Numerical modelling and simulation are powerful tools for analysis and design, and with the improvement of computational power and numerical methods they are being applied on complex phenomena and systems. This work shows examples of the application of a very sophisticated numerical method, namely the Spectral/hp element method, in the study of the flow inside bioinspired complex geometries. The two topics investigated are fluid dynamic instabilities in a constricted channel and flow inside a paediatric ventricular assist device were studied by means of computational fluid mechanics. The constricted channel is an idealized model of a nasal cavity, which is characterized by complex airway channels, and also bears some resemblance to a human artery in the presence of an atherosclerotic plaques. The paediatric ventricular assist device is an actual device, designed by the Bioengineering research group of the Heart Institute of the Medicine School of the University of São Paulo, which works as a pump that assists the left ventricle of patients waiting for transplantation. Therefore, the aim of this thesis is to contribute in the understanding of biological and bio-inspired geometries flows, using computational tools. Linear and nonlinear stability were carried out for the constricted channel. Three different flow regimes were investigated: symmetric steady flow, which is stable for low Reynolds number, asymmetric steady flow, which rises as a result of the primary bifurcation of the symmetric flow and pulsatile flow. Direct stability analysis was carried out to determine the unstable regions and the critical values for each flow regime. The physical mechanisms behind the transition processes were studied by means of direct numerical simulations to characterize the bifurcations. Since the bifurcations had subcritical behaviour, the relevance of non-normal growth in these flows was assessed. Dependence on phase, Reynolds number and spanwise wavenumber of optimal modes were extensively investigated in stable regions of the three flow regimes. Convective instabilities were also studied in order to comprehend the physical mechanisms which led the optimal modes to their maxima growth, and different convective mechanisms were found. The flow inside the paediatric ventricular assist device was analyzed by means of threedimensional numerical simulations. A computational model based on special boundaries conditions was developed to model the pulsatile flow. In this model, the opening and closure of the mitral valve and diaphragm were represented with the use of specially devised boundary conditions. The driving force and the flow direction of the diaphragm were defined by velocity distribution on the diaphragm wall, and the opening and closure of the mitral valve were performed by a velocity waveform which goes to zero in the systolic period. Flow patterns, velocity fields and time-average wall shear rate were analyzed to evaluate the performance of the device. / Modelagem e simulação numéricas são ferramentas poderosas para análise e design, e com a melhoria do poder computacional e dos métodos numéricos, eles estão sendo aplicados em fenômenos e sistemas complexos. Este trabalho mostra exemplos de aplicações de um método numérico sofisticado, o método dos elementos espectrais/hp, no estudo do escoamento dentro de geometrias complexas bio-inspiradas. Os dois tópicos investigados são: instabilidades em dinâmica de fluido em um canal constrito e o escoamento dentro de um dispositivo de assistência ventricular pediátrica. O canal constrito é um modelo idealizado de uma cavidade nasal, que é caracterizada por canais complexos da via aérea, e também tem semelhança com uma artéria humana na presença de placas ateroscleróticas. O dispositivo de assistência ventricular pediátrica é um dispositivo real, projetado pelo grupo de pesquisa de Bioengenharia do Instituto do Coração da Faculdade de Medicina da Universidade de São Paulo, que funciona como uma bomba que auxilia o ventrículo esquerdo dos pacientes à espera de transplante. Portanto, o objetivo desta tese é contribuir na compreensão de escoamentos em geometrias biológicas e bio-inspiradas, usando ferramentas computacionais. Análises de estabilidade linear e não linear foram feitas para um canal constrito. Três diferentes regimes de escoamento foram empregados: escoamento estacionário simétrico, que é estável para baixo número de Reynolds, escoamento assimétrico, o qual é resultado da primeira bifurcação do escoamento simétrico e escoamento pulsátil. Análise de estabilidade direta foi executada para determinar as regiões instáveis em cada regime de escoamento. Os mecanismos físicos por trás do processo de transição foram estudados por meio de simulação numérica direta para caracterizar as bifurcações. Uma vez que, as bifurcações tiveram um comportamento subcrítico, a relevância do crescimento não normal nestes escoamentos foi avaliado. Assim, dependência com a fase, número de Reynolds e número de onda do modo tridimensional foram extensivamente investigados em regiões estáveis para os três regimes de escoamento. Instabilidades convectivas foram também estudadas a fim de compreender os mecanismos físicos que conduzem os modos ótimos para seus crescimentos máximos, e diferentes mecanismos convectivos foram encontrados. O escoamento dentro do dispositivo de assistência ventricular pediátrico foi analisado por meios de simulações numéricas tridimensionais. Um modelo computacional baseado em condições de contorno especiais foi desenvolvido para modelar o escoamento pulsátil. Neste modelo, a abertura e fechamento da válvula mitral e diafragma foram representados com o uso de condições de contorno especialmente elaboradas. A força motora e o direcionamento do fluxo do diafragma foram definidos por uma distribuição de velocidades na parede do diafragma, e a abertura e fechamento da válvula mitral foram executadas por uma função de onda de velocidade que vai a zero no período sistólico. Padrões do escoamento, campos de velocidade e tensão de cisalhamento no tempo foram analisadas para avaliar o desempenho do dispositivo.

