aquí es pretén identificar en diferents fluxos canònics. Mitjançant la integració temporal de les equacions, resolutors de Newton-Krylov i el mètode iteratiu d'Arnoldi per a l'anàlisi d'estabilitat lineal, els diferents conjunts invariants siguin equilibris, equilibris relatius o òrbites periòdiques són acuradament calculats i continuats al llarg de l'espai de paràmetres per tal d'entendre els mecanismes involucrats en la transició. Des d'una perspectiva matemàtica, els sistemes dinàmics i la teoria de bifurcacions proporcionen el marc adequat per a comprendre les inestabilitats hidrodinàmiques i la transició a la turbulència des d'un punt de vista determinista. A més, l'ús de mètodes espectrals per a la discretització espaial resulta particularment convenient degut a la convergència exponencial de les solucions numèriques. En el primer treball, s'analitza l'inici de la transició del flux bidimensional de Poiseuille pla. En aquest cas, una nova família d'ones de Tollmien-Schlichting, que trenca la clàssica simetria de translació i reflexió, ha estat identificada i continuada al llarg de l'espai de paràmetres. A més, s'ha aclarit el rol d'una vella família d'ones viatgeres que en estudis previs no participava dels mecanismes de localització. A continuació, s'analitza la competició entre modes no lineals en el flux purament hidrodinàmic i també hidromagnètic de Taylor-Couette. Branques de solucions d'amplitud finita, en forma de modes mixtes, han estat identificades sorgint d'inestabilitats purament hidrodinàmiques i magnètiques. Aquestes interaccions de modes no lineals són eficientment calculades mitjançant dominis computacionals inclinats, enlloc dels clàssics ortogonals, permetent una reducció significativa dels recursos computacionals necessaris. Finalment, s'analitza la generalització dels fluxos extensibles entre plaques paral·leles que s'estiren i s'encongeixen biortogonalment. Sota la hipòtesi d'autosimilitud, s'identifiquen fluxos estacionaris tridimensionals de les equacions de Navier-Stokes i s'estenen al llarg de l'espai de paràmetres, tot estudiant totes les possibles configuracions d'acceleració de les plaques i trobant totes les bifurcacions existents. En finalitzar les exploracions s'han identificat un total de set famílies de solucions, algunes d'elles relacionades per simetries. La complexitat de la topologia d'aquests equilibris creix notablement en incrementar l'acceleració de les plaques, quan les diferents branques de solucions interaccionen per mitjà de bifurcacions de node-sella i punts de codimensió-2 en forma de bifurcacions de cúspide. Al marge de l'interès específic de cada un dels tres problemes estudiats, aquests també han servit com a demostració conceptual de l'aplicabilitat i idoneïtat dels mètodes i eines desenvolupats en el transcurs d'aquesta tesi, que poden ajudar a abordar un ampli ventall de problemes en una gran varietat de disciplines de la física i l'enginyeria.Ciència i tecnologia aeroespacial

A Spectral Eigenvalue Method for Multilayred Continuua

Abstract not provided