Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The present work is dedicated to perform the mathematical modeling and DNS and LES simulations of a three-dimensional, temporally evolving incompressible plane wake are performed, seeking to evidence differences in stability, transition and onset of both coherent and small scale structures, when the flow is subjected to random perturbations of different amplitudes. The perturbations are generated using the Random-Flow-Generator (RFG) technique, being imposed to the flow as initial conditions. The Navier-Stokes equations are solved in a prismatic domain, with periodic boundary conditions in all directions, using Fourier pseudospectral method. The invariants of the velocity gradient tensor, Q and R, are analyzed for random perturbations with magnitudes 10−3, 10−4 and 10−5, showing the onset of their characteristic teardrop correlation map. Moreover, maps of the second and third invariants of the rate-of-strain tensor, QS and RS, are shown, in order to evidence the differences in local flow strain and topological characteristics of the dissipation of kinetic energy. Isosurface plots of Q and QW, as well as vorticity contours are shown, allowing visual identification of the coherent structures and confirming patterns predicted by the invariant maps. / O presente trabalho se dedica a modelagem matemática e a simulações numéricas DNS e LES de uma esteira tridimensional, incompressível, em desenvolvimento temporal, buscando evidenciar diferenças na estabilidade, transição e no desenvolvimento de estruturas coerentes e de pequena escala, quando o escoamento é submetido a perturbações randômicas de diferentes amplitudes. As perturbações são geradas utilizando-se a técnica Random Flow Generator (RFG), sendo sobrepostas à condição inicial do escoamento. As equações de Navier-Stokes são resolvidas em um domínio prismático, com condições de contorno periódicas em todas as direções, utilizando-se o método pseudoespectral de Fourier. Os invariantes do tensor gradiente de velocidade, Q e R, são analisados para perturbações de magnitude 10−3, 10−4 and 10−5, mostrando a formação de uma correlação no formato de gota, característica da resolução das equações de Navier-Stokes. Além disso, são apresentados mapas do segundo e terceiro invariante do tensor taxa de deformação, QS e RS, a fim de evidenciar as diferenças locais no escoamento e as características topológicas na taxa de dissipação de energia cinética. Isosuperfícies de Q e QW, bem como contornos de vorticidade são apresentados, possibilitando a identificação visual das estruturas coerentes, e confirmando os padrões de estruturas previstos pelos mapas de invariância. / Mestre em Engenharia Mecânica

Accretion disks are ubiquitous in astrophysics. They are found in active galactic nuclei, around newly formed stars, around compact stellar objects, like black holes, neutron stars etc. When the ambient matter with sufficient initial angular momentum falls towards a central massive object, forming a disk shaped astrophysical structure, it is called an accretion disk. There are both ionized and neutral disks depending on their temperatures. Generally, in accretion disks, Gravitational force is balanced by the centrifugal force (due to the presence of angular momentum of the matter) and the forces due to gas pressure, radiation pressure and advection. Now, the matter to be accreted needs to lose angular momentum. For most of the accretion disks, the mass of the central object is much higher than the mass of the disk, giving rise to a dynamics governed by a central force. Therefore we can neglect the effect of self-gravity of the disk. Balancing the Newtonian gravitational force and centrifugal force leads to a Keplerian rotation profile of the accreting matter with the angular velocity ∼ r−3/2, where r is the distance from the central object. The Keplerian disk model is extremely useful to explain several flow classes (e.g. emission of soft X-ray in disks around stellar mass black holes). Due to the presence of differential rotation and hence shear viscosity, the matter can slowly lose its angular momentum and falls towards the central object. In this way, the accreting matter in the disk releases its gravitational potential energy and gives rise to luminosity that we observe. However, the molecular viscosity originated from the microscopic physics (due to the collisions between molecules) of the disk matter is not sufficient to explain the observed luminosity or accretion rate. For example, it can be shown that the temperature arisen from the dissipation of energy due to molecular viscosity (which is around 50000K for optical depth τ = 100) is much less than the temperature observed in these systems (around 107K). In my thesis, I have addressed the famous problem of infall of matter in astrophysical accretion disks. In general, the emphasis is given on the flows whose angular velocity decreases but specific angular momentum increases with increasing radial coordinate. Such flows, which are extensively seen in astrophysics, are Rayleigh-stable, but must be turbulent in order to explain observed data (observed temperature, as described above). Since the molecular viscosity is negligible in these systems, for a very large astrophysical length scale, Shakura and Sunyaev argued for turbulent viscosity for energy dissipation and hence to explain the infall of matter towards the central object. This idea is particularly attractive because of its high Reynolds number (Re ∼ 1014). However, the Keplerian disks, which are relevant to many astrophysical applications, are remarkably Rayleigh stable. Therefore, linear perturbation apparently cannot induce the onset of turbulence, and consequently cannot provide enough viscosity to transport matter inwards. The primary theme of my thesis is, how these accretion disks can be made turbulent in the first place to give rise to turbulent viscosity. With the application of Magnetorotational Instability (MRI) to Keplerian disks, Balbus and Hawley showed that initial seed, weak magnetic fields can lead to the velocity and magnetic field perturbations growing exponentially. Within a few rotation times, such exponential growth could reveal the onset of