As simulações de escoamentos em torno de geometrias de aplicações industriais (geometrias complexas), como por exemplo configurações de aeronaves com hipersustentadores defletidos, apresentam uma vasta gama de estruturas vorticais (complexidade do escoamento). A importância das interações entre as estruturas é grande para a correta previsão da dinâmica das estruturas vorticais presentes no escoamento, uma vez que estas interações ditam as características do processo de transferência de energia cinética turbulenta. Vale ressaltar que no processo de transferência de energia cinética turbulenta não temos uma única direção e sim a possibilidade de duas direções, que representam o processo de cascata direta ou clássica (a transferência de energia cinética turbulenta se dá das maiores estruturas vorticais para as menores - forward scatter ) e a cascata indireta (a transferência de energia cinética turbulenta que se dá das menores estruturas vorticais para as maiores - backscatter ). O balanço entre estes dois processos, direto e indireto, resulta na dominância do processo direto, ou seja, o processo dominante de transferência de energia se dá das maiores estruturas vorticais para as menores. Entretanto, ambos os processos devem estar presentes na solução numérica, para que esta seja capaz de prever de forma correta a dinâmica (interações entre estruturas vorticais de tamanhos variados) presente no escoamento. Os modelos convencionais utilizados no tratamento da turbulência (ou fechamento da turbulência), sejam do tipo RANS (Reynolds Average Navier Stokes ) ou do tipo LES (Large Eddy Simulation) apresentam limitações teóricas (modelo não é capaz de representar as interações entre todas as escalas presentes no escoamento) e práticas (necessidade de discretização espacial que aumenta significativamente o custo computacional). No caso dos modelos LES a malha nas proximidades de paredes sólidas deveriam ser extremamente refinadas, o que resulta em praticamente resolver todas as escalas, para representar os efeitos da cascata direta (forward scatter ) e da cascata indireta (backscatter ) de energia cinética turbulenta. Isto ocorre em decorrência do caráter dissipativo dos modelos submalha utilizadas nas formulações LES. Por este motivo, o presente trabalho tem por objetivo desenvolver uma metodologia para solução do escoamento turbulento que seja capaz de apresentar os processos de cascata direta e cascata indireta sem a necessidade de malhas extremamente refinadas. Para tanto, iremos utilizar as equações Navier-Stokes escritas em função das flutuações (flutuações resolvidas), sendo esta formulação baseada nos trabalhos de Morris et al. [1997], Labourasse e Sagaut [2002] e Batten et al. [2004b]. As equações são obtidas por meio da divisão dos campos em uma média temporal, flutuações resolvidas e flutuações submalha. Sendo a média temporal, obtida previamente por meio de uma solução RANS do escoamento, que no nosso caso é obtida com o modelo RANS SA-QCR2013 proposto por Mani et al. [2013]. As flutuações resolvidas são o resultado da solução numérica das equações obtidas com a discretização espacial dada pela malha utilizada. Por fim as flutuações submalha são introduzidas via modelo de Billson [2004] (modelo de sintetização ou reconstrução da turbulência). Esta formulação foi aplicada para solução do escoamento em um canal formado por paredes paralelas com Re? = 395 e Re? = 1000. Estes números de Reynolds foram escolhidos por existirem resultados obtidos via DNS ou até mesmo resultados experimentais disponíveis na literatura, os resultados são enconstrados em Moser et al. [1999], del Álamo et al. [2004] e Schultz e Flack [2013]. Os resultados obtidos com o modelo proposto mostraram que a cascata inversa (backscatter ) está presente em todas as regiões da camada limite (subcamada laminar, buffer layer e logarítmica) do canal, onde o pico de transferência ocorre, para os números de Reynolds avaliados, na região da buffer layer. Este comportamento foi observado nos resultados gerados por todas as malhas avaliadas, a diferenças entre as malhas está no refinamento na região próxima às paredes sólidas. O refinamento da malha na direção da altura do canal (normal às paredes sólidas) faz com que o balanço entre as taxas de dissipação de energia cinética turbulenta passe a indicar a dominância da cascata direta no processo de transferência de energia. Nas malhas menos refinadas na região próxima à parede temos o domínio da cascata indireta no processo de transferência de energia cinética turbulenta. A introdução das flutuações submalha via modelo de sintetização da turbulência leva a uma tendência de inverter o domínio da cascata inversa (backscatter ) nas malhas menos refinadas. Os resultados obtidos com a metodologia NLDE com flutuações turbulentas submalha introduzidas por meio de modelo de sintetização