A focus has been put on the stability characteristics of different flow types existing on air vehicles. Flow passing over wings and different junctions on an aircraft face numerous local features, ranging from different pressure gradients, to interacting boundary layers. Primarily, stability characteristics of flow over a wing subject to negative pressure gradient is studied. The current numerical study conforms to an experimental study conducted by Saric and coworkers, in their Arizona State University wind tunnel experiments. Within that framework, a passive control mechanism has been tested to delay transition of flow from laminar to turbulence. The same control approach has been studied here, in addition to underling mechanisms playing major roles in flow transition, such as nonlinear effects and secondary instabilities. Another common three-dimensional flow feature arises as a result of streamlines passing through a junction, the so called corner-flow. For instance, this flow can be formed in the junction between the wing and fuselage on a plane. A series of direct numerical simulations using linear Navier-Stokes equations have been performed to determine the optimal initial perturbation. Optimal refers to a perturbation which can gain the maximum energy from the flow over a period of time. Power iterations between direct and adjoint Navier- Stokes equations determine the optimal initial perturbation. In other words this method seeks to determine the worst case scenario in terms of perturbation growth. Determining the optimal initial condition can help improve the design of such surfaces in addition to possible control mechanisms. / <p>QC 20130604</p> / RECEPT

This thesis deals with the instabilities of the incompressible boundary-layer flow that is induced by a disk rotating in otherwise still fluid. The results presented are mostly limited to linear instabilities derived from direct numerical simulations (DNS) but with the objective that further work will focus on the nonlinear regime, providing greater insights into the transition route to turbulence. The numerical code Nek5000 has been chosen for the DNS using a spectral-element method in an effort to reduce spurious effects from low-order discretizations. Large-scale parallel simulations have been used to obtain the present results. The known similarity solution of the Navier–Stokes equation for the rotating-disk flow, also called the von Karman flow, is investigated and can be reproduced with good accuracy by the DNS. With the addition of small roughnesses on the disk surface, convective instabilities appear and data from the DNS are analysed and compared with experimental and theoretical data. A theoretical analysis is also presented using a local linear-stability approach, where two stability solvers have been developedbased on earlier work. A good correspondence between DNS and theory is found and the DNS results are found to explain well the behaviour of the experimental boundary layer within the range of Reynolds numbers for small amplitude (linear) disturbances. The comparison between the DNS and experimental results, presented for the first time here, shows that the DNS allows (for large azimuthal domains) a range of unstable azimuthal wavenumbers β to exist simultaneously with the dominantβ varying, which is not accounted for in local theory, where β is usually fixed for each Reynolds number at which the stability analysis is applied. Furthermore, the linear impulse response of the rotating-disk boundary layer is investigated using DNS. The local response is known to be absolutely unstable. The global response is found to be stable if the edge of the disk is assumed to be at infinity, and unstable if the domain is finite and the edge of the domain is placed such that there is a large enough pocket region for the absolute instability to develop. The global frequency of the flow is found to be determined by the edge Reynolds number. / <p>QC 20140708</p>

In this work, a DNS code was developed to investigate problems on transition in subsonic compressible boundary layer on a flat plate. Code validation tests were performed for linear and nonlinear stages of transition, on incompressible and compressible regimes. The focus of the present work is to investigate natural transition in subsonic boundary layers modeled by wave packets; and perform a preliminary study of transition induced by white noise. Three main problems were considered, namely, a DNS simulation and analysis of the ex- periment (MEDEIROS; GASTER, 1999b) of wave packet evolution on incompressible boundary layer, the influence of compressibility on wave packet evolution at subsonic Mach numbers and finally, a preliminary study of the evolution of a white noise perturbation in the boundary layer at Mach 0.2 and Mach 0.9. Comparisons between numerical and experimental results show remarkably good agreement in the linear and nonlinear stages, in both, spatial and Fourier spaces. A numerical simulation