The effect of single mode initial conditions at the interface of Rayleigh-Taylor(RT) mixing are experimentally examined utilizing the low Atwood number water channel facility at Texas A&M. The water channel convects two separated stratified flows and unifies them at the end of a splitter plate. The RT instability is attained by convecting a cold stream above a warmer stream. Average density calculations are based on long time average optical measurements. The water channel was modifified with a flapper fin like device at the end of the splitter plate which was actuated by a computer controlled servo motor. Other modifications to the experiment were implemented resulting in reduced uncertainty. The experiment examined five different modes in addition to the baseline: 2 cm, 3 cm, 4 cm, 6 cm, and 8 cm wavelengths. The mixing width growth rates were shown to be dependent on initial conditions. Additionally, it appears that the growth rates commence with terminal velocity and are observed to line up with the baseline case.

Le couplage entre processus physico-chimiques et le transfert de matière ou de chaleur peuvent donner lieu à des structures spatio-temporelles induites par des flux convectifs. Ces flux peuvent résulter de gradients de densité ou de tension superficielle et sont l'expression de la conversion d'énergie chimique en énergie mécanique. Quand la tension superficielle est à l'origine de ces mouvements, les effets correspondants sont connus sous le nom d'effet Marangoni. Ils jouent un rôle dans de nombreuses applications comme les procédés industriels d'extraction en amplifiant notablement la vitesse des processus de transfert. Les systèmes réels, trop complexes, doit être simplifiés par le développement de systèmes modèles afin d'établir au niveau fondamental la théorie sous-jacente à de telles dynamiques. Une succession de régimes dynamiques est observée lors de la dissolution d'une goutte de dichlorométhane (DCM) déposée sur une solution aqueuse de tensioactif (bromure de céthytriméthylammonium, CTAB). La succession remarquable de formes et de mouvements induits est déterminée par la concentration du tensioactif qui joue le rôle de paramètre de contrôle. A faible concentration en CTAB, un mouvement de translation ou des pulsations. Aux concentrations plus élevées, la goutte entre en rotation ou forme des structures polygonales. Bien que chimiquement simple, le système est complexe et implique plusieurs processus physico-chimiques : évaporation, solubilisation, transfert de tensioactifs, adsorption aux interfaces et agrégation. Les effets thermiques et de transport qui en résultent sont à l'origine des variations locales de tension interfaciale donnant lieu aux effets Marangoni. Nous nous sommes concentrés sur le comportement de la goutte quand la concentration en tensioactif conduit au régime de pulsation. Nous avons tout d'abord analysé le comportement de la goutte pendant la période d'induction qui précède le régime instable. L'analyse de la forme de la goutte corrélée à des mesures d'Imagerie par Vélocimétrie de Particules (PIV), ont montré que les flux créés par la dissolution du DCM limitent dans un premier temps l'adsorption du CTAB à l'interface eau/huile. L'instabilité ne démarre que lorsque la dissolution est réduite et que l'adsorption devient effective. La phase d'induction apparait comme une transition lente entre un coefficient d'étalement négatif (goutte ayant la forme d'une lentille) vers un coefficient d'étalement positif qui entraine l'expansion du film et les pulsations suivantes. Ces pulsations sont accompagnées par l'éjection de gouttelettes qui se forment à partir d'un bourrelet apparaissant au bord du film pendant la phase d'expansion. La rupture de ce bourrelet ressemble au phénomène connu sous le nom d'instabilité de Rayleigh-Plateau (RP). Cependant, la longueur d'onde caractéristique de formation des gouttelettes est deux fois plus faible que celle attendue dans le cas d'une instabilité de RP classique. L'origine de cet écart réside dans la modulation du bourrelet avant sa rupture. Cette modulation est en fait déterminée par des ondulations apparaissant à la surface du film et formant des rides en direction radiale. Ces rides pourraient être attribuées à un effet Marangoni thermique connu sous le nom d'instabilité de Bénard-Marangoni. Elles jouent également un rôle important dans la formation de la structure de démouillage hautement organisée décrite dans le dernier chapitre. L'ajout de CTAB dans la phase organique (goutte) donne lieu à des oscillations plus rapides qui, après une phase d'expansion de grande amplitude et l'éjection d'une couronne parfaite de gouttelettes, résultent lors de la phase de démouillage en une structure dont la forme rappelle une fleur. Une interprétation qualitative permettant d'identifier les principaux processus à l'œuvre et basée sur des mesures indépendantes de tension interfaciale apporte une explication des pulsations observées et de l'auto-organisation induite. / Far-from-equilibrium systems exhibit a wide variety of spatial and temporal patterns known as dissipative structures. The interplay between physico-chemical processes and mass or heat transfer can give rise to spatio-temporal structures induced by convective flows. These flows may