turbulenta apresentam boa concordância com os respectivos resultados obtidos via DNS e ou experimentais. / Simulations of flows around industrial geometries (complex geometries), such as configurations of aircraft with deployed high-lift surface, present a wide range of vortical structures (flow complexity). The importance of the interactions between the structures is great for the correct prediction of the dynamics of the vortical structures present in the flow since these interactions dictate the characteristics of the turbulent kinetic energy transfer process. It is noteworthy that in the process of transferring turbulent kinetic energy we do not have a single direction but the possibility of two directions, which represent the direct cascade or classical cascade process (the transfer of turbulent kinetic energy occurs from the large eddy to small eddy - the forward scatter) and the reverse cascade (the transfer of turbulent kinetic energy occurs from small eddy to the large eddy - backscatter). The net balance between these two processes, direct and reverse, results in the predominance of the direct process, that is, the dominant process of energy transfer occurs from the largest eddy to the smaller ones. However, both processes must be present in a numerical solution, so that it is able to predict correctly the dynamics (interactions between vortical structures of varying sizes) present in the flow. The conventional models used in turbulence treatment (or turbulence closure), whether of the RANS (Reynolds Average Navier Stokes) type or the LES (Large Eddy Simulation) type have theoretical limitations (model is not able to represent the interactions between the scales present in the flow) and practices (needs spatial discretization that signifcantly increases the computational cost). In the case of LES models, the mesh close to solid walls should be extremely refined, which results in practically resolving all scales to represent the effects of the forward scatter and the backscatter of turbulent kinetic energy. This is due to the dissipative character of the sub-grid models used in the LES formulations. For this reason, the present research effort aims to develop a methodology for solving turbulent flow, that is able to present both energy transfer process, forward scatter and backscatter without the need of extremely refined meshes. For this, we will use the Navier-Stokes equations written in function of the fluctuations (resolved fluctuations), being this formulation based on the works of Morris et al. [1997], Labourasse e Sagaut [2002] and Batten et al. [2004b]. The equations are obtained by dividing the fields into an average time, resolved fluctuations and sub-grid fluctuations. The time-averaged, obtained previously by means of a RANS solution of the flow, which in our case is obtained with the model RANS SA-QCR2013 proposed by Mani et al. [2013]. The resolved fluctuations are the result of the numerical solution of the equations obtained with the spatial discretization given by the mesh used. Finally, the sub-grid turbulence fluctuations are introduced via the model of Billson [2004] (model for synthesizing or reconstructing turbulence). This formulation was applied to solve of the flow in a channel formed by parallel walls at Re? = 395 and Re? = 1000. The reason to choose those Reynolds number is related to the fact that there are results obtained via DNS or even experimental results available in the literature, one can found those results in Moser et al. [1999], del Álamo et al. [2004] and Schultz e Flack [2013]. The results obtained with the proposed model showed that the backscatter is present in all regions of the boundary layer (lamellar layer, buffer layer, and log-layer) of the channel, where the transfer peak occurs, for the evaluated Reynolds numbers, in the region of the buffer layer. This behavior was observed in the results generated by all meshes evaluated, the differences between the meshes are in the refinement in the region near the solid walls. The refinement of the mesh in the direction of the channel height (normal to the solid walls) causes the balance between the rates of dissipation of turbulent kinetic energy to indicate the dominance of the direct cascade in the energy transfer process. In the less refined meshes in the region near the wall, we have the domain of the indirect cascade in the process of transfer of turbulent kinetic energy. The introduction of the sub-grid fluctuations via the turbulence synthesizing model leads to a tendency to invert the reverse cascade domain (backscatter) in the solutions obtained with the coarsest grid. The results obtained with the NLDE turbulence, in which we use a synthetic turbulence model to introduce subgrid turbulent fluctuations, show good agreement with DNS results and or experimental results.