of this experiment and the analysis carried out is not available in the literature for wave packets in the incompressible boundary layer. The nonlinear modal analysis performed established the existence of tuned fundamental and subharmonic resonance of H-type and K-type in the packet. Influence of compressibility in the wave packet evolution was here investigated in boundary layers at Mach 0.7 and Mach 0.9. There are no works reported in the literature on wave packets in compressible subsonic boundary layer. In the linear regime, the oblique modes were the most unstable for Mach > 0.7, as expected by the results of the literature. In the nonlinear regime, strong streaks were observed, associated with low frequency modes that eventually decay downstream. An isolated wave packet at Mach 0.9 showed nonlinear amplification only in the subharmonic band, which may be associated to H-type or detuned resonance. However this packet has a relatively stable character. On the other hand, at Mach 0.9 spanwise interaction of wave packet pairs were more unstable than the isolated case, because stable modes for the isolated packet evolution becomes unstable in the wave packet interaction. This scenario evidenced the presence of oblique transition. Finally, the nonlinear evolution of the same white noise disturbance at Mach 0.2 and Mach 0.9 were observed to be completely different. In the incompressible boundary layer localized lambda vortex structures were observed, that could be associated to the local presence of H-type and/or K-type resonance. In the compressible regime, longitudinal vortex structures distributed across the entire domain seemed to be linked to oblique transition. In the white noise evolution, compressibility seems to have a stronger effect than in the wave packet evolution. In the conditions considered, the wave packet interaction appear to be a better representation of white noise compressible transition scenario. / No presente trabalho, um código DNS (Direct Numerical Simulation) foi desenvolvido para abordar problemas de transição para turbulência em camada limite subsônica compressível em uma placa plana. Foram realizados testes de validação de código , nos regimes linear e não linear do processo de transição, nos regimes incompressível e compressível. O foco do presente trabalho é estudar transição natural modelada por meio de pacotes de onda em camada limite compressível subsônica, e realizar uma análise preliminar da transição induzida por ruído branco. Três assuntos principais foram considerados: uma simulação DNS e uma análise comparativa com o experimento (MEDEIROS; GASTER, 1999b) sobre a evolução de um pacote de ondas em camada limite incompressível, a influência da compressibilidade na evolução de pacotes de ondas no regime subsônico, e por último, um estudo preliminar da transição induzida por ruído branco em Mach 0.2 e Mach 0.9. As comparações realizadas entre a solução numérica e os dados experimentais mostram uma boa concordância, nos regimes linear e não linear, tanto no espaço físico quanto no espaço de Fourier. A simulação numérica deste experimento e a análise realizada neste trabalho, não são encontradas na literatura para o regime incompressível. A análise modal não linear aplicada aos resultados, permitiu identificar a presença das ressonâncias tipo H e tipo K no pacote de ondas. A influência da compressibilidade na evolução dos pacotes de onda foi estudada em Mach 0.7 e Mach 0.9. Na literatura não há trabalhos sobre pacotes de ondas no regime sub- sônico. No regime linear da transição, os modos oblíquos resultam ser os mais instáveis para Mach > 0.7, como era de esperar, de acordo com os resultados da literatura. No regime não linear, foram observadas estrias de moderada amplitude, associadas com modos de baixa frequência que acabam decaindo. O pacote de ondas em Mach 0.9 apresentou amplificação não linear somente na banda subharmônica, que pode ser associada com transição tipo H ou ressonância dessintonizada. No entanto, o comportamento geral neste regime é estabilizante. Por sua vez, a interação entre pacotes de ondas em Mach 0.9 mostrou um comportamento desestabilizante, pois a interação acaba gerando amplificação não linear em modos que decaem no pacote isolado. Os modos amplificados sugerem a presença do mecanismo de transição oblíqua. Finalmente, a evolução da mesma perturbação constituída por ruído branco em Mach 0.2 e Mach 0.9, resultaram ser completamente diferentes. Na camada limite incompressível foram observados vórtices tipo lambda, que poderiam ser gerados pela presença localizada das ressonâncias tipo H e/ou tipo K. No regime compressível foram observados vórtices distribuidos em todo o domínio, o que sugere a presença da transição oblíqua. Na transição gerada por ruído branco a compressibilidade teve uma influência maior que no pacote de ondas. Nas condições estudadas, a interação entre pacotes de ondas parece ser uma melhor representação do ruído branco no regime compressível.