result from density or surface tension gradients. They are the expression of the conversion from chemical into mechanical energy. When surface tension is the driving force, the corresponding effects are known as Marangoni effects. They are at play in numerous applications as extraction processes, oil recovery, and chemical reactors at all scales and noticeably modify transfer rates. The complexity of real systems deserves the development of model systems, essentials to settle, on a fundamental level, the theory governing the related dynamics. A succession of dynamical regimes is observed during the dissolution of a dichloromethane drop deposited on aqueous solutions of a cationic surfactant (cetyltrimethylammonium bromide, CTAB). The remarkable range of shapes and motion patterns that emerges is related to the surfactant concentration, which is used as a control parameter. For low surfactant concentrations, we observe translational motion and pulsations of the drop. At intermediate concentrations the drop transforms and starts to rotate. At higher concentrations polygonal shapes are observed. Although chemically simple and of easy implementation, the system is relatively complex and involves several processes: evaporation, solubilization, surfactant mass transfer, interfacial adsorption and self-aggregation. Thermal and transport effects induced are at the origin of local variations of interfacial tension leading to the Marangoni flows. In this thesis, we focused on the behavior of the dichloromethane drop when the aqueous surfactant concentration (0.5 mM) leads to the pulsating regime. At this concentration, we have first analyzed the behavior of the drop during the induction period that precedes the instable regime. Drop shape analysis, correlated to Particle Image Velocimetry (PIV) measurements, showed that dissolution flows initially hinder adsorption of CTAB at the water/oil interface. The instability is only triggered when dissolution is reduced and water/oil adsorption becomes effective. The induction period appears as a slow transition from an initial negative spreading coefficient (a lens shape drop) towards a positive spreading coefficient that triggers film expansion and following pulsations. These pulsations are accompanied by the ejection of smaller droplets which are formed from a toroidal rim that is created during the expanding phase of the drop. The break-up of this toroidal rim, resembles to what is known as the Rayleigh-Plateau (RP) instability. Nevertheless, the observed characteristic wavelength is a factor of 2 too small in respect to the classical RP instability. We have found the origin of this discrepancy in the fact that modulations that appear on the rim before it transforms into droplets are settled by deformations arising at the surface of the expanding film. They appear as wrinkles that form in the film and may be related to thermal Marangoni effects known as Benard-Marangoni instability. These wrinkles play an important part in the highly organized dewetting structure described in the last chapter of the thesis. The addition of CTAB also in the organic (drop) phase leads to faster pulsations which, after a very high amplitude expanding stage and the ejection of a perfect crown of droplets, result during the film receding stage in the formation of a pattern which symmetry is reminiscent of a flower. A qualitative interpretation aimed at identifying the main processes at play and based on independent surface tension data gives a consistent explanation of the observed pulsations and related self-organized patterns.

La faisabilite de la fusion magnetique est dependante de notre capacite a confiner l'energie des particules supra-thermiques liberees a haute energie par les reactions de fusion, dans les meilleures conditions de securite et d'efficacite. Dans ce but, il est necessaire de comprendre l'interaction entre les particules energetiques et le plasma thermo-nucleaire qui constitue l'environnement des reactions de fusion, afin de la controler. La these que nous presentons ici s'inscrit dans cet effort. Le coeur du travail mene est l'etude d'un type d'instabilite, le Beta Alfven Eigenmode (BAE), que peuvent exciter les particules energetiques, et dont on peut craindre qu'il degrade fortement non seulement le confinement des particules energetiques mais aussi le confinement du plasma dans sa globalite. Dans un premier temps, nous nous attacherons a decrire les caracteristiques de ce mode et nous deriverons sa relation de dispersion ainsi que sa structure. Dans une seconde partie, nous effectuerons l'etude de la stabilite lineaire de ce mode en presence de particules energetiques. Cette etude nous a permis de definir un critere analytique rendant compte de la capacite des particules energetiques a exciter le BAE. Ce critere sera discute et confronte aux resultats d'experiences menees durant la these. Cette etude lineaire presentant cependant quelques limites, il nous est apparu important de nous poser la question de la possibilite d'une modication de la stabilite du BAE liee a l'utilisation d'une description non-lineaire. Nous suggererons dans cette presentation un processus, verifie analytiquement et numeriquement, dont peut resulter l'existence d'etats meta-stables pour le BAE.