As simulações de escoamentos em torno de geometrias de aplicações industriais (geometrias complexas), como por exemplo configurações de aeronaves com hipersustentadores defletidos, apresentam uma vasta gama de estruturas vorticais (complexidade do escoamento). A importância das interações entre as estruturas é grande para a correta previsão da dinâmica das estruturas vorticais presentes no escoamento, uma vez que estas interações ditam as características do processo de transferência de energia cinética turbulenta. Vale ressaltar que no processo de transferência de energia cinética turbulenta não temos uma única direção e sim a possibilidade de duas direções, que representam o processo de cascata direta ou clássica (a transferência de energia cinética turbulenta se dá das maiores estruturas vorticais para as menores - forward scatter ) e a cascata indireta (a transferência de energia cinética turbulenta que se dá das menores estruturas vorticais para as maiores - backscatter ). O balanço entre estes dois processos, direto e indireto, resulta na dominância do processo direto, ou seja, o processo dominante de transferência de energia se dá das maiores estruturas vorticais para as menores. Entretanto, ambos os processos devem estar presentes na solução numérica, para que esta seja capaz de prever de forma correta a dinâmica (interações entre estruturas vorticais de tamanhos variados) presente no escoamento. Os modelos convencionais utilizados no tratamento da turbulência (ou fechamento da turbulência), sejam do tipo RANS (Reynolds Average Navier Stokes ) ou do tipo LES (Large Eddy Simulation) apresentam limitações teóricas (modelo não é capaz de representar as interações entre todas as escalas presentes no escoamento) e práticas (necessidade de discretização espacial que aumenta significativamente o custo computacional). No caso dos modelos LES a malha nas proximidades de paredes sólidas deveriam ser extremamente refinadas, o que resulta em praticamente resolver todas as escalas, para representar os efeitos da cascata direta (forward scatter ) e da cascata indireta (backscatter ) de energia cinética turbulenta. Isto ocorre em decorrência do caráter dissipativo dos modelos submalha utilizadas nas formulações LES. Por este motivo, o presente trabalho tem por objetivo desenvolver uma metodologia para solução do escoamento turbulento que seja capaz de apresentar os processos de cascata direta e cascata indireta sem a necessidade de malhas extremamente refinadas. Para tanto, iremos utilizar as equações Navier-Stokes escritas em função das flutuações (flutuações resolvidas), sendo esta formulação baseada nos trabalhos de Morris et al. [1997], Labourasse e Sagaut [2002] e Batten et al. [2004b]. As equações são obtidas por meio da divisão dos campos em uma média temporal, flutuações resolvidas e flutuações submalha. Sendo a média temporal, obtida previamente por meio de uma solução RANS do escoamento, que no nosso caso é obtida com o modelo RANS SA-QCR2013 proposto por Mani et al. [2013]. As flutuações resolvidas são o resultado da solução numérica das equações obtidas com a discretização espacial dada pela malha utilizada. Por fim as flutuações submalha são introduzidas via modelo de Billson [2004] (modelo de sintetização ou reconstrução da turbulência). Esta formulação foi aplicada para solução do escoamento em um canal formado por paredes paralelas com Re? = 395 e Re? = 1000. Estes números de Reynolds foram escolhidos por existirem resultados obtidos via DNS ou até mesmo resultados experimentais disponíveis na literatura, os resultados são enconstrados em Moser et al. [1999], del Álamo et al. [2004] e Schultz e Flack [2013]. Os resultados obtidos com o modelo proposto mostraram que a cascata inversa (backscatter ) está presente em todas as regiões da camada limite (subcamada laminar, buffer layer e logarítmica) do canal, onde o pico de transferência ocorre, para os números de Reynolds avaliados, na região da buffer layer. Este comportamento foi observado nos resultados gerados por todas as malhas avaliadas, a diferenças entre as malhas está no refinamento na região próxima às paredes sólidas. O refinamento da malha na direção da altura do canal (normal às paredes sólidas) faz com que o balanço entre as taxas de dissipação de energia cinética turbulenta passe a indicar a dominância da cascata direta no processo de transferência de energia. Nas malhas menos refinadas na região próxima à parede temos o domínio da cascata indireta no processo de transferência de energia cinética turbulenta. A introdução das flutuações submalha via modelo de sintetização da turbulência leva a uma tendência de inverter o domínio da cascata inversa (backscatter ) nas malhas menos refinadas. Os resultados obtidos com a metodologia NLDE com flutuações turbulentas submalha introduzidas por meio de modelo de sintetização turbulenta apresentam boa concordância com os respectivos resultados obtidos via DNS e ou experimentais. / Simulations of flows around industrial geometries (complex geometries), such as configurations of aircraft with deployed high-lift surface, present a wide range of vortical structures (flow complexity). The importance of the interactions between the structures is great for the correct prediction of the dynamics of the vortical structures present in the flow since these interactions dictate the characteristics of the turbulent kinetic energy transfer process. It is noteworthy that in the process of transferring turbulent kinetic energy we do not have a single direction but the possibility of two directions, which represent the direct cascade or classical cascade process (the transfer of turbulent kinetic energy occurs from the large eddy to small eddy - the forward scatter) and the reverse cascade (the transfer of turbulent kinetic energy occurs from small eddy to the large eddy - backscatter). The net balance between these two processes, direct and reverse, results in the predominance of the direct process, that is, the dominant process of energy transfer occurs from the largest eddy to the smaller ones. However, both processes must be present in a numerical solution, so that it is able to predict correctly the dynamics (interactions between vortical structures of varying sizes) present in the flow. The conventional models used in turbulence treatment (or turbulence closure), whether of the RANS (Reynolds Average Navier Stokes) type or the LES (Large Eddy Simulation) type have theoretical limitations (model is not able to represent the interactions between the scales present in