Compressible plane Couette flow is studied with superposed small perturbations. The steady mean flow is characterized by a non-uniform shear-rate and a varying temperature across the wall-normal direction for an appropriate perfect gas model. The studies are broadly into four main categories as said briefly below. Nonmodal transient growth studies and estimation of optimal perturbations have been made. The maximum amplification of perturbation energy over time, G max, is found to increase with Reynolds number Re, but decreases with Mach number M. More specifically, the optimal energy amplification Gopt (the supremum of G max over both the streamwise and spanwise wavenumbers) is maximum in the incompressible limit and decreases monotonically as M increases. The corresponding optimal streamwise wavenumber, αopt, is non-zero at M = 0, increases with increasing M, reaching a maximum for some value of M and then decreases, eventually becoming zero at high Mach numbers. While the pure streamwise vortices are the optimal patterns at high Mach numbers (in contrast to incompressible Couette flow), the modulated streamwise vortices are the optimal patterns for low-to-moderate values of the Mach number. Unlike in incompressible shear flows, the streamwise-independent modes in the present flow do not follow the scaling law G(t/Re) ~ Re2, the reasons for which are shown to be tied to the dominance of some terms (related to density and temperature fluctuations) in the linear stability operator. Based on a detailed nonmodal energy anlaysis, we show that the transient energy growth occurs due to the transfer of energy from the mean flow to perturbations via an inviscid algebraic instability. The decrease of transient growth with increasing Mach number is also shown to be tied to the decrease in the energy transferred from the mean flow (E1) in the same limit. The sharp decay of the viscous eigenfunctions with increasing Mach number is responsible for the decrease of E1 for the present mean flow. Linear stability and the non-modal transient energy growth in compressible plane Couette flow are investigated for the uniform shear flow with constant viscosity. For a given M, the critical Reynolds number (Re), the dominant instability (over all stream-wise wavenumbers, α) of each mean flow belongs different modes for a range of supersonic M. An analysis of perturbation energy reveals that the instability is primarily caused by an excess transfer of energy from mean-flow to perturbations. It is shown that the energy-transfer from mean-flow occurs close to the moving top-wall for “mode I” instability, whereas it occurs in the bulk of the flow domain for “mode II”.For the Non-modal transient growth anlaysis, it is shown that the maximum temporal amplification of perturbation energy, G max,, and the corresponding time-scale are significantly larger for the uniform shear case compared to those for its non-uniform counterpart. For α = 0, the linear stability operator can be partitioned into L ~ L ¯ L +Re2Lp is shown to have a negligibly small contribution to perturbation energy which is responsible for the validity of the well-known quadratic-scaling law in uniform shear flow: G(t/Re) ~ Re2 . In contrast , the dominance of Lp is responsible for the invalidity of this scaling-law in non-uniform shear flow. An inviscid reduced model, based on Ellignsen-Palm-type solution, has been shown to capture all salient features of transient energy growth of full viscous problem. For both modal and non-modal instability, the viscosity-stratification of the underlying mean flow would lead to a delayed transition in compressible Couette flow. Modal and nonmodal spatial growths of perturbations in compressible plane Couette flow are studied. The modal instability at a chosen set of parameters is caused by the scond least-decaying mode in the otherwise stable parameter setting. The eigenfunction is accurately computed using a three-domain spectral collocation method, and an anlysis of the energy contained in the least-decaying mode reveals that the instability is due to the work by the pressure fluctuations and an increased transfer of energy from mean flow. In the case of oblique modes the stability at higher spanwise wave number is due to higher thermal diffusion rate. At high frequency range there are disjoint regions of instability at chosen Reynolds number and Mach number. The stability characteristics in the inviscid limit is also presented. The increase in Mach number and frequency is found to further destabilize the unstable modes for the case of two-dimensional(2D) perturbations. The behaviors of the non-inflexional neutral modes are found to be similar to that of compressible boundary layer. A leading order viscous correction to the inviscid solution reveals that the neutral and unstable modes are destabilized by the no-slip enforced by viscosity. The viscosity has a dual role on the stable inviscid mode. A spatial transient growth studies have been performed and it is found that the transient amplification is of the order of Reynolds number for a superposition of stationary modes. The optimal perturbations are similar to the streamwise invariant perturbations in the temporal setting. Ellignsen & Palm solution for the spatial algebraic growth of stationary inviscid perturbation has been derived and found to agree well with the transient growth of viscous counterpart. This inviscid solution captures the features of streamwise vortices and streaks, which are observed as optimal viscous perturbations. The temporal secondary instability of most-unstable primary wave is also studied. The secondary growth-rate is many fold higher when compared with that of primary wave and found to be phase-locked. The fundamental mode is more unstable than subharmonic or detuned modes. The secondary growth is studied by varying the parameters such as β, Re, M and the detuning parameter.