Large-scale unsteady flow structures play an influential role in the dynamics of many practical flows, such as those found in gas turbine combustion chambers. This thesis is concerned primarily with large-scale unsteady structures that arise due to self-sustained hydrodynamic oscillations, also known as global hydrodynamic instability. Direct numerical simulation (DNS) of the Navier--Stokes equations in the low Mach number limit is used to obtain a steady base flow, and the most unstable direct and adjoint global modes. These are combined, using a structural sensitivity framework, to identify the region of the flow and the feedback mechanisms that are responsible for causing the global instability. Using a Lagrangian framework, the direct and adjoint global modes are also used to identify the regions of the flow where steady and unsteady control, such as a drag force or heat input, can suppress or promote the global instability. These tools are used to study a variety of reacting and non-reacting flows to build an understanding of the physical mechanisms that are responsible for global hydrodynamic instability in swirling diffusion flames. In a non-swirling lifted jet diffusion flame, two modes of global instability are found. The first mode is a high-frequency mode caused by the instability of the low-density jet shear layer in the premixing zone. The second mode is a low-frequency mode caused by an instability of the outer shear layer of the flame. Two types of swirling diffusion flames with vortex breakdown bubbles are considered. They show qualitatively similar behaviour to the lifted jet diffusion flames. The first type of flame is unstable to a low-frequency mode, with wavemaker located at the flame base. The second type of flame is unstable to a high-frequency mode, with wavemaker located at the upstream edge of the vortex breakdown bubble. Feedback from density perturbations is found to have a strong influence on the unstable modes in the reacting flows. The wavemaker of the high-frequency mode in the reacting flows is very similar to its non-reacting counterpart. The low-frequency mode, however, is only observed in the reacting flows. The presence of reaction increases the influence of changes in the base flow mixture fraction profiles on the eigenmode. This increased influence acts through the heat release term. These results emphasize the possibility that non-reacting simulations and experiments may not always capture the important instability mechanisms of reacting flows, and highlight the importance of including heat release terms in stability analyses of reacting flows.

In the case of cold accretion disk, coupling between charge neutral gas and magnetic field is too weak such that the magneto-rotational instability will be less effective or even stop working. In such a situation it is of prime interest to investigate the pure hydrodynamic turbulence and transport phenomenon. As the Reynolds number increases, the relative importance of the non-linear term in the hydrodynamic equation increases and in the case of accretion disk where molecular viscosity is too small the Reynolds number is large enough for the non-linear term to bring new effects. We investigate a scenario, the ‘weakly non-linear’ evolution of amplitude of linear mode when the flow is bounded by two parallel walls. The unperturbed flow is similar to plane Couette flow but with Coriolis force included in the hydrodynamic equation. Although there is no exponentially growing eigenmode, due to self-interaction the least stable eigenmode will grow in an intermediate phase. Later on this will lead to higher order non-linearity and plausible turbulence. Although the non-linear term in the hydrodynamic equation is energy conserving, within the weakly non-linear analysis it is possible to define a lower bound of the energy needed for flow to transform to turbulent phase. Such an unstable phase is possible only if the Reynolds number ≥ 103−4. In Chapter-2 we set up equation of amplitude for the hydrodynamic perturbation and study the effect of weak non-linear evolution of linear mode for general angular momentum distribution, where Keplerian disk is obtained as a special case. As we know that to explain observed hard X-rays the choice of Keplerian angular momentum profile is not adequate, we consider the sub-Keplerian regime of the disk. In Chapter-3 we assume that the cooling mechanism is dominated by bremsstrahlung process (without any strict knowledge of the magnetic field structure).We show that in a range of Shakura-Sunyaev viscosity 0.2 ≥ α ≥ 0.0005, flow behavior varies widely, particularly by means of the size of disk, efficiency of cooling and corresponding temperatures of ions and electrons. We also show that the disk around a rotating black hole is hotter compared to that around a Schwarzschild black hole, rendering a larger difference between ion and electron temperatures in the former case. We finally reproduce the observed luminosities(L) of two extreme cases—the under-fed AGNs and quasars and ultra-luminous X-ray sources at different combinations of mass accretion rate, ratio of specific heats, Shakura-Sunyaev viscosity parameter and Kerr parameter. In Chapter-4 we investigate the viscous two temperature accretion disk flows around rotating blackholes. We describe the global solution of accretion flows, unlike that in Chapter-3, with a sub-Keplerian angular momentum profile, by solving the underlying conservation equations including explicit cooling processes self-consistently. Bremsstrahlung, synchrotron and inverse comptonization of soft photons are considered as possible cooling mechanisms. We focus on the set of solutions for sub-Eddington, Eddington and super-Eddington mass accretion rates around Schwarzschild and Kerr black holes with a Kerr parameter 0.998. We analyse various phases of advection–general advective paradigm to radiatively inefficient paradigm. The solution may potentially explain the hard X-rays and γ-rays emitted from AGNs and X-ray binaries. We also compare the solutions for two different regimes of viscosity. We finally reproduce the observed luminosities of the under-fed AGNs and quasars, ultra-luminous X-ray sources at different combinations of input parameters such as mass accretion rate and ratio of specific heats.