the flow) and practices (needs spatial discretization that signifcantly increases the computational cost). In the case of LES models, the mesh close to solid walls should be extremely refined, which results in practically resolving all scales to represent the effects of the forward scatter and the backscatter of turbulent kinetic energy. This is due to the dissipative character of the sub-grid models used in the LES formulations. For this reason, the present research effort aims to develop a methodology for solving turbulent flow, that is able to present both energy transfer process, forward scatter and backscatter without the need of extremely refined meshes. For this, we will use the Navier-Stokes equations written in function of the fluctuations (resolved fluctuations), being this formulation based on the works of Morris et al. [1997], Labourasse e Sagaut [2002] and Batten et al. [2004b]. The equations are obtained by dividing the fields into an average time, resolved fluctuations and sub-grid fluctuations. The time-averaged, obtained previously by means of a RANS solution of the flow, which in our case is obtained with the model RANS SA-QCR2013 proposed by Mani et al. [2013]. The resolved fluctuations are the result of the numerical solution of the equations obtained with the spatial discretization given by the mesh used. Finally, the sub-grid turbulence fluctuations are introduced via the model of Billson [2004] (model for synthesizing or reconstructing turbulence). This formulation was applied to solve of the flow in a channel formed by parallel walls at Re? = 395 and Re? = 1000. The reason to choose those Reynolds number is related to the fact that there are results obtained via DNS or even experimental results available in the literature, one can found those results in Moser et al. [1999], del Álamo et al. [2004] and Schultz e Flack [2013]. The results obtained with the proposed model showed that the backscatter is present in all regions of the boundary layer (lamellar layer, buffer layer, and log-layer) of the channel, where the transfer peak occurs, for the evaluated Reynolds numbers, in the region of the buffer layer. This behavior was observed in the results generated by all meshes evaluated, the differences between the meshes are in the refinement in the region near the solid walls. The refinement of the mesh in the direction of the channel height (normal to the solid walls) causes the balance between the rates of dissipation of turbulent kinetic energy to indicate the dominance of the direct cascade in the energy transfer process. In the less refined meshes in the region near the wall, we have the domain of the indirect cascade in the process of transfer of turbulent kinetic energy. The introduction of the sub-grid fluctuations via the turbulence synthesizing model leads to a tendency to invert the reverse cascade domain (backscatter) in the solutions obtained with the coarsest grid. The results obtained with the NLDE turbulence, in which we use a synthetic turbulence model to introduce subgrid turbulent fluctuations, show good agreement with DNS results and or experimental results.

Particle laden turbulent flows find applications in many industrial processes such as energy conversion, air pollution control etc. In these types of flows, there are strong coupling between the turbulent fluctuations in the fluid velocity fields, and the fluctuating velocities of the particles. In order to analyze the stresses and the heat and mass transfer properties in turbulent suspensions, it is necessary to have a good understanding of not just the mean flow of the gas and particles, but also of the fluctuations in the two phases. The coupling is a two-way coupling; the fluid turbulence contributes to the velocity fluctuations in the particles, and conversely, the particle velocity fluctuations generate fluctuations in the fluid. Two-phase flow models capture these interactions only in an indirect way, usually through a ‘particle pressure’ term for the particle phase. In the present work the effect of fluid velocity fluctuations on the dynamics of the particles in a turbulent gas-solid suspension is analyzed in the low Reynolds number and high Stokes number limit, where the particle relaxation time is long compared to the correlation time for the fluid velocity fluctuations. The direct numerical simulation (DNS) is used for solving the Navier-Stokes equations for the fluid, the particles are modeled as hard spheres which undergo elastic collisions. A one-way coupling algorithm is used where the force exerted by the fluid on the particles is incorporated, but not the reverse force exerted by the particles on the fluid. This is because the main focus of our study is to examine the effect of the fluid turbulence on the particle fluctuations, and we are interested in examining whether a Langevin model with random forcing can accurately capture the effect of fluid turbulence on the particle phase. First, the turbulent flow in a plane Couette is analyzed. Though this is a model flow which is not encountered often in applications, it is easier to analyze because the turbulent velocity fluctuations are maximum at the center of the channel, in contrast to the Poiseuille flow, where the velocity fluctuations are maximum at a location between the center and the wall. Also, in a Couette flow, the wall-normal and the spanwise root mean square velocities are nearly a constant in the central region in the channel, and the percentage variation in the stream-wise velocity fluctuations is also less than that in a pressure driven Poiseuille flow. Therefore, it is possible to treat the central region as a region with homogeneous, but anisotropic, fluid velocity fluctuations and with a linear mean velocity variation. The particle mean and root mean square fluctuating velocities, as well as the probability distribution function for the fluid velocity fluctuations and the distribution of acceleration of the particles in the central region of the Couette, which comprises about 20% of the entire channel have been studied. It is found that the distribution of particle velocities is very different from a Gaussian, especially in the span-wise and wall-normal directions. However, the distribution of the acceleration fluctuation on the particles is found to be close to a Gaussian, though the distribution is highly anisotropic and there is a correlation between the fluctuations in the flow and gradient directions. The non-Gaussian nature of the fluid velocity fluctuations is found to be due to