In this work, a DNS code was developed to investigate problems on transition in subsonic compressible boundary layer on a flat plate. Code validation tests were performed for linear and nonlinear stages of transition, on incompressible and compressible regimes. The focus of the present work is to investigate natural transition in subsonic boundary layers modeled by wave packets; and perform a preliminary study of transition induced by white noise. Three main problems were considered, namely, a DNS simulation and analysis of the ex- periment (MEDEIROS; GASTER, 1999b) of wave packet evolution on incompressible boundary layer, the influence of compressibility on wave packet evolution at subsonic Mach numbers and finally, a preliminary study of the evolution of a white noise perturbation in the boundary layer at Mach 0.2 and Mach 0.9. Comparisons between numerical and experimental results show remarkably good agreement in the linear and nonlinear stages, in both, spatial and Fourier spaces. A numerical simulation of this experiment and the analysis carried out is not available in the literature for wave packets in the incompressible boundary layer. The nonlinear modal analysis performed established the existence of tuned fundamental and subharmonic resonance of H-type and K-type in the packet. Influence of compressibility in the wave packet evolution was here investigated in boundary layers at Mach 0.7 and Mach 0.9. There are no works reported in the literature on wave packets in compressible subsonic boundary layer. In the linear regime, the oblique modes were the most unstable for Mach > 0.7, as expected by the results of the literature. In the nonlinear regime, strong streaks were observed, associated with low frequency modes that eventually decay downstream. An isolated wave packet at Mach 0.9 showed nonlinear amplification only in the subharmonic band, which may be associated to H-type or detuned resonance. However this packet has a relatively stable character. On the other hand, at Mach 0.9 spanwise interaction of wave packet pairs were more unstable than the isolated case, because stable modes for the isolated packet evolution becomes unstable in the wave packet interaction. This scenario evidenced the presence of oblique transition. Finally, the nonlinear evolution of the same white noise disturbance at Mach 0.2 and Mach 0.9 were observed to be completely different. In the incompressible boundary layer localized lambda vortex structures were observed, that could be associated to the local presence of H-type and/or K-type resonance. In the compressible regime, longitudinal vortex structures distributed across the entire domain seemed to be linked to oblique transition. In the white noise evolution, compressibility seems to have a stronger effect than in the wave packet evolution. In the conditions considered, the wave packet interaction appear to be a better representation of white noise compressible transition scenario. / No presente trabalho, um código DNS (Direct Numerical Simulation) foi desenvolvido para abordar problemas de transição para turbulência em camada limite subsônica compressível em uma placa plana. Foram realizados testes de validação de código , nos regimes linear e não linear do processo de transição, nos regimes incompressível e compressível. O foco do presente trabalho é estudar transição natural modelada por meio de pacotes de onda em camada limite compressível subsônica, e realizar uma análise preliminar da transição induzida por ruído branco. Três assuntos principais foram considerados: uma simulação DNS e uma análise comparativa com o experimento (MEDEIROS; GASTER, 1999b) sobre a evolução de um pacote de ondas em camada limite incompressível, a influência da compressibilidade na evolução de pacotes de ondas no regime subsônico, e por último, um estudo preliminar da transição induzida por ruído branco em Mach 0.2 e Mach 0.9. As comparações realizadas entre a solução numérica e os dados experimentais mostram uma boa concordância, nos regimes linear e não linear, tanto no espaço físico quanto no espaço de Fourier. A simulação numérica deste experimento e a análise realizada neste trabalho, não são encontradas na literatura para o regime incompressível. A análise modal não linear aplicada aos resultados, permitiu identificar a presença das ressonâncias tipo H e tipo K no pacote de ondas. A influência da compressibilidade na evolução dos pacotes de onda foi estudada em Mach 0.7 e Mach 0.9. Na literatura não há trabalhos sobre pacotes de ondas no regime sub- sônico. No regime linear da transição, os modos oblíquos resultam ser os mais instáveis para Mach > 0.7, como era de esperar, de acordo com os resultados da literatura. No regime não linear, foram observadas estrias de moderada amplitude, associadas com modos de baixa frequência que acabam decaindo. O pacote de ondas em Mach 0.9 apresentou amplificação não linear somente na banda subharmônica, que pode ser associada com transição tipo H ou ressonância dessintonizada. No entanto, o comportamento geral neste regime é estabilizante. Por sua vez, a interação entre pacotes de ondas em Mach 0.9 mostrou um comportamento desestabilizante, pois a interação acaba gerando amplificação não linear em modos que decaem no pacote isolado. Os modos amplificados sugerem a presença do mecanismo de transição oblíqua. Finalmente, a evolução da mesma perturbação constituída por ruído branco em Mach 0.2 e Mach 0.9, resultaram ser completamente diferentes. Na camada limite incompressível foram observados vórtices tipo lambda, que poderiam ser gerados pela presença localizada das ressonâncias tipo H e/ou tipo K. No regime compressível foram observados vórtices distribuidos em todo o domínio, o que sugere a presença da transição oblíqua. Na transição gerada por ruído branco a compressibilidade teve uma influência maior que no pacote de ondas. Nas condições estudadas, a interação entre pacotes de ondas parece ser uma melhor representação do ruído branco no regime compressível.