La dynamique instable des jets tournants est étudiée, en tenant compte des variations de masse volumique au sein de l'écoulement. Un code de simulation numérique directe permettant de résoudre les équations de Navier-Stokes à masse volumique variable a été développé, en utilisant une méthode originale et efficace pour résoudre le champs de pression. Analytiquement, deux modes instables bidimensionnels ont été mis en évidence, et sont identifiés comme des modes de Couette-Taylor et de Rayleigh-Taylor, ainsi qu'un troisième mode tridimensionnel, du à un couplage de vitesse. La dynamique instable de cet écoulement résulte d'une compétition entre ces trois modes, et les simulations numériques montrent que ces modes perdurent non linéairement. Ensuite, le comportement spatio-temporel de cette instabilité est étudiée par simulation numérique directe, et il a été montré qu'il existe une transition vers des modes absolument instables, sous l'effet du rapport de densité s ainsi que du taux de rotation q. Cette dynamique est également étudiée expérimentalement au travers de plusieurs méthodes de mesures, et la présence de mode globaux auto-entretenus est mise en évidence qui sont en bon accord avec les résultats numériques. Finalement, le phénomène de l'éclatement tourbillonnaire est étudié, et montre le rôle prépondérant de la viscosité réelle. En effet, l'éclatement tourbillonnaire est un mécanisme permettant de soulager le système de l'intensification de la vorticité, au travers de la viscosité, alors qu'il n'apparaît pas en traitant les équations d'Euler tronquées. / The unstable dynamics of a swirling jet flow is studied, including density variations within the flow. A direct numerical simulation method was developed to solve variable density Navier-Stokes equations, using an accurate and efficient pressure solver. Analitically, two unstable bi-dimensionnal modes are highlighted, and are identified as Couette-Taylor and Rayleigh-Taylor modes. A three-dimensionnal mode is also highlighted, wich is created by the shear. Numerical simulations show that those modes are nonlinearly persistant. Then, the spatio-temporal instability behaviour is studied numerically, and show that the instability undergoes to a convective/absolute transition with density ratio s and rotation rate q. This dynamic is also studied experiementally through different methods, and Global selfsustained modes are highlighted wich are in ggod agreement with numerical results. Finally, the vortex breakdown phenomenon is studied, and show the crucial role of real viscosity. Indeed, the vorticity intensification is relaxed through the viscosity effect, while it is not treating the truncated Euler Equations.