inter-particle collisions induced by the large particle velocity fluctuations in the flow direction. Another interesting result is a comparison of the distribution of the acceleration on a particle due to the fluid velocity fluctuation at the particle position, and the distribution of the ratio of fluid velocity fluctuation to the viscous relaxation time in the fluid. The comparison shows that these two distributions are almost identical, indicating that the fluid velocity fluctuations are not correlated over time scales comparable to the relaxation time of a particle. This result is important because it indicates that in order to model the fluctuating force on the particle, it is sufficient to obtain the variance of the force distribution from the variance of the fluid velocity distribution function. Finally, the correlation time for the acceleration correlations is calculated along the trajectory of a particle. The correlation time is found to be of the same magnitude as the correlation time for the fluid velocity in an Eulerian reference frame, and much smaller than the viscous relaxation time and the time between collisions of the particles. All of these results indicate that the effect of the turbulent fluid velocity fluctuations can be accurately represented by an anisotropic Gaussian white noise. The above results are used to formulate a ‘fluctuating force’ model for the particle phase alone, where the force exerted by the fluid turbulent velocity fluctuations is modeled as random Gaussian white noise, which is incorporated into the equation of motion for the particles. The variance of the distribution function for the fluctuating force distribution is obtained from the variance of the local turbulent fluid velocity fluctuations, assuming linear Stokes drag law. The force distribution is anisotropic, and it has a non-zero correlation between the flow and gradient directions. It is found that the results of the fluctuating force simulations are in quantitative agreement with the results of the complete DNS, both for the particle concentration and variances of the particle velocity fluctuations, at relatively low volume fractions where the viscous relaxation time is small compared to the time between collisions, as well as at higher volume fractions where the time between collisions is small compared to the viscous relaxation time. The simulations are also able to predict the velocity distributions in the center of the Couette, even in cases where the velocity distribution is very different from a Gaussian distribution. The fluctuating force model is applied to the turbulent flow of a gas-particle suspension in a vertical channel in the limit of high Stokes number. In contrast to the Couette flow analyzed the fluid velocity variances in the different directions in the channel are highly non-homogeneous, and they exhibit a significant variation across the channel. First, we analyze the fluctuating particle velocity and acceleration distributions at different locations across the channel using direct numerical simulation. The distributions are found to be non-Gaussian near the center of the channel, and they exhibit significant skewness. The time correlations of the fluid velocity fluctuations and the acceleration fluctuations on the particles are evaluated and compared. Unlike the case of Couette flow it is found that the time correlation functions for the fluid in the fixed Eulerian frame are not in agreement with the time correlation of the acceleration on the particles. However, the time correlations of the particle acceleration are in good agreement with the velocity time correlations in the fluid in a ‘moving Eulerian’ reference frame, moving with the mean velocity of the fluid. The fluctuating force simulations are used to model the particle phase, where the force on the particles due to the fluid velocity fluctuations are substituted by random white noise in the equations for the particle motion. The random noise is assumed to be Gaussian and anisotropic. The variances of the fluctuating force are calculated form the fluid velocity fluctuations in a moving Eulerian reference frame using DNS. The results from the fluctuating force simulations are then compared with the results obtained from DNS. Quantitative agreement between the two simulations are obtained provided the particle viscous relaxation time is at least five times larger than the fluid integral time. The interactions between the solid particles and the fluid turbulence have been investigated experimentally in a vertical fully developed channel flow of air and solid particles. Experiments are conducted at low volume fraction for which viscous relaxation time of the particle is expected to be lower than the particle particle collision time, as well as at moderately high volume fraction where the particle particle collision time is expected to be lower than the particle relaxation time. Velocity statistics of both the particle and gas phases are obtained using high spatial resolution Particle Image Velocimetry (PIV) system. It is observed that at low solid volume fraction, the particle root mean square velocities and the velocity distribution are in good agreement with those predicted by the fluctuating force simulation, provided the polydispersity in the particle size distribution is incorporated in the fluctuating force simulations. In this case, the modification of turbulence in the center of the channel due to the particles is small. At much higher volume fraction, the mean gas flow is significantly affected by the presence of particles, and the mean flow is no longer symmetric about the center line of the channel. Simultaneously, there is also a significant change in the volume fraction across the channel, and the volume fraction is also not symmetric about the center line. This seems to indicate that there is a spontaneous instability of the symmetric volume fraction and velocity profiles, giving rise to a region of high fluid velocity and high particle volume fraction coexisting with a region of low gas velocity and low particle volume fraction. There is some recirculation of the gas within the channel, and the gas phase turbulence intensity is significantly enhanced when the velocity and volume fraction profiles become asymmetric. As we have considered only one way coupling in the computation of the particle laden flow it is expected that the particle statistics obtained for this condition can not be predicted by our fluctuating force model due to modification of the gas phase statistics.