turbulence. Since then, MRI has been a widely accepted mechanism to explain origin of instability and hence transport of matter in accretion disks. Note that for flows having strong magnetic fields, where the magnetic field is tightly coupled with the flow, MRI is not expected to work. Hence, it is very clear that the MRI is bounded in a small regime of parameter values when the field is also weak. It has been well established by several works that transient growth (TG) can reveal nonlinearity and transition to turbulence at a sub-critical Re. Such a sub-critical transition to turbulence was invoked to explain colder, purely hydrodynamic accretion flows, e.g. quiescent cataclysmic variables, proto-planetary and star-forming disks, the outer region of the disks in active galactic nuclei etc. Baroclinic instability is another plausible source for vigorous turbulence in colder accretion disks. Note that while hotter flows are expected to be ionized enough to produce weak magnetic fields therein and subsequent MRI, colder flows may remain to be practically neutral in charge and hence any instability and turbulence therein must be hydrodynamic. However, in the absence of magnetic effects, the Coriolis force does not allow any significant TG in accretion disks in three dimensions, independent of Re, while in pure two dimensions, TG could be large at large Re. However, a pure two-dimensional flow is a very idealistic case. Nevertheless, in the presence of magnetic field, even in three dimensions, TG could be very large (Coriolis effects could not suppress the growth). Hence, in a real three-dimensional flow, it is very important to explore magnetic TG. However, as mentioned above, the charge neutral Rayleigh-stable astrophysical flows have hardly any magnetic field (e.g. protoplanetary disks, quiescent cataclysmic variables etc.). Also, the hydrodynamic Rayleigh-stable Taylor-Couette flows and plane Couette flows in the laboratory experiments are seen to be turbulent without the presence of any magnetic field, while they are shown to be stable in linear stability analysis. It is a century old unsolved problem to explain hydrodynamically, the linear instability of Couette flows and other Rayleigh-stable Flows, which are observed to be turbulent, starting from laboratory experiments to astrophysical observations. Therefore, as in one hand, the hydrodynamic instability of the astrophysical accretion flows and laboratory shear flows (e.g. Rayleighstable Taylor-Couette flow, plane Couette flow etc.) has to be understood, on the other hand, the magnetohydrodynamic (MHD) instability of the hotter flows has also to be investigated to understand the nature of MHD instability clearly, whether it arises due to MRI or TG. I have investigated the effect of stochastic noise (which is generated by the shearing motion of the disk layers) on the hydrodynamics and magnetohydrodynamics of accretion disks and explain how stochastic noise can make accretion Disks turbulent. It is found that such stochastically driven flows exhibit large temporal and spatial correlations of perturbations, and hence large energy dissipations of perturbation with time, which presumably generates instability and turbulence. I have also given in my thesis, a plausible resolution of the hydrodynamic turbulence problem of the accretion flows and laboratory shear flows (as discussed above) from pure hydrodynamics, invoking the idea of Brownian motion of particles. I have shown that in any shear flow, very likely, the stochastic noise is generated due to thermal fluctuations. Therefore, the shear flows must be studied including the effect of stochastically driving force and hence the governing equations should not be deterministic. Incorporating the effects of noise in the study of the above mentioned shear flows, I have shown in my thesis that hydrodynamic Rayleigh-stable flows and plane Couette flows can be linearly unstable. I have also investigated the importance of transient growth over magnetorotational instability (MRI) to produce turbulence in accretion disks. Balbus and Hawley asserted that the MRI is the fastest weak field instability in accretion disks. However, they used only the plane wave perturbations to study the instability problem. I have shown that for the flows with high Reynolds number, which are indeed the case for astrophysical accretion disks, transient growth can make the system nonlinear much faster than MRI and can be a plausible primary source of turbulence, using the shearing mode perturbations. Therefore, this thesis provides a plausible resolution of hydrodynamic turbulence observed in astrophysical accretion disks and some laboratory shear flows, such as, Rayleigh-stable Taylor-Couette flows and plane Couette flows. Moreover, this thesis also provides a clear understanding of MHD turbulence for astrophysical accretion disks.

Tato numerická studie je zaměřená na axisymetrickou pulzní trysku při zachování relativně nízkých Reynoldsových čísel a její fyzikální podstatu, která dosud nebyla zcela vysvětlena. Hlavním cílem práce bylo prozkoumat a zhodnotit vliv přidání periodického komponentu rychlosti ke stacionární složce rychlosti. Nejdříve byl řešen stacionární případ, poté byla do simulace přidána pulzace a byla vytvořena nestacionární simulace. Numerické řešení stacionárního případu bylo ověřeno pomocí asymptotického řešení, které předložil Hermann Schlichting [44]. Přesnost tohoto analytické řešení byla opravena na základě experimentálních poznatků Andradeho a Tsiena [1]. Pomocí této korekce je zmenšena oblast singularity řešení v blízkosti počátku proudění. Z matematického pohledu se v podstatě jedná korekcí prvního řádu, což bylo dokázáno Revueltou a spol [36]. Samotné analytické řešení bylo vytvořeno v MATLABu zatímco pro numerické řešení byl použit software Ansys Fluent. Při numerické simulaci byly Navier-Stokesovi rovnice integrovány ve své plné formě za pomoci algoritmu založeném na tzv. rovnici korekce tlaku. Pulzační tryska byla poté řešena pro různé parametry tak, aby bylo možné zhodnotit vliv jednotlivých parametrů na evoluci takto modulovaného proudu. Nakonec byla posouzena možná aplikace pulzních trysek v průmyslu s ohledem na možnost snížení emisí v průběhu spalovacího procesu.