Os mecanismos de instabilidade hidrodinâmica têm um papel importante no processo da transição do escoamento de laminar para turbulento. A análise da instabilidade hidrodinâmica em uma cavidade com tampa deslizante foi realizada através da decomposição em modos globais (biglobal) para avaliar o efeito da compressibilidade neste fenômeno. O escoamento base foi obtido através de simulação numérica direta (DNS). Para tal, foi desenvolvido um código DNS compressível com discretização espacial por diferenças finitas compactas de alta resolução espectral e capacidade de processamento paralelo, com um método de decomposição de domínio que mantém a precisão das diferenças finitas compactas. O escoamento base é usado para montar o problema de autovalor oriundo das equações de Navier-Stokes linearizadas para a perturbação, discretizadas por diferenças finitas explícitas. O uso de diferenças finitas em conjunto com a implementação em matrizes esparsas reduz sensivelmente o uso de memória. Através do algoritmo de Arnoldi, a ordem do problema de autovalor é reduzida e os autovalores de interesse são recuperados. Os resultados indicam o efeito estabilizante da compressibilidade nos modos dominantes da cavidade e revelam modos inerentes ao escoamento compressível, para os quais a compressibilidade tem efeito desestabilizante. Dentre estes modos compressíveis, estão presentes modos de propagação sonora em dutos e modos relacionados à geração de som na cavidade. / Hydrodynamic instability mechanisms play an important role in laminar to turbulent transition. Hydrodynamic instability analysis of a lid-driven cavity flow was performed by global mode decomposition (biglobal) to evaluate compressibility effects on this phenomenon. The basic flow was calculated by direct numerical simulation (DNS). A compressible DNS code was developed with spectral-like compact finite difference spatial discretization. The code allows parallel processing with a domain decomposition method that preserves the compact finite difference accuracy. The basic flow is used to form the eigenvalue problem associated to the linear Navier- Stokes equations for the perturbation, which were discretized by an explicit finite difference scheme. The combination of sparse matrix techniques and finite difference discretization leads to a significant memory reduction. The order of the eigenvalue problem was reduced using the Arnoldi algorithm and the eigenvalues of interest were calculated. Results show the stabilizing effect of compressibility on the leading modes and reveal some modes intrinsic to compressible flow, for which compressibility has a destabilizing effect. Among these compressible modes, there are some related to sound propagation in ducts and to sound generation inside the cavity.

A poluição sonora é um problema central de uma grande diversidade de aplicações industriais. Na engenharia, podemos citar diversos casos que geram ruído, como exemplos os trens, automóveis, rotores de helicópteros e o ruído aerodinâmico das aeronaves, o qual se divide em ruído gerado pelos motores a jato e a estruturas da aeronave. No presente momento o ruído dos motores aeronáuticos, principalmente os jatos, atingiu níveis de ruídos semelhantes às estruturas da aeronave, como por exemplo, eslates, flaps e trens de pouso. Desta forma, as autoridades de transporte aéreo estão exigindo também redução no ruído das estruturas. O presente trabalho apresenta a verificação das potencialidades e limitações do software CFD++, programa este adquirido pela EMBRAER e inserido como parte do projeto Aeronave Silenciosa, para assim poder compreender de uma melhor maneira o fenômeno da aeroacústica, e deste modo, poder contribuir para a redução do ruído externo das aeronaves. Para verificar as potencialidades e limitações do CFD++, foi proposto investigar o mecanismo de som do eslate. Tal fenômeno é devido ao deslocamento da camada limite no intradorso do eslate a partir de onde se desenvolve a camada de mistura, foco do presente trabalho. / Noise pollution is a central problem of a wide variety of industrial applications. In engineering, cite several cases that generate noise, as examples trains, automobiles, rotors of helicopters and the noise generated by aircraft, which is divided into noise generated by jet engines and airframe. At present the noise of aircraft engines, largely the jets reached noise levels similar structures, such as slat, flaps and landing gear. Thus, the air transport authorities are also demanding a reduction in noise of the structures of airframe. This report presents the verification of potentialities and limitations of CFD++, a program acquired by EMBRAER and inserted as part of the Silent Aircraft, so they can understand better how the phenomenon of aeroacoustics, and thus able to reduce contribute external noise from aircraft. To check the potentialities and limitations of CFD++, was proposed to investigate the mechanism of sound generated by the slat. This phenomenon is due to the displacement of the boundary layer on the lower surface of the slat from which the mixed layer develops. The mixing layer is the focus of this work.