Control of the transition of laminar flow to turbulence would result in lower drag and reduced energy consumption in many engineering applications. A spectral state-space model of linearised plane Poiseuille flow with wall transpiration ac¬tuation and wall shear measurements is developed from the Navier-Stokes and continuity equations, and optimal controllers are synthesized and assessed in sim¬ulations of the flow. The polynomial-form collocation model with control by rate of change of wall-normal velocity is shown to be consistent with previous interpo¬lating models with control by wall-normal velocity. Previous methods of applying the Dirichlet and Neumann boundary conditions to Chebyshev series are shown to be not strictly valid. A partly novel method provides the best numerical behaviour after preconditioning. Two test cases representing the earliest stages of the transition are consid¬ered, and linear quadratic regulators (LQR) and estimators (LQE) are synthesized. Finer discretisation is required for convergence of estimators. A novel estimator covariance weighting improves estimator transient convergence. Initial conditions which generate the highest subsequent transient energy are calculated. Non-linear open- and closed-loop simulations, using an independently derived finite-volume Navier-Stokes solver modified to work in terms of perturbations, agree with linear simulations for small perturbations. Although the transpiration considered is zero net mass flow, large amounts of fluid are required locally. At larger perturbations the flow saturates. State feedback controllers continue to stabilise the flow, but estimators may overshoot and occasionally output feedback destabilises the flow. Actuation by simultaneous wall-normal and tangential transpiration is derived. There are indications that control via tangential actuation produces lower highest transient energy, although requiring larger control effort. State feedback controllers are also synthesized which minimise upper bounds on the highest transient energy and control effort. The performance of these controllers is similar to that of the optimal controllers.

Tyrolean type water-intake structures are widely used on mountain rivers to provide water to hydropower stations. The main concern encountered in these kinds of structures is the amount of water diverted from the main channel and sediment carried by this flow. The diverted flow should not be less than the design discharge of the hydropower station and the amount of the sediment entering the hydraulic system should be minimum. In this study a physical model of a Tyrolean weir was constructed at the Hydromechanics Laboratory and by varying the dimensions of the various components of this system / the length, the inclination angle of the rack and the distance between the rack bars, the amount of water and sediment entering the system were measured for a wide range of discharges. The experiments were conducted in two stages. In the first stage the tests were made with only water, and in the second stage, water and sediment having different gradation were used. Applying dimensional analysis to the related parameters of the system dimensionless terms were defined for water capture efficiency and discharge coefficient of the system, and their variations with the relevant parameters were plotted. Using these diagrams one can determine the amount of water to be diverted by a Tyrolean weir of known geometry and main channel flow.

Tyrolean type water-intake structures are commonly used on mountain rivers to supply water to hydropower stations. The amount of water to be diverted from the main channel is the major concern in these kind of structures and should not be less than the design discharge. In this study a physical model of a Tyrolean type water-intake structure was built at the laboratory and the diverted flow from the main channel through the intake structure having steel racks and perforated plates of different types were measured. The experiments were conducted in two stages. In the first stage the tests were carried out with only steel racks having three different bar openings and slopes, and in the second stage, perforated screens of three different circular openings and screen slopes were used. Applying dimensional analysis to the related parameters of the system the dimensionless terms were defined for the water capture efficiency and discharge coefficient of the system, and their variations with the relevant parameters were plotted. Using these diagrams one can determine the amount of water to be diverted by a Tyrolean weir of known geometry and main channel discharge.

This study is an experimental research to formulate the discharge over sharp-crested rectangular weirs. Firstly, a series of measurements on different weir heights were conducted to find the minimum weir height for which channel bed friction has no effect on discharge capacity. After determining the appropriate weir height, weir width was reduced to collect data on discharge-water head over weir relationship for a variety of different weir openings. Then, the data was analyzed through regression analysis along with utilization of global optimization technique to reach the desired formulation for the discharge. By taking advantage of a newly-introduced &ldquo / weir velocity&rdquo / concept, a simple function was eventually detected for the discharge where no discharge coefficient was involved. The behavior of the weir velocity function obtained in the present study illustrates the transition between the fully contracted and partially contracted weirs. In addition, the proposed weir velocity formulation is simple and robust to calculate the discharge for full range of weir widths.

A generalized non-dimensional mathematical expression has been developed to describe the rating relation of depth and discharge for intermediate and high streamflow of natural and controlled streams. The expressions have been tested against observations from forty-three stations in West-Central Florida. The intermediate-flow region model has also been validated using data from thirty additional stations in the study area. The proposed model for the intermediate flow is a log-linear equation with zero intercept and the proposed model for the high-flow region is a log-linear equation with a variable intercept. The models are normalized by the depth and discharge values at 10 percent exceedance using data published by the U.S. Geological Survey. For un-gauged applications, Q10 and d10 were derived from a relationship shown to be reasonably well correlated to the watershed drainage area with a correlation coefficient of 0.94 for Q10 and 0.86 for d10. The average relative error for this parameter set shows that, for the intermediate-flow range, better than 50% agreement with the USGS rating data can be expected for about 86% of the stations and for the high-flow range, better than 50% for 44% of the stations. Testing the model outside West Central Florida, in some stations at North Florida, and South Alabama and Georgia, show some reasonable relative errors but not as good as the results obtained for West Central Florida. Using a model with a different slope, developed specific for those particular stations improved the results significantly.

Στην παρούσα διατριβή μελετάται η ροή βιομαγνητικών ρευστών υπό την επίδραση μαγνητικού πεδίου. Ως βιομαγνητικό ορίζεται ένα ρευστό το οποίο υπάρχει σε έναν έμβιο οργανισμό και η ροή του επηρεάζεται πάντοτε από την παρουσία μαγνητικού πεδίου. Χαρακτηριστικό βιομαγνητικό ρευστό θεωρείται το αίμα και αυτό χρησιμοποιείται για να δωθούν τιμές στις παραμέτρους που εμφανίζονται στα προβλήματα που μελετώνται.... / - / The flow of biomagnetic fluids in the presence of an applied magnetic field is studied in the present thesis. As biomagnetic is defined a fluid that exists in a living creature (biofluid) and its flow is affected by the presence of a magnetic field. The most characteristic biofluid is the blood. The Newtonian viscous laminar incompressible blood flow is considered in the present thesis for the estimation of the parameters appearing in the problems under consideration. An introduction is made at the first chapter of the thesis concerning fundamental concepts of the magnetic fluids such as the magnetization and equilibrium flow. Experimental applications in the biomedicine are also given as well as the mathematical model describing the flow of biological fluids under the influence of an applied magnetic field. In order to investigate the effect of the magnetic field in the next three chapters basic flow problems of biomagnetic fluid (blood) are studied. In the second chapter the flow over a stretching sheet under the influence of an applied magnetic field is studied. The physical problem is described by a coupled system of non linear partial differential equations (pdes) with their appropriate boundary conditions. For the variation of the magnetization with the temperature and/or the magnetic field intensity two cases are considered (I and II). The arising system describing the physical problem is transformed into corresponding coupled systems of non linear ordinary differential equations (ods) after the introduction of proper non dimensional variables. For the numerical solution, finite differences are used for the case I, whereas a spectral method with Chebyshev polynomials is also used for the case II. It is apparent that the application of the magnetic field increases the skin friction and the pressure on the surface, whereas the heat transfer is reducing. A comparison is also made between the two numerical methods used in the case II. The efficiency and the accuracy of the spectral method over against the finite differences method are demonstrated. The superiority of the spectral method is apparent especially when high accuracy solution is desired. In the third chapter the fundamental problem of the biomagnetic fluid flow taking place in a rectangular duct under the influence of an applied magnetic field is studied. For the numerical solution of the problem, which is described by a coupled and non linear system of PDEs, with their appropriate boundary conditions, the stream function-vorticity formulation is adopted and the solution is obtained developing an efficient numerical technique based on the upwind finite differences joint with a line by line implicit method. Results concerning the velocity and temperature field, skin friction and rate of heat transfer indicate that the presence of magnetic field appreciable influence the flow field. The three dimensional, fully developed flow of a biomagnetic fluid in an impermeable rectangular duct under the influence of an applied magnetic field is numerically studied in the fourth chapter. The system of the partial differential equations, resulting after the introduction of appropriate non-dimensional variables, is solved applying an efficient numerical technique based on a pressure-linked pseudotransient method on a collocated grid. Results concerning the existence and the uniqueness of the solution are also given. The obtained results, for different values for the parameters entering into the problem under consideration, show that the flow is appreciably influenced by the presence of the magnetic field in the sense of reduction of the axial velocity and the formation of two vortices at the transverse plane. These first results indicate that the magnetic field significantly influences the blood flow and encourage further study in more complex geometries, oscillatory flow or including the non-Newtonian behaviour of blood in order to demonstrate applications in biomechanics and biomedicine.