Os mecanismos de instabilidade hidrodinâmica têm um papel importante no processo da transição do escoamento de laminar para turbulento. A análise da instabilidade hidrodinâmica em uma cavidade com tampa deslizante foi realizada através da decomposição em modos globais (biglobal) para avaliar o efeito da compressibilidade neste fenômeno. O escoamento base foi obtido através de simulação numérica direta (DNS). Para tal, foi desenvolvido um código DNS compressível com discretização espacial por diferenças finitas compactas de alta resolução espectral e capacidade de processamento paralelo, com um método de decomposição de domínio que mantém a precisão das diferenças finitas compactas. O escoamento base é usado para montar o problema de autovalor oriundo das equações de Navier-Stokes linearizadas para a perturbação, discretizadas por diferenças finitas explícitas. O uso de diferenças finitas em conjunto com a implementação em matrizes esparsas reduz sensivelmente o uso de memória. Através do algoritmo de Arnoldi, a ordem do problema de autovalor é reduzida e os autovalores de interesse são recuperados. Os resultados indicam o efeito estabilizante da compressibilidade nos modos dominantes da cavidade e revelam modos inerentes ao escoamento compressível, para os quais a compressibilidade tem efeito desestabilizante. Dentre estes modos compressíveis, estão presentes modos de propagação sonora em dutos e modos relacionados à geração de som na cavidade. / Hydrodynamic instability mechanisms play an important role in laminar to turbulent transition. Hydrodynamic instability analysis of a lid-driven cavity flow was performed by global mode decomposition (biglobal) to evaluate compressibility effects on this phenomenon. The basic flow was calculated by direct numerical simulation (DNS). A compressible DNS code was developed with spectral-like compact finite difference spatial discretization. The code allows parallel processing with a domain decomposition method that preserves the compact finite difference accuracy. The basic flow is used to form the eigenvalue problem associated to the linear Navier- Stokes equations for the perturbation, which were discretized by an explicit finite difference scheme. The combination of sparse matrix techniques and finite difference discretization leads to a significant memory reduction. The order of the eigenvalue problem was reduced using the Arnoldi algorithm and the eigenvalues of interest were calculated. Results show the stabilizing effect of compressibility on the leading modes and reveal some modes intrinsic to compressible flow, for which compressibility has a destabilizing effect. Among these compressible modes, there are some related to sound propagation in ducts and to sound generation inside the cavity.

[pt] A estabilidade do escoamento estratificado de ar e água, sujeito a perturbações periódicas no nível de líquido, é investigada numericamente em um duto horizontal. Selecionou-se o Modelo de Dois Fluidos unidimensional para a simulação do escoamento. As equações de conservação de massa e de quantidade de movimento linear para as fases gás e líquido são discretizadas de acordo com o método dos Volumes Finitos. O acoplamento entre as equações é resolvido sequencialmente com uma versão modificada do método PRIME. Perturbações no nível de líquido foram introduzidas de maneira controlada na entrada da tubulação. A evolução dessas perturbações, ao longo da tubulação, é analisada e os resultados são comparados com as previsões fornecidas por um modelo baseado na teoria linear de Kelvin-Helmholtz. A velocidade de propagação, a frequência e o número de onda das perturbações apresentam excelente concordância entre a simulação e modelo, indicando que, de fato, ambas abordagens são capazes de prever características fundamentais dessas ondas. As taxas de crescimento previstas pelo modelo e as obtidas na simulação, também, foram comparadas apresentando razoável concordância. Os resultados mostram que a frequência da perturbação tem influência na taxa de amplificação e que ondas com frequências mais altas tendem a serem mais instáveis. Para tubulações longas, efeitos não lineares podem ser observados em regiões afastadas da entrada da tubulação. Nesse estágio é possível observar alterações no mecanismo de crescimento das perturbações. / [en] The stability of stratified air-water flow under the influence of periodic disturbances in the liquid level is investigated numerically for a horizontal pipe. One-dimensional two-fluid model is used for flow simulation. Conservation equations for mass and linear momentum are discretized for both phases using a finite volume method. Coupling between equations is achieved by sequentially solving a modified version of PRIME method. Controlled disturbances are introduced in the flow by oscillating the liquid level at the pipe inlet. The evolution of the generated disturbances along the pipe is analyzed and the results are compared with predictions given by a model based on linear theory of Kelvin-Helmholtz (KH). An excellent agreement is obtained for velocity, frequency and wave number of the perturbations. This indicates that both approaches are capable to predict the fundamental characteristics of the waves. Amplifications rates predicted by simulation and model have been also compared and the results show a reasonable agreement. It is found that the frequency of the perturbations plays a role in the amplification rate. For increasingly higher frequencies the disturbances tends to be more unstable. The analysis is extended for long pipes, in such cases the growth rates changes at locations far from the inlet. It is conjectured that non linear mechanisms are related to observations.